87 Search Results for "Wang, Yusu"

Volume

LIPIcs, Volume 129

35th International Symposium on Computational Geometry (SoCG 2019)

SoCG 2019, June 18-21, 2019, Portland, Oregon, USA

Editors: Gill Barequet and Yusu Wang

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)

Copy BibTex To Clipboard

@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)

Copy BibTex To Clipboard

@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.2020.15
The Reeb Graph Edit Distance Is Universal

Authors: Ulrich Bauer, Claudia Landi, and Facundo Mémoli

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

Abstract
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.
Cite as

Ulrich Bauer, Claudia Landi, and Facundo Mémoli. The Reeb Graph Edit Distance Is Universal. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 15:1-15:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{bauer_et_al:LIPIcs.SoCG.2020.15,
  author =	{Bauer, Ulrich and Landi, Claudia and M\'{e}moli, Facundo},
  title =	{{The Reeb Graph Edit Distance Is Universal}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{15:1--15:16},
  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.2020.15},
  URN =		{urn:nbn:de:0030-drops-121730},
  doi =		{10.4230/LIPIcs.SoCG.2020.15},
  annote =	{Keywords: Reeb graphs, topological descriptors, edit distance, interleaving distance}
}
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)

Copy BibTex To Clipboard

@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)

Copy BibTex To Clipboard

@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)

Copy BibTex To Clipboard

@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)

Copy BibTex To Clipboard

@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

Gill Barequet and Yusu Wang. LIPIcs, Volume 129, SoCG'19, Complete Volume. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@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

Gill Barequet and Yusu Wang. Front Matter, Table of Contents, Preface, Conference Organization. In 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)

Copy BibTex To Clipboard

@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
Invited Talk
DOI: 10.4230/LIPIcs.SoCG.2019.1
A Geometric Data Structure from Neuroscience (Invited Talk)

Authors: Sanjoy Dasgupta

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

Abstract
An intriguing geometric primitive, "expand-and-sparsify", has been found in the olfactory system of the fly and several other organisms. It maps an input vector to a much higher-dimensional sparse representation, using a random linear transformation followed by winner-take-all thresholding. I will show that this representation has a variety of formal properties, such as locality preservation, that make it an attractive data structure for algorithms and machine learning. In particular, mimicking the fly’s circuitry yields algorithms for similarity search and for novelty detection that have provable guarantees as well as having practical performance that is competitive with state-of-the-art methods. This talk is based on work with Saket Navlakha (Salk Institute), Chuck Stevens (Salk Institute), and Chris Tosh (Columbia).
Cite as

Sanjoy Dasgupta. A Geometric Data Structure from Neuroscience (Invited Talk). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, p. 1:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{dasgupta:LIPIcs.SoCG.2019.1,
  author =	{Dasgupta, Sanjoy},
  title =	{{A Geometric Data Structure from Neuroscience}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{1:1--1:1},
  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.2019.1},
  URN =		{urn:nbn:de:0030-drops-104055},
  doi =		{10.4230/LIPIcs.SoCG.2019.1},
  annote =	{Keywords: Geometric data structure, algorithm design, neuroscience}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.SoCG.2019.2
Some Geometric and Computational Challenges Arising in Structural Molecular Biology (Invited Talk)

Authors: Bruce R. Donald

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

Abstract
Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous geometry to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials, one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited geometric information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions. We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. For more information please see Y. Qi et al. Jour. Mol. Biol. 2018; 430(18 Pt B):3412-3426. doi: 10.1016/j.jmb.2018.06.022. Ultimately, this method has parallels to other branches of geometric computing that balance discrete and continuous representations, including physical geometric algorithms, robotics, computational geometry, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.
Cite as

Bruce R. Donald. Some Geometric and Computational Challenges Arising in Structural Molecular Biology (Invited Talk). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 2:1-2:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{donald:LIPIcs.SoCG.2019.2,
  author =	{Donald, Bruce R.},
  title =	{{Some Geometric and Computational Challenges Arising in Structural Molecular Biology}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{2:1--2:2},
  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.2019.2},
  URN =		{urn:nbn:de:0030-drops-104065},
  doi =		{10.4230/LIPIcs.SoCG.2019.2},
  annote =	{Keywords: Geometric computing, combutational biology, Continuous Interdomain Orientation Distributions of Protein Conformations}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.3
A New Lower Bound for Semigroup Orthogonal Range Searching

Authors: Peyman Afshani

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

Abstract
We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle’s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao’s influential result had shown that the problem is already non-trivial in one dimension [Yao, 1982]: using m units of space, the query time Q(n) must be Omega(alpha(m,n) + n/(m-n+1)) where alpha(*,*) is the inverse Ackermann’s function, a very slowly growing function. In d dimensions, Bernard Chazelle [Chazelle, 1990] proved that the query time must be Q(n) = Omega((log_beta n)^{d-1}) where beta = 2m/n. Chazelle’s lower bound is known to be tight for when space consumption is "high" i.e., m = Omega(n log^{d+epsilon}n). We have two main results. The first is a lower bound that shows Chazelle’s lower bound was not tight for "low space": we prove that we must have m Q(n) = Omega(n (log n log log n)^{d-1}). Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions.
Cite as

Peyman Afshani. A New Lower Bound for Semigroup Orthogonal Range Searching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 3:1-3:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{afshani:LIPIcs.SoCG.2019.3,
  author =	{Afshani, Peyman},
  title =	{{A New Lower Bound for Semigroup Orthogonal Range Searching}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{3:1--3:14},
  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.2019.3},
  URN =		{urn:nbn:de:0030-drops-104075},
  doi =		{10.4230/LIPIcs.SoCG.2019.3},
  annote =	{Keywords: Data Structures, Range Searching, Lower bounds}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.4
Independent Range Sampling, Revisited Again

Authors: Peyman Afshani and Jeff M. Phillips

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

Abstract
We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer k, we can extract k independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight. This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries. We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to n^{2/3}, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem.
Cite as

Peyman Afshani and Jeff M. Phillips. Independent Range Sampling, Revisited Again. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 4:1-4:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{afshani_et_al:LIPIcs.SoCG.2019.4,
  author =	{Afshani, Peyman and Phillips, Jeff M.},
  title =	{{Independent Range Sampling, Revisited Again}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{4:1--4:13},
  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.2019.4},
  URN =		{urn:nbn:de:0030-drops-104088},
  doi =		{10.4230/LIPIcs.SoCG.2019.4},
  annote =	{Keywords: Range Searching, Data Structures, Sampling}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.5
An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications

Authors: Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl

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

Abstract
In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in R^d and if D >= 1 is an integer, then there is a d-variate polynomial P of degree at most D so that each connected component of R^d \ Z(P) intersects O(n/D^{d-g}) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently - the expected running time of our algorithm is linear in |S|. Our approach exploits the technique of quantifier elimination combined with that of epsilon-samples. We present four applications of our result. The first is a data structure for answering point-enclosure queries among a family of semi-algebraic sets in R^d in O(log n) time, with storage complexity and expected preprocessing time of O(n^{d+epsilon}). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(n^{t+epsilon}) storage and expected preprocessing time, where t > 0 is an integer that depends on d and the description complexity of the ranges. The third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in R^{d} in O(log^2 n) time, with O(n^{d+epsilon}) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.
Cite as

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 5:1-5:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.5,
  author =	{Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Zahl, Joshua},
  title =	{{An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{5:1--5:14},
  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.2019.5},
  URN =		{urn:nbn:de:0030-drops-104096},
  doi =		{10.4230/LIPIcs.SoCG.2019.5},
  annote =	{Keywords: Polynomial partitioning, quantifier elimination, semi-algebraic range spaces, epsilon-samples}
}
  • Refine by Author
  • 19 Wang, Yusu
  • 7 Dey, Tamal K.
  • 4 Agarwal, Pankaj K.
  • 4 Mémoli, Facundo
  • 3 Barequet, Gill
  • Show More...

  • Refine by Classification
  • 46 Theory of computation → Computational geometry
  • 14 Theory of computation → Design and analysis of algorithms
  • 6 Mathematics of computing → Combinatoric problems
  • 6 Mathematics of computing → Graph algorithms
  • 6 Theory of computation → Randomness, geometry and discrete structures
  • Show More...

  • Refine by Keyword
  • 5 Topological Data Analysis
  • 4 Fréchet distance
  • 4 persistent homology
  • 3 Data Structures
  • 3 Persistent homology
  • Show More...

  • Refine by Type
  • 86 document
  • 1 volume

  • Refine by Publication Year
  • 71 2019
  • 4 2015
  • 4 2017
  • 3 2020
  • 2 2018
  • Show More...

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact