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Documents authored by Wang, Yu

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Wang, Yu

Document
DOI: 10.4230/LIPIcs.SoCG.2019.57
Distribution-Sensitive Bounds on Relative Approximations of Geometric Ranges

Authors: Yufei Tao and Yu Wang

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
A family R of ranges and a set X of points, all in R^d, together define a range space (X, R|_X), where R|_X = {X cap h | h in R}. We want to find a structure to estimate the quantity |X cap h|/|X| for any range h in R with the (rho, epsilon)-guarantee: (i) if |X cap h|/|X| > rho, the estimate must have a relative error epsilon; (ii) otherwise, the estimate must have an absolute error rho epsilon. The objective is to minimize the size of the structure. Currently, the dominant solution is to compute a relative (rho, epsilon)-approximation, which is a subset of X with O~(lambda/(rho epsilon^2)) points, where lambda is the VC-dimension of (X, R|_X), and O~ hides polylog factors. This paper shows a more general bound sensitive to the content of X. We give a structure that stores O(log (1/rho)) integers plus O~(theta * (lambda/epsilon^2)) points of X, where theta - called the disagreement coefficient - measures how much the ranges differ from each other in their intersections with X. The value of theta is between 1 and 1/rho, such that our space bound is never worse than that of relative (rho, epsilon)-approximations, but we improve the latter’s 1/rho term whenever theta = o(1/(rho log (1/rho))). We also prove that, in the worst case, summaries with the (rho, 1/2)-guarantee must consume Omega(theta) words even for d = 2 and lambda <=3. We then constrain R to be the set of halfspaces in R^d for a constant d, and prove the existence of structures with o(1/(rho epsilon^2)) size offering (rho,epsilon)-guarantees, when X is generated from various stochastic distributions. This is the first formal justification on why the term 1/rho is not compulsory for "realistic" inputs.
Cite as

Yufei Tao and Yu Wang. Distribution-Sensitive Bounds on Relative Approximations of Geometric Ranges. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{tao_et_al:LIPIcs.SoCG.2019.57,
  author =	{Tao, Yufei and Wang, Yu},
  title =	{{Distribution-Sensitive Bounds on Relative Approximations of Geometric Ranges}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{57:1--57:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.57},
  URN =		{urn:nbn:de:0030-drops-104617},
  doi =		{10.4230/LIPIcs.SoCG.2019.57},
  annote =	{Keywords: Relative Approximation, Disagreement Coefficient, Data Summary}
}

Wang, Yujiang

Document
DOI: 10.4230/OASIcs.ICCSW.2018.7
The iBUG Eye Segmentation Dataset

Authors: Bingnan Luo, Jie Shen, Yujiang Wang, and Maja Pantic

Published in: OASIcs, Volume 66, 2018 Imperial College Computing Student Workshop (ICCSW 2018)

Abstract
This paper presents the first dataset for eye segmentation in low resolution images. Although eye segmentation has long been a vital preprocessing step in biometric applications, this work is the first to focus on low resolutions image that can be expected from a consumer-grade camera under conventional human-computer interaction and / or video-chat scenarios. Existing eye datasets have multiple limitations, including: (a) datasets only contain high resolution images; (b) datasets did not include enough pose variations; (c) a utility landmark ground truth did not be provided; (d) high accurate pixel-level ground truths had not be given. Our dataset meets all the above conditions and requirements for different segmentation methods. Besides, a baseline experiment has been performed on our dataset to evaluate the performances of landmark models (Active Appearance Model, Ensemble Regression Tree and Supervised Descent Method) and deep semantic segmentation models (Atrous convolutional neural network with conditional random field). Since the novelty of our dataset is to segment the iris and the sclera areas, we evaluate above models on sclera and iris only respectively in order to indicate the feasibility on eye-partial segmentation tasks. In conclusion, based on our dataset, deep segmentation methods performed better in terms of IOU-based ROC curves and it showed potential abilities on low-resolution eye segmentation task.
Cite as

Bingnan Luo, Jie Shen, Yujiang Wang, and Maja Pantic. The iBUG Eye Segmentation Dataset. In 2018 Imperial College Computing Student Workshop (ICCSW 2018). Open Access Series in Informatics (OASIcs), Volume 66, pp. 7:1-7:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:OASIcs.ICCSW.2018.7,
  author =	{Luo, Bingnan and Shen, Jie and Wang, Yujiang and Pantic, Maja},
  title =	{{The iBUG Eye Segmentation Dataset}},
  booktitle =	{2018 Imperial College Computing Student Workshop (ICCSW 2018)},
  pages =	{7:1--7:9},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-097-2},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{66},
  editor =	{Pirovano, Edoardo and Graversen, Eva},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ICCSW.2018.7},
  URN =		{urn:nbn:de:0030-drops-101883},
  doi =		{10.4230/OASIcs.ICCSW.2018.7},
  annote =	{Keywords: dataset, eye, segmentation, landmark, pixel-level}
}

Wang, Zhengyu

Document
DOI: 10.4230/LIPIcs.CCC.2015.412
Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

Authors: Fu Li, Iddo Tzameret, and Zhengyu Wang

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Abstract
Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e. Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexity, and among a handful of fundamental hardness questions in complexity theory by and large. Non-commutative arithmetic formulas, on the other hand, constitute a quite weak computational model, for which exponential-size lower bounds were shown already back in 1991 by Nisan [STOC 1991], using a particularly transparent argument. In this work we show that Frege lower bounds in fact follow from corresponding size lower bounds on non-commutative formulas computing certain polynomials (and that such lower bounds on non-commutative formulas must exist, unless NP=coNP). More precisely, we demonstrate a natural association between tautologies T to non-commutative polynomials p, such that: (*) if T has a polynomial-size Frege proof then p has a polynomial-size non-commutative arithmetic formula; and conversely, when T is a DNF, if p has a polynomial-size non-commutative arithmetic formula over GF(2) then T has a Frege proof of quasi-polynomial size. The argument is a characterization of Frege proofs as non-commutative formulas: we show that the Frege system is (quasi-)polynomially equivalent to a non-commutative Ideal Proof System (IPS), following the recent work of Grochow and Pitassi [FOCS 2014] that introduced a propositional proof system in which proofs are arithmetic circuits, and the work in [Tzameret 2011] that considered adding the commutator as an axiom in algebraic propositional proof systems. This gives a characterization of propositional Frege proofs in terms of (non-commutative) arithmetic formulas that is tighter than (the formula version of IPS) in Grochow and Pitassi [FOCS 2014], in the following sense: (i) The non-commutative IPS is polynomial-time checkable - whereas the original IPS was checkable in probabilistic polynomial-time; and (ii) Frege proofs unconditionally quasi-polynomially simulate the non-commutative IPS - whereas Frege was shown to efficiently simulate IPS only assuming that the decidability of PIT for (commutative) arithmetic formulas by polynomial-size circuits is efficiently provable in Frege.
Cite as

Fu Li, Iddo Tzameret, and Zhengyu Wang. Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 412-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{li_et_al:LIPIcs.CCC.2015.412,
  author =	{Li, Fu and Tzameret, Iddo and Wang, Zhengyu},
  title =	{{Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{412--432},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.412},
  URN =		{urn:nbn:de:0030-drops-50585},
  doi =		{10.4230/LIPIcs.CCC.2015.412},
  annote =	{Keywords: Proof complexity, algebraic complexity, arithmetic circuits, Frege, non-commutative formulas}
}
Document
DOI: 10.4230/DagSemProc.05391.9
Verification of Solutions for Almost Linear Complementarity Problems

Authors: Götz Alefeld and Zhengyu Wang

Published in: Dagstuhl Seminar Proceedings, Volume 5391, Algebraic and Numerical Algorithms and Computer-assisted Proofs (2006)

Abstract
We present a computational enclosure method for the solution of a class of nonlinear complementarity problems. The procedure also delivers a proof for the uniqueness of the solution.
Cite as

Götz Alefeld and Zhengyu Wang. Verification of Solutions for Almost Linear Complementarity Problems. In Algebraic and Numerical Algorithms and Computer-assisted Proofs. Dagstuhl Seminar Proceedings, Volume 5391, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{alefeld_et_al:DagSemProc.05391.9,
  author =	{Alefeld, G\"{o}tz and Wang, Zhengyu},
  title =	{{Verification of Solutions for  Almost Linear  Complementarity Problems}},
  booktitle =	{Algebraic and Numerical Algorithms and Computer-assisted Proofs},
  pages =	{1--21},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{5391},
  editor =	{Bruno Buchberger and Shin'ichi Oishi and Michael Plum and Sigfried M. Rump},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05391.9},
  URN =		{urn:nbn:de:0030-drops-4431},
  doi =		{10.4230/DagSemProc.05391.9},
  annote =	{Keywords: Complementarity problems, verification of solutions}
}

Wang, Yue

Document
DOI: 10.4230/DagSemProc.07151.2
Discovery of Sensor Network Layout using Connectivity Information

Authors: Jie Gao, Sol Lederer, and Yue Wang

Published in: Dagstuhl Seminar Proceedings, Volume 7151, Geometry in Sensor Networks (2007)

Abstract
We propose a distributed algorithm to discover and recover the layout of a large sensor network having a complex shape. As sensor network deployments grow large in size and become non-uniform, localization algorithms suffer from ``flip'' ambiguities---where a part of the network folds on top of another while keeping all edge length measurements preserved. We explore the high-order topological information in a sensor field to prevent incorrect flips and accurately recover the shape of the sensor network. We select landmarks on network boundaries with sufficient density, construct the landmark Voronoi diagram and its dual combinatorial Delaunay complex on these landmarks. The key insight is that when the landmarks are dense enough to capture the local geometric complexity, the combinatorial Delaunay complex is globally rigid and has a unique realization in the plane. An embedding by simply gluing the Delaunay triangles properly derives a faithful network layout, which consequently leads to a practical and sufficiently accurate localization algorithm. We prove the global rigidity of the combinatorial Delaunay complex in the case of a continuous geometric region. Simulation results on discrete networks show surprisingly good results, while multi-dimensional scaling and rubberband representation perform poorly or not at all in recovering the network layout. This is joint work with Sol Lederer and Yue Wang.
Cite as

Jie Gao, Sol Lederer, and Yue Wang. Discovery of Sensor Network Layout using Connectivity Information. In Geometry in Sensor Networks. Dagstuhl Seminar Proceedings, Volume 7151, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{gao_et_al:DagSemProc.07151.2,
  author =	{Gao, Jie and Lederer, Sol and Wang, Yue},
  title =	{{Discovery of Sensor Network Layout using Connectivity Information}},
  booktitle =	{Geometry in Sensor Networks},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7151},
  editor =	{Subhash Suri and Roger Wattenhofer and Peter Widmayer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07151.2},
  URN =		{urn:nbn:de:0030-drops-11149},
  doi =		{10.4230/DagSemProc.07151.2},
  annote =	{Keywords: Sensor Networks, Localization, Delaunay complex, Rigidity}
}

Wang, Yusu

Document
DOI: 10.4230/DagRep.14.2.206
Applied and Combinatorial Topology (Dagstuhl Seminar 24092)

Authors: Paweł Dłotko, Dmitry Feichtner-Kozlov, Anastasios Stefanou, Yusu Wang, and Jan F Senge

Published in: Dagstuhl Reports, Volume 14, Issue 2 (2024)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 24092 "Applied and Combinatorial Topology". The last twenty years of rapid development of Topological Data Analysis (TDA) have shown the need to analyze the shape of data to better understand the data. Since an explosion of new ideas in 2000’ including those of Persistent Homology and Mapper Algorithms, the community rushed to solve detailed theoretical questions related to the existing invariants. However, topology and geometry still have much to offer to the data science community. New tools and techniques are within reach, waiting to be brought over the fence to enrich our understanding and potential to analyze data. At the same time, the fields of Discrete Morse Theory (DMT) and Combinatorial Topology (CT) are developed in parallel with no strong connection to data-intensive TDA or to other statistical pipelines (e.g. machine learning). This Dagstuhl Seminar brought together a number of experts in Discrete Morse Theory, Combinatorial Topology, Topological Data Analysis, and Statistics to (i) enhance the existing interactions between these fields on the one hand, and (ii) discuss the possibilities of adopting new invariants from algebra, geometry, and topology; in particular inspired by continuous and discrete Morse theory and combinatorial topology; to analyze and better understand the notion of shape of the data. The different talks in the seminar included both introductory talks as well as current research expositions and proved fruitful for the open problem and break-out sessions. The topics that were discussed included 1) algorithmic aspects for efficient computation as well as Morse theoretic approximations 2) topological information gain of multiparameter persistence 3) understanding the magnitude function and its relation to graph problems.
Cite as

Paweł Dłotko, Dmitry Feichtner-Kozlov, Anastasios Stefanou, Yusu Wang, and Jan F Senge. Applied and Combinatorial Topology (Dagstuhl Seminar 24092). In Dagstuhl Reports, Volume 14, Issue 2, pp. 206-239, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{dlotko_et_al:DagRep.14.2.206,
  author =	{D{\l}otko, Pawe{\l} and Feichtner-Kozlov, Dmitry and Stefanou, Anastasios and Wang, Yusu and Senge, Jan F},
  title =	{{Applied and Combinatorial Topology (Dagstuhl Seminar 24092)}},
  pages =	{206--239},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2024},
  volume =	{14},
  number =	{2},
  editor =	{D{\l}otko, Pawe{\l} and Feichtner-Kozlov, Dmitry and Stefanou, Anastasios and Wang, Yusu and Senge, Jan F},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.2.206},
  URN =		{urn:nbn:de:0030-drops-205059},
  doi =		{10.4230/DagRep.14.2.206},
  annote =	{Keywords: Applied Topology, Topological Data Analysis, Discrete Morse Theory, Combinatorial Topology, Statistics}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.37
A Generalization of the Persistent Laplacian to Simplicial Maps

Authors: Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
The (combinatorial) graph Laplacian is a fundamental object in the analysis of, and optimization on, graphs. Via a topological view, this operator can be extended to a simplicial complex K and therefore offers a way to perform "signal processing" on p-(co)chains of K. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes K ↪ L connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we significantly expand the scope of the persistent Laplacian by generalizing it to a pair of weighted simplicial complexes connected by a weight preserving simplicial map f: K → L. Such a simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets, i.e. cases in which it does not hold that K ⊂ L. However, the simplicial map setting is much more challenging than the inclusion setting since the underlying algebraic structure is much more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint positive semi-definite operator to a given subspace. As a consequence of this relation, we prove that the qth persistent Betti number of the simplicial map f: K → L equals the nullity of the qth persistent Laplacian Δ_q^{K,L}. We then propose an algorithm for finding the matrix representation of Δ_q^{K,L} which in turn yields a fundamentally different algorithm for computing the qth persistent Betti number of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under weight-preserving simplicial maps and establish monotonicity results for their eigenvalues.
Cite as

Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang. A Generalization of the Persistent Laplacian to Simplicial Maps. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gulen_et_al:LIPIcs.SoCG.2023.37,
  author =	{G\"{u}len, Aziz Burak and M\'{e}moli, Facundo and Wan, Zhengchao and Wang, Yusu},
  title =	{{A Generalization of the Persistent Laplacian to Simplicial Maps}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{37:1--37:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.37},
  URN =		{urn:nbn:de:0030-drops-178877},
  doi =		{10.4230/LIPIcs.SoCG.2023.37},
  annote =	{Keywords: combinatorial Laplacian, persistent Laplacian, Schur complement, persistent homology, persistent Betti number}
}
Document
DOI: 10.4230/LIPIcs.SEA.2021.14
Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams

Authors: Samantha Chen and Yusu Wang

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Abstract
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.
Cite as

Samantha Chen and Yusu Wang. Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.SEA.2021.14,
  author =	{Chen, Samantha and Wang, Yusu},
  title =	{{Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.14},
  URN =		{urn:nbn:de:0030-drops-137861},
  doi =		{10.4230/LIPIcs.SEA.2021.14},
  annote =	{Keywords: persistence diagrams, approximation algorithms, Wasserstein distance, optimal transport}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.26
Elder-Rule-Staircodes for Augmented Metric Spaces

Authors: Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.
Cite as

Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang. Elder-Rule-Staircodes for Augmented Metric Spaces. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cai_et_al:LIPIcs.SoCG.2020.26,
  author =	{Cai, Chen and Kim, Woojin and M\'{e}moli, Facundo and Wang, Yusu},
  title =	{{Elder-Rule-Staircodes for Augmented Metric Spaces}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.26},
  URN =		{urn:nbn:de:0030-drops-121848},
  doi =		{10.4230/LIPIcs.SoCG.2020.26},
  annote =	{Keywords: Persistent homology, Multiparameter persistence, Barcodes, Elder rule, Hierarchical clustering, Graded Betti numbers}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.36
An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology

Authors: Tamal K. Dey, Tianqi Li, and Yusu Wang

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their persistence. These concepts are well suited for spaces that are not directed. As a result, one needs a concept of homology that accommodates orientations in input space. Path-homology developed for directed graphs by Grigoryan, Lin, Muranov and Yau has been effectively adapted for this purpose recently by Chowdhury and Mémoli. They also give an algorithm to compute this path-homology. Our main contribution in this paper is an algorithm that computes this path-homology and its persistence more efficiently for the 1-dimensional (H₁) case. In developing such an algorithm, we discover various structures and their efficient computations that aid computing the 1-dimensional path-homology. We implement our algorithm and present some preliminary experimental results.
Cite as

Tamal K. Dey, Tianqi Li, and Yusu Wang. An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2020.36,
  author =	{Dey, Tamal K. and Li, Tianqi and Wang, Yusu},
  title =	{{An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{36:1--36:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.36},
  URN =		{urn:nbn:de:0030-drops-121944},
  doi =		{10.4230/LIPIcs.SoCG.2020.36},
  annote =	{Keywords: computational topology, directed graph, path homology, persistent path homology}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.29
Local Cliques in ER-Perturbed Random Geometric Graphs

Authors: Matthew Kahle, Minghao Tian, and Yusu Wang

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
We study a random graph model introduced in [Srinivasan Parthasarathy et al., 2017] where one adds Erdős - Rényi (ER) type perturbation to a random geometric graph. More precisely, assume G_X^* is a random geometric graph sampled from a nice measure on a metric space X = (X,d). An ER-perturbed random geometric graph G^(p,q) is generated by removing each existing edge from G_X^* with probability p, while inserting each non-existent edge to G_X^* with probability q. We consider a localized version of clique number for G^(p,q): Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. We show that the edge clique number presents two fundamentally different types of behaviors in G^(p,q), depending on which "type" of randomness it is generated from. As an application of the above results, we show that by a simple filtering process based on the edge clique number, we can recover the shortest-path metric of the random geometric graph G_X^* within a multiplicative factor of 3 from an ER-perturbed observed graph G^(p,q), for a significantly wider range of insertion probability q than what is required in [Srinivasan Parthasarathy et al., 2017].
Cite as

Matthew Kahle, Minghao Tian, and Yusu Wang. Local Cliques in ER-Perturbed Random Geometric Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kahle_et_al:LIPIcs.ISAAC.2019.29,
  author =	{Kahle, Matthew and Tian, Minghao and Wang, Yusu},
  title =	{{Local Cliques in ER-Perturbed Random Geometric Graphs}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{29:1--29:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.29},
  URN =		{urn:nbn:de:0030-drops-115253},
  doi =		{10.4230/LIPIcs.ISAAC.2019.29},
  annote =	{Keywords: random graphs, random geometric graphs, edge clique number, the probabilistic method, metric recovery}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.83
FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees

Authors: Elena Farahbakhsh Touli and Yusu Wang

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
The Gromov-Hausdorff distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Any finite tree with positive edge weight can be viewed as a metric tree where the weight is treated as edge length and the metric is the induced shortest path metric in the tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP-hard to approximate the Gromov-Hausdorff distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximate the Gromov-Hausdorff distance between two general metric trees within a multiplicative factor of 14. Interestingly, the development of our algorithm is made possible by a connection between the Gromov-Hausdorff distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary (it is a variant of the Reeb graphs and contour trees), and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we re-define the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a fixed-parameter tractable algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the Gromov-Hausdorff distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of O(sqrt{n}) even for trees with unit edge length.
Cite as

Elena Farahbakhsh Touli and Yusu Wang. FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{farahbakhshtouli_et_al:LIPIcs.ESA.2019.83,
  author =	{Farahbakhsh Touli, Elena and Wang, Yusu},
  title =	{{FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{83:1--83:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.83},
  URN =		{urn:nbn:de:0030-drops-112048},
  doi =		{10.4230/LIPIcs.ESA.2019.83},
  annote =	{Keywords: Gromov-Hausdorff distance, Interleaving distance, Merge trees}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.SoCG.2019
LIPIcs, Volume 129, SoCG'19, Complete Volume

Authors: Gill Barequet and Yusu Wang

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
LIPIcs, Volume 129, SoCG'19, Complete Volume
Cite as

35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{barequet_et_al:LIPIcs.SoCG.2019,
  title =	{{LIPIcs, Volume 129, SoCG'19, Complete Volume}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019},
  URN =		{urn:nbn:de:0030-drops-105562},
  doi =		{10.4230/LIPIcs.SoCG.2019},
  annote =	{Keywords: Theory of computation, Computational geometry, Design and analysis of algorithms, Mathematics of computing, Combinatorics, Graph algortihms}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.SoCG.2019.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Gill Barequet and Yusu Wang

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barequet_et_al:LIPIcs.SoCG.2019.0,
  author =	{Barequet, Gill and Wang, Yusu},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{0:i--0:xvi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.0},
  URN =		{urn:nbn:de:0030-drops-104047},
  doi =		{10.4230/LIPIcs.SoCG.2019.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.3
Vietoris-Rips and Cech Complexes of Metric Gluings

Authors: Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.
Cite as

Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. Vietoris-Rips and Cech Complexes of Metric Gluings. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adamaszek_et_al:LIPIcs.SoCG.2018.3,
  author =	{Adamaszek, Michal and Adams, Henry and Gasparovic, Ellen and Gommel, Maria and Purvine, Emilie and Sazdanovic, Radmila and Wang, Bei and Wang, Yusu and Ziegelmeier, Lori},
  title =	{{Vietoris-Rips and Cech Complexes of Metric Gluings}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.3},
  URN =		{urn:nbn:de:0030-drops-87162},
  doi =		{10.4230/LIPIcs.SoCG.2018.3},
  annote =	{Keywords: Vietoris-Rips and Cech complexes, metric space gluings and wedge sums, metric graphs, persistent homology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.31
Graph Reconstruction by Discrete Morse Theory

Authors: Tamal K. Dey, Jiayuan Wang, and Yusu Wang

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.
Cite as

Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Graph Reconstruction by Discrete Morse Theory. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2018.31,
  author =	{Dey, Tamal K. and Wang, Jiayuan and Wang, Yusu},
  title =	{{Graph Reconstruction by Discrete Morse Theory}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{31:1--31:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.31},
  URN =		{urn:nbn:de:0030-drops-87443},
  doi =		{10.4230/LIPIcs.SoCG.2018.31},
  annote =	{Keywords: graph reconstruction, discrete Morse theory, persistence}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.23
Declutter and Resample: Towards Parameter Free Denoising

Authors: Mickael Buchet, Tamal K. Dey, Jiayuan Wang, and Yusu Wang

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice.
Cite as

Mickael Buchet, Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Declutter and Resample: Towards Parameter Free Denoising. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buchet_et_al:LIPIcs.SoCG.2017.23,
  author =	{Buchet, Mickael and Dey, Tamal K. and Wang, Jiayuan and Wang, Yusu},
  title =	{{Declutter and Resample: Towards Parameter Free Denoising}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.23},
  URN =		{urn:nbn:de:0030-drops-72133},
  doi =		{10.4230/LIPIcs.SoCG.2017.23},
  annote =	{Keywords: denoising, parameter free, k-distance,compact sets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.36
Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers

Authors: Tamal K. Dey, Facundo Mémoli, and Yusu Wang

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called nerve of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex N(U) of a path-connected cover U of a domain X cannot be richer than that of the domain X itself. Intuitively, this result means that no new H_1-homology class can be "created" under a natural map from X to the nerve complex N(U). Equipping X with a pseudometric d, we further refine this result and characterize the classes of H_1(X) that may survive in the nerve complex using the notion of size of the covering elements in U. These fundamental results about nerve complexes then lead to an analysis of the H_1-homology of Reeb spaces, mappers and multiscale mappers. The analysis of H_1-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.
Cite as

Tamal K. Dey, Facundo Mémoli, and Yusu Wang. Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2017.36,
  author =	{Dey, Tamal K. and M\'{e}moli, Facundo and Wang, Yusu},
  title =	{{Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.36},
  URN =		{urn:nbn:de:0030-drops-72220},
  doi =		{10.4230/LIPIcs.SoCG.2017.36},
  annote =	{Keywords: Topology, Nerves, Mapper, Multiscale Mapper, Reeb Spaces}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.53
A Quest to Unravel the Metric Structure Behind Perturbed Networks

Authors: Srinivasan Parthasarathy, David Sivakoff, Minghao Tian, and Yusu Wang

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially continuous) hidden domain or process. Subsequent analysis, processing, and inferences are then performed on this observed graph. In this paper we advocate the perspective that an observed graph is often a noisy version of some discretized 1-skeleton of a hidden domain, and specifically we will consider the following natural network model: We assume that there is a true graph G^* which is a certain proximity graph for points sampled from a hidden domain X; while the observed graph G is an Erdos-Renyi type perturbed version of G^*. Our network model is related to, and slightly generalizes, the much-celebrated small-world network model originally proposed by Watts and Strogatz. However, the main question we aim to answer is orthogonal to the usual studies of network models (which often focuses on characterizing / predicting behaviors and properties of real-world networks). Specifically, we aim to recover the metric structure of G^* (which reflects that of the hidden space X as we will show) from the observed graph G. Our main result is that a simple filtering process based on the Jaccard index can recover this metric within a multiplicative factor of 2 under our network model. Our work makes one step towards the general question of inferring structure of a hidden space from its observed noisy graph representation. In addition, our results also provide a theoretical understanding for Jaccard-Index-based denoising approaches.
Cite as

Srinivasan Parthasarathy, David Sivakoff, Minghao Tian, and Yusu Wang. A Quest to Unravel the Metric Structure Behind Perturbed Networks. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{parthasarathy_et_al:LIPIcs.SoCG.2017.53,
  author =	{Parthasarathy, Srinivasan and Sivakoff, David and Tian, Minghao and Wang, Yusu},
  title =	{{A Quest to Unravel the Metric Structure Behind Perturbed Networks}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.53},
  URN =		{urn:nbn:de:0030-drops-72112},
  doi =		{10.4230/LIPIcs.SoCG.2017.53},
  annote =	{Keywords: metric structure, Erd\"{o}s-R\'{e}nyi perturbation, graphs, doubling measure}
}
Document
Multimedia Contribution
DOI: 10.4230/LIPIcs.SoCG.2017.65
Cardiac Trabeculae Segmentation: an Application of Computational Topology (Multimedia Contribution)

Authors: Chao Chen, Dimitris Metaxas, Yusu Wang, and Pengxiang Wu

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
In this video, we present a research project on cardiac trabeculae segmentation. Trabeculae are fine muscle columns within human ventricles whose both ends are attached to the wall. Extracting these structures are very challenging even with state-of-the-art image segmentation techniques. We observed that these structures form natural topological handles. Based on such observation, we developed a topological approach, which employs advanced computational topology methods and achieve high quality segmentation results.
Cite as

Chao Chen, Dimitris Metaxas, Yusu Wang, and Pengxiang Wu. Cardiac Trabeculae Segmentation: an Application of Computational Topology (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 65:1-65:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.SoCG.2017.65,
  author =	{Chen, Chao and Metaxas, Dimitris and Wang, Yusu and Wu, Pengxiang},
  title =	{{Cardiac Trabeculae Segmentation: an Application of Computational Topology}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{65:1--65:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.65},
  URN =		{urn:nbn:de:0030-drops-72429},
  doi =		{10.4230/LIPIcs.SoCG.2017.65},
  annote =	{Keywords: image segmentation, trabeculae, persistent homology, homology localization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.35
SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse

Authors: Tamal K. Dey, Dayu Shi, and Yusu Wang

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on P indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice.
Cite as

Tamal K. Dey, Dayu Shi, and Yusu Wang. SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dey_et_al:LIPIcs.ESA.2016.35,
  author =	{Dey, Tamal K. and Shi, Dayu and Wang, Yusu},
  title =	{{SimBa: An Efficient Tool for Approximating  Rips-Filtration Persistence via Simplicial Batch-Collapse}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.35},
  URN =		{urn:nbn:de:0030-drops-63869},
  doi =		{10.4230/LIPIcs.ESA.2016.35},
  annote =	{Keywords: Rips filtration, Homology groups, Persistence, Topological data analysis}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.461
Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs

Authors: Ulrich Bauer, Elizabeth Munch, and Yusu Wang

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov-Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. Our result also implies the bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
Cite as

Ulrich Bauer, Elizabeth Munch, and Yusu Wang. Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 461-475, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bauer_et_al:LIPIcs.SOCG.2015.461,
  author =	{Bauer, Ulrich and Munch, Elizabeth and Wang, Yusu},
  title =	{{Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{461--475},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.461},
  URN =		{urn:nbn:de:0030-drops-51467},
  doi =		{10.4230/LIPIcs.SOCG.2015.461},
  annote =	{Keywords: Reeb graph, interleaving distance, functional distortion distance}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.491
Comparing Graphs via Persistence Distortion

Authors: Tamal K. Dey, Dayu Shi, and Yusu Wang

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the non-linear structure hidden behind the data. In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea. This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. Our persistence-distortion distance has two properties not shared by previous methods: First, it is stable against the perturbations of the input graph metrics. Second, it is a continuous distance measure, in the sense that it is defined on an alignment of the underlying spaces of input graphs, instead of merely their nodes. This makes our persistence-distortion distance robust against, for example, different discretizations of the same underlying graph. Despite considering the input graphs as continuous spaces, that is, taking all points into account, we show that we can compute the persistence-distortion distance in polynomial time. The time complexity for the discrete case where only graph nodes are considered is much faster.
Cite as

Tamal K. Dey, Dayu Shi, and Yusu Wang. Comparing Graphs via Persistence Distortion. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 491-506, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dey_et_al:LIPIcs.SOCG.2015.491,
  author =	{Dey, Tamal K. and Shi, Dayu and Wang, Yusu},
  title =	{{Comparing Graphs via Persistence Distortion}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{491--506},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.491},
  URN =		{urn:nbn:de:0030-drops-51285},
  doi =		{10.4230/LIPIcs.SOCG.2015.491},
  annote =	{Keywords: Graph matching, metric graphs, persistence distortion, topological method}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.796
Maintaining Contour Trees of Dynamic Terrains

Authors: Pankaj K. Agarwal, Thomas Mølhave, Morten Revsbæk, Issam Safa, Yusu Wang, and Jungwoo Yang

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
We study the problem of maintaining the contour tree T of a terrain Sigma, represented as a triangulated xy-monotone surface, as the heights of its vertices vary continuously with time. We characterize the combinatorial changes in T and how they relate to topological changes in Sigma. We present a kinetic data structure (KDS) for maintaining T efficiently. It maintains certificates that fail, i.e., an event occurs, only when the heights of two adjacent vertices become equal or two saddle vertices appear on the same contour. Assuming that the heights of two vertices of Sigma become equal only O(1) times and these instances can be computed in O(1) time, the KDS processes O(kappa + n) events, where n is the number of vertices in Sigma and kappa is the number of events at which the combinatorial structure of T changes, and processes each event in O(log n) time. The KDS can be extended to maintain an augmented contour tree and a join/split tree.
Cite as

Pankaj K. Agarwal, Thomas Mølhave, Morten Revsbæk, Issam Safa, Yusu Wang, and Jungwoo Yang. Maintaining Contour Trees of Dynamic Terrains. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 796-811, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{agarwal_et_al:LIPIcs.SOCG.2015.796,
  author =	{Agarwal, Pankaj K. and M{\o}lhave, Thomas and Revsb{\ae}k, Morten and Safa, Issam and Wang, Yusu and Yang, Jungwoo},
  title =	{{Maintaining Contour Trees of Dynamic Terrains}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{796--811},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.796},
  URN =		{urn:nbn:de:0030-drops-51406},
  doi =		{10.4230/LIPIcs.SOCG.2015.796},
  annote =	{Keywords: Contour tree, dynamic terrain, kinetic data structure}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.827
Topological Analysis of Scalar Fields with Outliers

Authors: Mickaël Buchet, Frédéric Chazal, Tamal K. Dey, Fengtao Fan, Steve Y. Oudot, and Yusu Wang

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
Given a real-valued function f defined over a manifold M embedded in R^d, we are interested in recovering structural information about f from the sole information of its values on a finite sample P. Existing methods provide approximation to the persistence diagram of f when geometric noise and functional noise are bounded. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice. We propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees and experimental results on the quality of our approximation of the sampled scalar field.
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Mickaël Buchet, Frédéric Chazal, Tamal K. Dey, Fengtao Fan, Steve Y. Oudot, and Yusu Wang. Topological Analysis of Scalar Fields with Outliers. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 827-841, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{buchet_et_al:LIPIcs.SOCG.2015.827,
  author =	{Buchet, Micka\"{e}l and Chazal, Fr\'{e}d\'{e}ric and Dey, Tamal K. and Fan, Fengtao and Oudot, Steve Y. and Wang, Yusu},
  title =	{{Topological Analysis of Scalar Fields with Outliers}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{827--841},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.827},
  URN =		{urn:nbn:de:0030-drops-51052},
  doi =		{10.4230/LIPIcs.SOCG.2015.827},
  annote =	{Keywords: Persistent Homology, Topological Data Analysis, Scalar Field Analysis, Nested Rips Filtration, Distance to a Measure}
}

Wang, Yuting

Document
DOI: 10.4230/LIPIcs.TLCA.2015.332
A Proof-theoretic Characterization of Independence in Type Theory

Authors: Yuting Wang and Kaustuv Chaudhuri

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract
For lambda-terms constructed freely from a type signature in a type theory such as LF, there is a simple inductive subordination relation that is used to control type-formation. There is a related—but not precisely complementary—notion of independence that asserts that the inhabitants of the function space tau_1 --> tau_2 depend vacuously on their arguments. Independence has many practical reasoning applications in logical frameworks, such as pruning variable dependencies or transporting theorems and proofs between type signatures. However, independence is usually not given a formal interpretation. Instead, it is generally implemented in an ad hoc and uncertified fashion. We propose a formal definition of independence and give a proof-theoretic characterization of it by: (1) representing the inference rules of a given type theory and a closed type signature as a theory of intuitionistic predicate logic, (2) showing that typing derivations in this signature are adequately represented by a focused sequent calculus for this logic, and (3) defining independence in terms of strengthening for intuitionistic sequents. This scheme is then formalized in a meta-logic, called G, that can represent the sequent calculus as an inductive definition, so the relevant strengthening lemmas can be given explicit inductive proofs. We present an algorithm for automatically deriving the strengthening lemmas and their proofs in G.
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Yuting Wang and Kaustuv Chaudhuri. A Proof-theoretic Characterization of Independence in Type Theory. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 332-346, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{wang_et_al:LIPIcs.TLCA.2015.332,
  author =	{Wang, Yuting and Chaudhuri, Kaustuv},
  title =	{{A Proof-theoretic Characterization of Independence in Type Theory}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{332--346},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.332},
  URN =		{urn:nbn:de:0030-drops-51736},
  doi =		{10.4230/LIPIcs.TLCA.2015.332},
  annote =	{Keywords: subordination; independence; sequent calculus; focusing; strengthening}
}

Wang, Jiayuan

Document
DOI: 10.4230/LIPIcs.SoCG.2018.31
Graph Reconstruction by Discrete Morse Theory

Authors: Tamal K. Dey, Jiayuan Wang, and Yusu Wang

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.
Cite as

Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Graph Reconstruction by Discrete Morse Theory. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2018.31,
  author =	{Dey, Tamal K. and Wang, Jiayuan and Wang, Yusu},
  title =	{{Graph Reconstruction by Discrete Morse Theory}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{31:1--31:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.31},
  URN =		{urn:nbn:de:0030-drops-87443},
  doi =		{10.4230/LIPIcs.SoCG.2018.31},
  annote =	{Keywords: graph reconstruction, discrete Morse theory, persistence}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.23
Declutter and Resample: Towards Parameter Free Denoising

Authors: Mickael Buchet, Tamal K. Dey, Jiayuan Wang, and Yusu Wang

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice.
Cite as

Mickael Buchet, Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Declutter and Resample: Towards Parameter Free Denoising. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buchet_et_al:LIPIcs.SoCG.2017.23,
  author =	{Buchet, Mickael and Dey, Tamal K. and Wang, Jiayuan and Wang, Yusu},
  title =	{{Declutter and Resample: Towards Parameter Free Denoising}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.23},
  URN =		{urn:nbn:de:0030-drops-72133},
  doi =		{10.4230/LIPIcs.SoCG.2017.23},
  annote =	{Keywords: denoising, parameter free, k-distance,compact sets}
}

Wang, Yuyi

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.45
A Simple Distributed Algorithm for Sparse Fractional Covering and Packing Problems

Authors: Qian Li, Minghui Ouyang, and Yuyi Wang

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
This paper presents a distributed algorithm in the CONGEST model that achieves a (1+ε)-approximation for row-sparse fractional covering problems (RS-FCP) and the dual column-sparse fraction packing problems (CS-FPP). Compared with the best-known (1+ε)-approximation CONGEST algorithm for RS-FCP/CS-FPP developed by Kuhn, Moscibroda, and Wattenhofer (SODA'06), our algorithm is not only much simpler but also significantly improves the dependency on ε.
Cite as

Qian Li, Minghui Ouyang, and Yuyi Wang. A Simple Distributed Algorithm for Sparse Fractional Covering and Packing Problems. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 45:1-45:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ISAAC.2024.45,
  author =	{Li, Qian and Ouyang, Minghui and Wang, Yuyi},
  title =	{{A Simple Distributed Algorithm for Sparse Fractional Covering and Packing Problems}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{45:1--45:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.45},
  URN =		{urn:nbn:de:0030-drops-221726},
  doi =		{10.4230/LIPIcs.ISAAC.2024.45},
  annote =	{Keywords: CONGEST model, row-sparse fractional covering, column-sparse fractional packing, positive linear programming, simple algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.61
The k-Server Problem with Delays on the Uniform Metric Space

Authors: Predrag Krnetić, Darya Melnyk, Yuyi Wang, and Roger Wattenhofer

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
In this paper, we present tight bounds for the k-server problem with delays in the uniform metric space. The problem is defined on n+k nodes in the uniform metric space which can issue requests over time. These requests can be served directly or with some delay using k servers, by moving a server to the corresponding node with an open request. The task is to find an online algorithm that can serve the requests while minimizing the total moving and delay costs. We first provide a lower bound by showing that the competitive ratio of any deterministic online algorithm cannot be better than (2k+1) in the clairvoyant setting. We will then show that conservative algorithms (without delay) can be equipped with an accumulative delay function such that all such algorithms become (2k+1)-competitive in the non-clairvoyant setting. Together, the two bounds establish a tight result for both, the clairvoyant and the non-clairvoyant settings.
Cite as

Predrag Krnetić, Darya Melnyk, Yuyi Wang, and Roger Wattenhofer. The k-Server Problem with Delays on the Uniform Metric Space. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{krnetic_et_al:LIPIcs.ISAAC.2020.61,
  author =	{Krneti\'{c}, Predrag and Melnyk, Darya and Wang, Yuyi and Wattenhofer, Roger},
  title =	{{The k-Server Problem with Delays on the Uniform Metric Space}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{61:1--61:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.61},
  URN =		{urn:nbn:de:0030-drops-134056},
  doi =		{10.4230/LIPIcs.ISAAC.2020.61},
  annote =	{Keywords: Online k-Server, Paging, Delayed Service, Conservative Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.16
Algorithmic Channel Design

Authors: Georgia Avarikioti, Yuyi Wang, and Roger Wattenhofer

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
Payment networks, also known as channels, are a most promising solution to the throughput problem of cryptocurrencies. In this paper we study the design of capital-efficient payment networks, offline as well as online variants. We want to know how to compute an efficient payment network topology, how capital should be assigned to the individual edges, and how to decide which transactions to accept. Towards this end, we present a flurry of interesting results, basic but generally applicable insights on the one hand, and hardness results and approximation algorithms on the other hand.
Cite as

Georgia Avarikioti, Yuyi Wang, and Roger Wattenhofer. Algorithmic Channel Design. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{avarikioti_et_al:LIPIcs.ISAAC.2018.16,
  author =	{Avarikioti, Georgia and Wang, Yuyi and Wattenhofer, Roger},
  title =	{{Algorithmic Channel Design}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{16:1--16:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.16},
  URN =		{urn:nbn:de:0030-drops-99648},
  doi =		{10.4230/LIPIcs.ISAAC.2018.16},
  annote =	{Keywords: blockchain, payment channels, layer 2 solution, network design, payment hubs, routing}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.62
Impatient Online Matching

Authors: Xingwu Liu, Zhida Pan, Yuyi Wang, and Roger Wattenhofer

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
We investigate the problem of Min-cost Perfect Matching with Delays (MPMD) in which requests are pairwise matched in an online fashion with the objective to minimize the sum of space cost and time cost. Though linear-MPMD (i.e., time cost is linear in delay) has been thoroughly studied in the literature, it does not well model impatient requests that are common in practice. Thus, we propose convex-MPMD where time cost functions are convex, capturing the situation where time cost increases faster and faster. Since the existing algorithms for linear-MPMD are not competitive any more, we devise a new deterministic algorithm for convex-MPMD problems. For a large class of convex time cost functions, our algorithm achieves a competitive ratio of O(k) on any k-point uniform metric space. Moreover, our deterministic algorithm is asymptotically optimal, which uncover a substantial difference between convex-MPMD and linear-MPMD which allows a deterministic algorithm with constant competitive ratio on any uniform metric space.
Cite as

Xingwu Liu, Zhida Pan, Yuyi Wang, and Roger Wattenhofer. Impatient Online Matching. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 62:1-62:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{liu_et_al:LIPIcs.ISAAC.2018.62,
  author =	{Liu, Xingwu and Pan, Zhida and Wang, Yuyi and Wattenhofer, Roger},
  title =	{{Impatient Online Matching}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{62:1--62:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.62},
  URN =		{urn:nbn:de:0030-drops-100107},
  doi =		{10.4230/LIPIcs.ISAAC.2018.62},
  annote =	{Keywords: online algorithm, online matching, convex function, competitive analysis, lower bound}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.1
Min-Cost Bipartite Perfect Matching with Delays

Authors: Itai Ashlagi, Yossi Azar, Moses Charikar, Ashish Chiplunkar, Ofir Geri, Haim Kaplan, Rahul Makhijani, Yuyi Wang, and Roger Wattenhofer

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
In the min-cost bipartite perfect matching with delays (MBPMD) problem, requests arrive online at points of a finite metric space. Each request is either positive or negative and has to be matched to a request of opposite polarity. As opposed to traditional online matching problems, the algorithm does not have to serve requests as they arrive, and may choose to match them later at a cost. Our objective is to minimize the sum of the distances between matched pairs of requests (the connection cost) and the sum of the waiting times of the requests (the delay cost). This objective exhibits a natural tradeoff between minimizing the distances and the cost of waiting for better matches. This tradeoff appears in many real-life scenarios, notably, ride-sharing platforms. MBPMD is related to its non-bipartite variant, min-cost perfect matching with delays (MPMD), in which each request can be matched to any other request. MPMD was introduced by Emek et al. (STOC'16), who showed an O(log^2(n)+log(Delta))-competitive randomized algorithm on n-point metric spaces with aspect ratio Delta. Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD. We get a lower bound of Omega(sqrt(log(n)/log(log(n)))) on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from Omega(sqrt(log(n))) (shown by Azar et al., SODA'17) to Omega(log(n)/log(log(n))), thus, almost matching their upper bound of O(log(n)). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis that improves the competitive ratio to O(log(n)). The key ingredient of the algorithm is an O(h)-competitive randomized algorithm for MBPMD on weighted trees of height h. Third, we provide an O(h)-competitive deterministic algorithm for MBPMD on weighted trees of height h. This algorithm is obtained by adapting the algorithm for MPMD by Azar et al. to the apparently more complicated bipartite setting.
Cite as

Itai Ashlagi, Yossi Azar, Moses Charikar, Ashish Chiplunkar, Ofir Geri, Haim Kaplan, Rahul Makhijani, Yuyi Wang, and Roger Wattenhofer. Min-Cost Bipartite Perfect Matching with Delays. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ashlagi_et_al:LIPIcs.APPROX-RANDOM.2017.1,
  author =	{Ashlagi, Itai and Azar, Yossi and Charikar, Moses and Chiplunkar, Ashish and Geri, Ofir and Kaplan, Haim and Makhijani, Rahul and Wang, Yuyi and Wattenhofer, Roger},
  title =	{{Min-Cost Bipartite Perfect Matching with Delays}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.1},
  URN =		{urn:nbn:de:0030-drops-75509},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.1},
  annote =	{Keywords: online algorithms with delayed service, bipartite matching, competitive analysis}
}

Wang, Yuyan

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.97
Relational Algorithms for k-Means Clustering

Authors: Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
This paper gives a k-means approximation algorithm that is efficient in the relational algorithms model. This is an algorithm that operates directly on a relational database without performing a join to convert it to a matrix whose rows represent the data points. The running time is potentially exponentially smaller than N, the number of data points to be clustered that the relational database represents. Few relational algorithms are known and this paper offers techniques for designing relational algorithms as well as characterizing their limitations. We show that given two data points as cluster centers, if we cluster points according to their closest centers, it is NP-Hard to approximate the number of points in the clusters on a general relational input. This is trivial for conventional data inputs and this result exemplifies that standard algorithmic techniques may not be directly applied when designing an efficient relational algorithm. This paper then introduces a new method that leverages rejection sampling and the k-means++ algorithm to construct a O(1)-approximate k-means solution.
Cite as

Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang. Relational Algorithms for k-Means Clustering. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 97:1-97:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{moseley_et_al:LIPIcs.ICALP.2021.97,
  author =	{Moseley, Benjamin and Pruhs, Kirk and Samadian, Alireza and Wang, Yuyan},
  title =	{{Relational Algorithms for k-Means Clustering}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{97:1--97:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.97},
  URN =		{urn:nbn:de:0030-drops-141668},
  doi =		{10.4230/LIPIcs.ICALP.2021.97},
  annote =	{Keywords: k-means, clustering, approximation, big-data, databases}
}

Wang, Dingyu

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.104
Non-Mergeable Sketching for Cardinality Estimation

Authors: Seth Pettie, Dingyu Wang, and Longhui Yin

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable, which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance. We evaluate sketching schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies 5m bits and whose relative variance is 2/m (standard error √{2/m}) has an MVP of 10. Our contributions are as follows. - Cohen [Edith Cohen, 2015] and Ting [Daniel Ting, 2014] independently discovered what we call the {Martingale transform} for converting a mergeable sketch into a non-mergeable sketch. We present a simpler way to analyze the limiting MVP of Martingale-type sketches. - Pettie and Wang proved that the Fishmonger sketch [Seth Pettie and Dingyu Wang, 2021] has the best MVP, H₀/I₀ ≈ 1.98, among a class of mergeable sketches called "linearizable" sketches. (H₀ and I₀ are precisely defined constants.) We prove that the Martingale transform is optimal in the non-mergeable world, and that Martingale Fishmonger in particular is optimal among linearizable sketches, with an MVP of H₀/2 ≈ 1.63. E.g., this is circumstantial evidence that to achieve 1% standard error, we cannot do better than a 2 kilobyte sketch. - Martingale Fishmonger is neither simple nor practical. We develop a new mergeable sketch called Curtain that strikes a nice balance between simplicity and efficiency, and prove that Martingale Curtain has limiting MVP≈ 2.31. It can be updated with O(1) memory accesses and it has lower empirical variance than Martingale LogLog, a practical non-mergeable version of HyperLogLog.
Cite as

Seth Pettie, Dingyu Wang, and Longhui Yin. Non-Mergeable Sketching for Cardinality Estimation. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 104:1-104:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pettie_et_al:LIPIcs.ICALP.2021.104,
  author =	{Pettie, Seth and Wang, Dingyu and Yin, Longhui},
  title =	{{Non-Mergeable Sketching for Cardinality Estimation}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{104:1--104:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.104},
  URN =		{urn:nbn:de:0030-drops-141731},
  doi =		{10.4230/LIPIcs.ICALP.2021.104},
  annote =	{Keywords: Cardinality Estimation, Sketching}
}

Wang, Yiyuan

Document
DOI: 10.4230/LIPIcs.CP.2023.41
Improving Local Search for Pseudo Boolean Optimization by Fragile Scoring Function and Deep Optimization

Authors: Wenbo Zhou, Yujiao Zhao, Yiyuan Wang, Shaowei Cai, Shimao Wang, Xinyu Wang, and Minghao Yin

Published in: LIPIcs, Volume 280, 29th International Conference on Principles and Practice of Constraint Programming (CP 2023)

Abstract
Pseudo-Boolean optimization (PBO) is usually used to model combinatorial optimization problems, especially for some real-world applications. Despite its significant importance in both theory and applications, there are few works on using local search to solve PBO. This paper develops a novel local search framework for PBO, which has three main ideas. First, we design a two-level selection strategy to evaluate all candidate variables. Second, we propose a novel deep optimization strategy to disturb some search spaces. Third, a sampling flipping method is applied to help the algorithm jump out of local optimum. Experimental results show that the proposed algorithms outperform three state-of-the-art PBO algorithms on most instances.
Cite as

Wenbo Zhou, Yujiao Zhao, Yiyuan Wang, Shaowei Cai, Shimao Wang, Xinyu Wang, and Minghao Yin. Improving Local Search for Pseudo Boolean Optimization by Fragile Scoring Function and Deep Optimization. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{zhou_et_al:LIPIcs.CP.2023.41,
  author =	{Zhou, Wenbo and Zhao, Yujiao and Wang, Yiyuan and Cai, Shaowei and Wang, Shimao and Wang, Xinyu and Yin, Minghao},
  title =	{{Improving Local Search for Pseudo Boolean Optimization by Fragile Scoring Function and Deep Optimization}},
  booktitle =	{29th International Conference on Principles and Practice of Constraint Programming (CP 2023)},
  pages =	{41:1--41:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-300-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{280},
  editor =	{Yap, Roland H. C.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2023.41},
  URN =		{urn:nbn:de:0030-drops-190784},
  doi =		{10.4230/LIPIcs.CP.2023.41},
  annote =	{Keywords: Local Search, Pseudo-Boolean Optimization, Deep Optimization}
}
Document
DOI: 10.4230/LIPIcs.CP.2021.39
Improving Local Search for Minimum Weighted Connected Dominating Set Problem by Inner-Layer Local Search

Authors: Bohan Li, Kai Wang, Yiyuan Wang, and Shaowei Cai

Published in: LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)

Abstract
The minimum weighted connected dominating set (MWCDS) problem is an important variant of connected dominating set problems with wide applications, especially in heterogenous networks and gene regulatory networks. In the paper, we develop a nested local search algorithm called NestedLS for solving MWCDS on classic benchmarks and massive graphs. In this local search framework, we propose two novel ideas to make it effective by utilizing previous search information. First, we design the restart based smoothing mechanism as a diversification method to escape from local optimal. Second, we propose a novel inner-layer local search method to enlarge the candidate removal set, which can be modelled as an optimized version of spanning tree problem. Moreover, inner-layer local search method is a general method for maintaining the connectivity constraint when dealing with massive graphs. Experimental results show that NestedLS outperforms state-of-the-art meta-heuristic algorithms on most instances.
Cite as

Bohan Li, Kai Wang, Yiyuan Wang, and Shaowei Cai. Improving Local Search for Minimum Weighted Connected Dominating Set Problem by Inner-Layer Local Search. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{li_et_al:LIPIcs.CP.2021.39,
  author =	{Li, Bohan and Wang, Kai and Wang, Yiyuan and Cai, Shaowei},
  title =	{{Improving Local Search for Minimum Weighted Connected Dominating Set Problem by Inner-Layer Local Search}},
  booktitle =	{27th International Conference on Principles and Practice of Constraint Programming (CP 2021)},
  pages =	{39:1--39:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-211-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{210},
  editor =	{Michel, Laurent D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.39},
  URN =		{urn:nbn:de:0030-drops-153304},
  doi =		{10.4230/LIPIcs.CP.2021.39},
  annote =	{Keywords: Operations Research, NP-hard Problem, Local Search, Weighted Connected Dominating Set Problem}
}
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