84 Search Results for "Factor, Michael"

Document
DOI: 10.4230/LIPIcs.ICDT.2024.21
Subgraph Enumeration in Optimal I/O Complexity

Authors: Shiyuan Deng and Yufei Tao

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
Given a massive data graph G = (V, E) and a small pattern graph Q, the goal of subgraph enumeration is to list all the subgraphs of G isomorphic to Q. In the external memory (EM) model, it is well-known that every indivisible algorithm must perform Ω({|E|^ρ}/{M^{ρ-1} B}) I/Os in the worst case, where M represents the number of words in (internal) memory, B denotes the number of words in a disk block, and ρ is the fractional edge covering number of Q. It has been a longstanding open problem to design an algorithm to match this lower bound. The state of the art is an algorithm in ICDT'23 that achieves an I/O complexity of O({|E|^ρ}/{M^{ρ-1} B} log_{M/B} |E|/B) with high probability. In this paper, we remove the log_{M/B} |E|/B factor, thereby settling the open problem when randomization is permitted.
Cite as

Shiyuan Deng and Yufei Tao. Subgraph Enumeration in Optimal I/O Complexity. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{deng_et_al:LIPIcs.ICDT.2024.21,
  author =	{Deng, Shiyuan and Tao, Yufei},
  title =	{{Subgraph Enumeration in Optimal I/O Complexity}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{21:1--21:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.21},
  URN =		{urn:nbn:de:0030-drops-198033},
  doi =		{10.4230/LIPIcs.ICDT.2024.21},
  annote =	{Keywords: Subgraph Enumeration, Conjunctive Queries, External Memory, Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2024.23
Join Sampling Under Acyclic Degree Constraints and (Cyclic) Subgraph Sampling

Authors: Ru Wang and Yufei Tao

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
Given a (natural) join with an acyclic set of degree constraints (the join itself does not need to be acyclic), we show how to draw a uniformly random sample from the join result in O(polymat/max{1, OUT}) expected time (assuming data complexity) after a preprocessing phase of O(IN) expected time, where IN, OUT, and polymat are the join’s input size, output size, and polymatroid bound, respectively. This compares favorably with the state of the art (Deng et al. and Kim et al., both in PODS'23), which states that, in the absence of degree constraints, a uniformly random sample can be drawn in Õ(AGM/max{1, OUT}) expected time after a preprocessing phase of Õ(IN) expected time, where AGM is the join’s AGM bound and Õ(.) hides a polylog(IN) factor. Our algorithm applies to every join supported by the solutions of Deng et al. and Kim et al. Furthermore, since the polymatroid bound is at most the AGM bound, our performance guarantees are never worse, but can be considerably better, than those of Deng et al. and Kim et al. We then utilize our techniques to tackle directed subgraph sampling, a problem that has extensive database applications and bears close relevance to joins. Let G = (V, E) be a directed data graph where each vertex has an out-degree at most λ, and let P be a directed pattern graph with a constant number of vertices. The objective is to uniformly sample an occurrence of P in G. The problem can be modeled as join sampling with input size IN = Θ(|E|) but, whenever P contains cycles, the converted join has cyclic degree constraints. We show that it is always possible to throw away certain degree constraints such that (i) the remaining constraints are acyclic and (ii) the new join has asymptotically the same polymatroid bound polymat as the old one. Combining this finding with our new join sampling solution yields an algorithm to sample from the original (cyclic) join (thereby yielding a uniformly random occurrence of P) in O(polymat/max{1, OUT}) expected time after O(|E|) expected-time preprocessing, where OUT is the number of occurrences.
Cite as

Ru Wang and Yufei Tao. Join Sampling Under Acyclic Degree Constraints and (Cyclic) Subgraph Sampling. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{wang_et_al:LIPIcs.ICDT.2024.23,
  author =	{Wang, Ru and Tao, Yufei},
  title =	{{Join Sampling Under Acyclic Degree Constraints and (Cyclic) Subgraph Sampling}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.23},
  URN =		{urn:nbn:de:0030-drops-198054},
  doi =		{10.4230/LIPIcs.ICDT.2024.23},
  annote =	{Keywords: Join Sampling, Subgraph Sampling, Degree Constraints, Polymatroid Bounds}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2024.24
Finding Smallest Witnesses for Conjunctive Queries

Authors: Xiao Hu and Stavros Sintos

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
A witness is a sub-database that preserves the query results of the original database but of much smaller size. It has wide applications in query rewriting and debugging, query explanation, IoT analytics, multi-layer network routing, etc. In this paper, we study the smallest witness problem (SWP) for the class of conjunctive queries (CQs) without self-joins. We first establish the dichotomy that SWP for a CQ can be computed in polynomial time if and only if it has head-cluster property, unless P = NP. We next turn to the approximated version by relaxing the size of a witness from being minimum. We surprisingly find that the head-domination property - that has been identified for the deletion propagation problem [Kimelfeld et al., 2012] - can also precisely capture the hardness of the approximated smallest witness problem. In polynomial time, SWP for any CQ with head-domination property can be approximated within a constant factor, while SWP for any CQ without such a property cannot be approximated within a logarithmic factor, unless P = NP. We further explore efficient approximation algorithms for CQs without head-domination property: (1) we show a trivial algorithm which achieves a polynomially large approximation ratio for general CQs; (2) for any CQ with only one non-output attribute, such as star CQs, we show a greedy algorithm with a logarithmic approximation ratio; (3) for line CQs, which contain at least two non-output attributes, we relate SWP problem to the directed steiner forest problem, whose algorithms can be applied to line CQs directly. Meanwhile, we establish a much higher lower bound, exponentially larger than the logarithmic lower bound obtained above. It remains open to close the gap between the lower and upper bound of the approximated SWP for CQs without head-domination property.
Cite as

Xiao Hu and Stavros Sintos. Finding Smallest Witnesses for Conjunctive Queries. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{hu_et_al:LIPIcs.ICDT.2024.24,
  author =	{Hu, Xiao and Sintos, Stavros},
  title =	{{Finding Smallest Witnesses for Conjunctive Queries}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.24},
  URN =		{urn:nbn:de:0030-drops-198066},
  doi =		{10.4230/LIPIcs.ICDT.2024.24},
  annote =	{Keywords: conjunctive query, smallest witness, head-cluster, head-domination}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.6
Facility Location in the Sublinear Geometric Model

Authors: Morteza Monemizadeh

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
In the sublinear geometric model, we are provided with an oracle access to a point set P of n points in a bounded discrete space [Δ]², where Δ = n^O(1) is a polynomially bounded number in n. That is, we do not have direct access to the points, but we can make certain types of queries and there is an oracle that responds to our queries. The type of queries that we assume we can make in this paper, are range counting queries where ranges are axis-aligned rectangles (that are basic primitives in database [Srikanta Tirthapura and David P. Woodruff, 2012; Bentley, 1975; Mark de Berg et al., 2008], computational geometry [Pankaj K. Agarwal, 2004; Pankaj K. Agarwal et al., 1996; Boris Aronov et al., 2010; Boris Aronov et al., 2009], and machine learning [Menachem Sadigurschi and Uri Stemmer, 2021; Long and Tan, 1998; Michael J. Kearns and Umesh V. Vazirani, 1995; Michael J. Kearns and Umesh V. Vazirani, 1994]). The oracle then answers these queries by returning the number of points that are in queried ranges. Let {Alg} be an algorithm that (exactly or approximately) solves a problem 𝒫 in the sublinear geometric model. The query complexity of Alg is measured in terms of the number of queries that Alg makes to solve 𝒫. In this paper, we study the complexity of the (uniform) Euclidean facility location problem in the sublinear geometric model. We develop a randomized sublinear algorithm that with high probability, (1+ε)-approximates the cost of the Euclidean facility location problem of the point set P in the sublinear geometric model using Õ(√n) range counting queries. We complement this result by showing that approximating the cost of the Euclidean facility location problem within o(log(n))-factor in the sublinear geometric model using the sampling strategy that we propose for our sublinear algorithm needs Ω̃(n^{1/4}) RangeCount queries. We leave it as an open problem whether such a polynomial lower bound on the number of RangeCount queries exists for any randomized sublinear algorithm that approximates the cost of the facility location problem within a constant factor.
Cite as

Morteza Monemizadeh. Facility Location in the Sublinear Geometric Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{monemizadeh:LIPIcs.APPROX/RANDOM.2023.6,
  author =	{Monemizadeh, Morteza},
  title =	{{Facility Location in the Sublinear Geometric Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{6:1--6:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.6},
  URN =		{urn:nbn:de:0030-drops-188315},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.6},
  annote =	{Keywords: Facility Location, Sublinear Geometric Model, Range Counting Queries}
}
Document
DOI: 10.4230/LIPIcs.TQC.2023.7
Improved Algorithm and Lower Bound for Variable Time Quantum Search

Authors: Andris Ambainis, Martins Kokainis, and Jevgēnijs Vihrovs

Published in: LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

Abstract
We study variable time search, a form of quantum search where queries to different items take different time. Our first result is a new quantum algorithm that performs variable time search with complexity O(√Tlog n) where T = ∑_{i = 1}ⁿ t_i² with t_i denoting the time to check the i^th item. Our second result is a quantum lower bound of Ω(√{Tlog T}). Both the algorithm and the lower bound improve over previously known results by a factor of √{log T} but the algorithm is also substantially simpler than the previously known quantum algorithms.
Cite as

Andris Ambainis, Martins Kokainis, and Jevgēnijs Vihrovs. Improved Algorithm and Lower Bound for Variable Time Quantum Search. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{ambainis_et_al:LIPIcs.TQC.2023.7,
  author =	{Ambainis, Andris and Kokainis, Martins and Vihrovs, Jevg\={e}nijs},
  title =	{{Improved Algorithm and Lower Bound for Variable Time Quantum Search}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{7:1--7:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-283-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{266},
  editor =	{Fawzi, Omar and Walter, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.7},
  URN =		{urn:nbn:de:0030-drops-183177},
  doi =		{10.4230/LIPIcs.TQC.2023.7},
  annote =	{Keywords: quantum search, amplitude amplification}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.18
Tight Correlation Bounds for Circuits Between AC0 and TC0

Authors: Vinayak M. Kumar

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We initiate the study of generalized AC⁰ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight ≥ k (up to negations of the input bits), which we denote GC⁰(k). The gate set of this class includes biased LTFs like the k-OR (outputs 1 iff ≥ k bits are 1) and k-AND (outputs 0 iff ≥ k bits are 0), and thus can be seen as an interpolation between AC⁰ and TC⁰. We establish a tight multi-switching lemma for GC⁰(k) circuits, which bounds the probability that several depth-2 GC⁰(k) circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-d size-s AC⁰ circuits lifts to depth-d size-s^{.99} GC⁰(.01 log s) circuits with no loss in parameters (other than hidden constants). Our result has the following applications: - Size-2^Ω(n^{1/d}) depth-d GC⁰(Ω(n^{1/d})) circuits do not correlate with parity (extending a result of Håstad (SICOMP, 2014)). - Size-n^Ω(log n) GC⁰(Ω(log² n)) circuits with n^{.249} arbitrary threshold gates or n^{.499} arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). - There is a seed length O((log m)^{d-1}log(m/ε)log log(m)) pseudorandom generator against size-m depth-d GC⁰(log m) circuits, matching the AC⁰ lower bound of Håstad up to a log log m factor (extending a result of Lyu (CCC, 2022)). - Size-m GC⁰(log m) circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).
Cite as

Vinayak M. Kumar. Tight Correlation Bounds for Circuits Between AC0 and TC0. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 18:1-18:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{kumar:LIPIcs.CCC.2023.18,
  author =	{Kumar, Vinayak M.},
  title =	{{Tight Correlation Bounds for Circuits Between AC0 and TC0}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{18:1--18:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.18},
  URN =		{urn:nbn:de:0030-drops-182885},
  doi =		{10.4230/LIPIcs.CCC.2023.18},
  annote =	{Keywords: AC⁰, TC⁰, Switching Lemma, Lower Bounds, Correlation Bounds, Circuit Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.15
Efficient Two-Parameter Persistence Computation via Cohomology

Authors: Ulrich Bauer, Fabian Lenzen, and Michael Lesnick

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

Abstract
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (ph), which dramatically improves the speed and scalability of ph computations for Vietoris-Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris-Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter ph. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter ph that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.
Cite as

Ulrich Bauer, Fabian Lenzen, and Michael Lesnick. Efficient Two-Parameter Persistence Computation via Cohomology. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{bauer_et_al:LIPIcs.SoCG.2023.15,
  author =	{Bauer, Ulrich and Lenzen, Fabian and Lesnick, Michael},
  title =	{{Efficient Two-Parameter Persistence Computation via Cohomology}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{15:1--15:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.15},
  URN =		{urn:nbn:de:0030-drops-178656},
  doi =		{10.4230/LIPIcs.SoCG.2023.15},
  annote =	{Keywords: Persistent homology, persistent cohomology, two-parameter persistence, clearing}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.32
Lower Bounds on Retroactive Data Structures

Authors: Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Jayson Lynch

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that has a data structure where operations run in O(T(n,m)) time per operation, but any partially retroactive version of that data structure requires T(n,m)⋅m^{1-o(1)} worst-case time per operation, where n is the size of the data structure at any time and m is the number of operations. Any data structure with operations running in O(T(n,m)) time per operation can be converted (via the "rollback method") into a partially retroactive data structure running in O(T(n,m)⋅m) time per operation, so our lower bound is tight up to an m^o(1) factor common in fine-grained complexity.
Cite as

Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Jayson Lynch. Lower Bounds on Retroactive Data Structures. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{chung_et_al:LIPIcs.ISAAC.2022.32,
  author =	{Chung, Lily and Demaine, Erik D. and Hendrickson, Dylan and Lynch, Jayson},
  title =	{{Lower Bounds on Retroactive Data Structures}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{32:1--32:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.32},
  URN =		{urn:nbn:de:0030-drops-173171},
  doi =		{10.4230/LIPIcs.ISAAC.2022.32},
  annote =	{Keywords: Retroactivity, time travel, rollback, fine-grained complexity}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.51
(1+ε)-Approximate Shortest Paths in Dynamic Streams

Authors: Michael Elkin and Chhaya Trehan

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ-1 versus n^{1+1/κ}, for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1+ε in the stretch for weighted graphs, and an additional factor of log^{O(1)}n in the space.) The only existing solution of the second type uses n^{1/2 - O(1/κ)} passes over the stream (for space O(n^{1+1/κ})), and applies only to unweighted graphs. In this paper we show that (1+ε)-approximate single-source shortest paths can be computed with Õ(n^{1+1/κ}) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n^{1/κ}). We achieve these results by devising efficient dynamic streaming constructions of (1 + ε, β)-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [S. Ganguly, 2007; Graham Cormode and D. Firmani, 2013], we believe that it is of independent interest.
Cite as

Michael Elkin and Chhaya Trehan. (1+ε)-Approximate Shortest Paths in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 51:1-51:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{elkin_et_al:LIPIcs.APPROX/RANDOM.2022.51,
  author =	{Elkin, Michael and Trehan, Chhaya},
  title =	{{(1+\epsilon)-Approximate Shortest Paths in Dynamic Streams}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{51:1--51:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.51},
  URN =		{urn:nbn:de:0030-drops-171733},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.51},
  annote =	{Keywords: Shortest Paths, Dynamic Streams, Approximate Distances, Spanners, Hopsets}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.15
Pseudorandom Generators, Resolution and Heavy Width

Authors: Dmitry Sokolov

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson [Michael Alekhnovich et al., 2004] we call a pseudorandom generator ℱ:{0, 1}ⁿ → {0, 1}^m hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement b ∉ Im(ℱ) for any string b ∈ {0, 1}^m. In [Michael Alekhnovich et al., 2004] the authors suggested the "functional encoding" of the considered statement for Nisan-Wigderson generator that allows the introduction of "local" extension variables. These extension variables may potentially significantly increase the power of the proof system. In [Michael Alekhnovich et al., 2004] authors gave a lower bound of exp[Ω(n²/{m⋅2^{2^Δ}})] on the length of Resolution proofs where Δ is the degree of the dependency graph of the generator. This lower bound meets the barrier for the restriction technique. In this paper, we introduce a "heavy width" measure for Resolution that allows us to show a lower bound of exp[n²/{m 2^𝒪(εΔ)}] on the length of Resolution proofs of the considered statement for the Nisan-Wigderson generator. This gives an exponential lower bound up to Δ := log^{2 - δ} n (the bigger degree the more extension variables we can use). In [Michael Alekhnovich et al., 2004] authors left an open problem to get rid of scaling factor 2^{2^Δ}, it is a solution to this open problem.
Cite as

Dmitry Sokolov. Pseudorandom Generators, Resolution and Heavy Width. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{sokolov:LIPIcs.CCC.2022.15,
  author =	{Sokolov, Dmitry},
  title =	{{Pseudorandom Generators, Resolution and Heavy Width}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{15:1--15:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.15},
  URN =		{urn:nbn:de:0030-drops-165770},
  doi =		{10.4230/LIPIcs.CCC.2022.15},
  annote =	{Keywords: proof complexity, pseudorandom generators, resolution, lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.86
Near-Optimal Decremental Hopsets with Applications

Authors: Jakub Łącki and Yasamin Nazari

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Given a weighted undirected graph G = (V,E,w), a hopset H of hopbound β and stretch (1+ε) is a set of edges such that for any pair of nodes u, v ∈ V, there is a path in G ∪ H of at most β hops, whose length is within a (1+ε) factor from the distance between u and v in G. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynomial) hopbound of 2^{log^{Ω(1)} n} [Bernstein, FOCS'09; HKN, FOCS'14; Chechik, FOCS'18]. By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain (2k-1)(1+ε)-approximate all-pairs shortest paths (for any constant k ≥ 2), in Õ(n^{1/k}) amortized update time and O(k) query time. This improves (by a polynomial factor) over the update-time of the best previously known decremental algorithm in the constant query time regime. Moreover, it improves over the result of [Chechik, FOCS'18] that has a query time of O(log log(nW)), where W is the aspect ratio, and the amortized update time is n^{1/k}⋅(1/ε)^{Õ(√{log n})}). For sparse graphs our construction nearly matches the best known static running time / query time tradeoff. We also obtain near-optimal bounds for maintaining approximate multi-source shortest paths and distance sketches, and get improved bounds for approximate single-source shortest paths. Our algorithms are randomized and our bounds hold with high probability against an oblivious adversary.
Cite as

Jakub Łącki and Yasamin Nazari. Near-Optimal Decremental Hopsets with Applications. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{lacki_et_al:LIPIcs.ICALP.2022.86,
  author =	{{\L}\k{a}cki, Jakub and Nazari, Yasamin},
  title =	{{Near-Optimal Decremental Hopsets with Applications}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{86:1--86:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.86},
  URN =		{urn:nbn:de:0030-drops-164277},
  doi =		{10.4230/LIPIcs.ICALP.2022.86},
  annote =	{Keywords: Dynamic Algorithms, Data Structures, Shortest Paths, Hopsets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.2
On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem

Authors: Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.
Cite as

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2,
  author =	{Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung},
  title =	{{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{2:1--2:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2},
  URN =		{urn:nbn:de:0030-drops-160109},
  doi =		{10.4230/LIPIcs.SoCG.2022.2},
  annote =	{Keywords: Approximation, Motion Planning, Scheduling}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.7
Minimum-Error Triangulations for Sea Surface Reconstruction

Authors: Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, and Heiko Röglin

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al. (Computers & Geosciences, 157:104920, 2021) who have suggested to learn a triangulation D of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by D to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay (k-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem. In geometric terms, the input to our problem consists of two sets of points in ℝ² with elevations: a set 𝒮 that is to be triangulated, and a set ℛ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in ℛ to the triangulated surface that is obtained by interpolating elevations of 𝒮 linearly in each triangle. Our goal is to find the triangulation of 𝒮 that has minimum error with respect to ℛ. In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless P = NP. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to k-OD triangulations for k ≤ 7. In particular, instances for which the number of connected components of the so-called k-OD fixed-edge graph is small can be solved within few seconds.
Cite as

Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, and Heiko Röglin. Minimum-Error Triangulations for Sea Surface Reconstruction. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{arutyunova_et_al:LIPIcs.SoCG.2022.7,
  author =	{Arutyunova, Anna and Driemel, Anne and Haunert, Jan-Henrik and Haverkort, Herman and Kusche, J\"{u}rgen and Langetepe, Elmar and Mayer, Philip and Mutzel, Petra and R\"{o}glin, Heiko},
  title =	{{Minimum-Error Triangulations for Sea Surface Reconstruction}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{7:1--7:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.7},
  URN =		{urn:nbn:de:0030-drops-160155},
  doi =		{10.4230/LIPIcs.SoCG.2022.7},
  annote =	{Keywords: Minimum-Error Triangulation, k-Order Delaunay Triangulations, Data dependent Triangulations, Sea Surface Reconstruction, fixed-Edge Graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.14
Quasi-Universality of Reeb Graph Distances

Authors: Ulrich Bauer, Håvard Bakke Bjerkevik, and Benedikt Fluhr

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We establish bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.
Cite as

Ulrich Bauer, Håvard Bakke Bjerkevik, and Benedikt Fluhr. Quasi-Universality of Reeb Graph Distances. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{bauer_et_al:LIPIcs.SoCG.2022.14,
  author =	{Bauer, Ulrich and Bjerkevik, H\r{a}vard Bakke and Fluhr, Benedikt},
  title =	{{Quasi-Universality of Reeb Graph Distances}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.14},
  URN =		{urn:nbn:de:0030-drops-160221},
  doi =		{10.4230/LIPIcs.SoCG.2022.14},
  annote =	{Keywords: Reeb graphs, contour trees, merge trees, distances, universality, interleaving distance, functional distortion distance, functional contortion distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.18
Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

Authors: Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, and Galyna Livshyts

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
The combinatorial diameter diam(P) of a polytope P is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an n-dimensional polytope P defined by the intersection of m i.i.d. half-spaces whose normals are chosen uniformly from the sphere, we show that diam(P) is Ω(n m^{1/(n-1)}) and O(n² m^{1/(n-1)} + n⁵ 4ⁿ) with high probability when m ≥ 2^{Ω(n)}. For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when m is large, where we rely on the Θ(n² m^{1/(n-1)}) bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these "shadows paths" together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope P^∘, corresponding to a random convex hull, by showing the relation diam(P) ≥ (n-1)(diam(P^∘)-2). We then prove that the shortest path between any "nearly" antipodal pair vertices of P^∘ has length Ω(m^{1/(n-1)}).
Cite as

Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, and Galyna Livshyts. Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{bonnet_et_al:LIPIcs.SoCG.2022.18,
  author =	{Bonnet, Gilles and Dadush, Daniel and Grupel, Uri and Huiberts, Sophie and Livshyts, Galyna},
  title =	{{Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.18},
  URN =		{urn:nbn:de:0030-drops-160269},
  doi =		{10.4230/LIPIcs.SoCG.2022.18},
  annote =	{Keywords: Random Polytopes, Combinatorial Diameter, Hirsch Conjecture}
}
  • Refine by Author
  • 6 Dinitz, Michael
  • 5 Lampis, Michael
  • 3 Paschos, Vangelis Th.
  • 2 Bauer, Ulrich
  • 2 Ben-Sasson, Eli
  • Show More...

  • Refine by Classification
  • 10 Theory of computation → Computational geometry
  • 6 Theory of computation → Approximation algorithms analysis
  • 5 Mathematics of computing → Graph algorithms
  • 5 Theory of computation → Design and analysis of algorithms
  • 4 Theory of computation → Sparsification and spanners
  • Show More...

  • Refine by Keyword
  • 7 Approximation Algorithms
  • 6 approximation algorithms
  • 4 Clustering
  • 2 Approximation
  • 2 Approximation algorithms
  • Show More...

  • Refine by Type
  • 84 document

  • Refine by Publication Year
  • 16 2019
  • 15 2022
  • 14 2016
  • 6 2017
  • 5 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact