15 Search Results for "Peng, Richard"

Document
DOI: 10.4230/LIPIcs.ISAAC.2023.18
Distance Queries over Dynamic Interval Graphs

Authors: Jingbang Chen, Meng He, J. Ian Munro, Richard Peng, Kaiyu Wu, and Daniel J. Zhang

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
We design the first dynamic distance oracles for interval graphs, which are intersection graphs of a set of intervals on the real line, and for proper interval graphs, which are intersection graphs of a set of intervals in which no interval is properly contained in another. For proper interval graphs, we design a linear space data structure which supports distance queries (computing the distance between two query vertices) and vertex insertion or deletion in O(lg n) worst-case time, where n is the number of vertices currently in G. Under incremental (insertion only) or decremental (deletion only) settings, we design linear space data structures that support distance queries in O(lg n) worst-case time and vertex insertion or deletion in O(lg n) amortized time, where n is the maximum number of vertices in the graph. Under fully dynamic settings, we design a data structure that represents an interval graph G in O(n) words of space to support distance queries in O(n lg n/S(n)) worst-case time and vertex insertion or deletion in O(S(n)+lg n) worst-case time, where n is the number of vertices currently in G and S(n) is an arbitrary function that satisfies S(n) = Ω(1) and S(n) = O(n). This implies an O(n)-word solution with O(√{nlg n})-time support for both distance queries and updates. All four data structures can answer shortest path queries by reporting the vertices in the shortest path between two query vertices in O(lg n) worst-case time per vertex. We also study the hardness of supporting distance queries under updates over an intersection graph of 3D axis-aligned line segments, which generalizes our problem to 3D. Finally, we solve the problem of computing the diameter of a dynamic connected interval graph.
Cite as

Jingbang Chen, Meng He, J. Ian Munro, Richard Peng, Kaiyu Wu, and Daniel J. Zhang. Distance Queries over Dynamic Interval Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 18:1-18:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2023.18,
  author =	{Chen, Jingbang and He, Meng and Munro, J. Ian and Peng, Richard and Wu, Kaiyu and Zhang, Daniel J.},
  title =	{{Distance Queries over Dynamic Interval Graphs}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{18:1--18:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.18},
  URN =		{urn:nbn:de:0030-drops-193207},
  doi =		{10.4230/LIPIcs.ISAAC.2023.18},
  annote =	{Keywords: interval graph, proper interval graph, intersection graph, geometric intersection graph, distance oracle, distance query, shortest path query, dynamic graph}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.104
A Simple Boosting Framework for Transshipment

Authors: Goran Zuzic

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Transshipment is an important generalization of both the shortest path problem and the optimal transport problem. The task asks to route a demand using a flow of minimum cost over (uncapacitated) edges. Transshipment has recently received extensive attention in theoretical computer science as it is the centerpiece of all modern theoretical breakthroughs in parallel and distributed (approximate) shortest-path computation, a classic and well-studied problem. The key advantage of transshipment over shortest paths is the so-called boosting property: one can often boost a crude approximate solution to a (near-optimal) (1+ε)-approximate solution. However, our understanding of this phenomenon is limited: it is not clear which approximators can be boosted. Moreover, all current boosting frameworks are built with a specific type of approximator in mind and are relatively complicated. The main takeaway of our paper is conceptual: any black-box oracle that computes an approximate dual solution can be boosted to an (1+ε)-approximator. This decouples and simplifies all known near-optimal (1+ε)-approximate transshipment and shortest paths results: they all (implicitly) construct approximate dual solutions and boost them. We provide a very simple analysis based on the multiplicative weights framework. Furthermore, to keep the paper completely self-contained, we provide a new (and arguably much simpler) analysis of multiplicative weights that leverages well-known optimization tools to bypass the ad-hoc calculations used in the standard analyses.
Cite as

Goran Zuzic. A Simple Boosting Framework for Transshipment. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 104:1-104:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{zuzic:LIPIcs.ESA.2023.104,
  author =	{Zuzic, Goran},
  title =	{{A Simple Boosting Framework for Transshipment}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{104:1--104:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.104},
  URN =		{urn:nbn:de:0030-drops-187570},
  doi =		{10.4230/LIPIcs.ESA.2023.104},
  annote =	{Keywords: mixed continuous-discrete optimization, boosting, multiplicative weights, theoretical computer science, shortest path, parallel algorithms}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.ICALP.2023.2
An Almost-Linear Time Algorithm for Maximum Flow and More (Invited Talk)

Authors: Rasmus Kyng

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
In this talk, I will explain a new algorithm for computing exact maximum and minimum-cost flows in almost-linear time, settling the time complexity of these basic graph problems up to subpolynomial factors. Our algorithm uses a novel interior point method that builds the optimal flow as a sequence of approximate minimum-ratio cycles, each of which is computed and processed very efficiently using a new dynamic data structure. By well-known reductions, our result implies almost-linear time algorithms for several problems including bipartite matching, optimal transport, and undirected vertex connectivity. Our framework also extends to minimizing general edge-separable convex functions to high accuracy, yielding the first almost-linear time algorithms for many other problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and isotonic regression. This talk is based on joint work with Li Chen, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva [Chen et al., 2022]. Our result appeared in FOCS'22 and won the FOCS best paper award.
Cite as

Rasmus Kyng. An Almost-Linear Time Algorithm for Maximum Flow and More (Invited Talk). In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, p. 2:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kyng:LIPIcs.ICALP.2023.2,
  author =	{Kyng, Rasmus},
  title =	{{An Almost-Linear Time Algorithm for Maximum Flow and More}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.2},
  URN =		{urn:nbn:de:0030-drops-180543},
  doi =		{10.4230/LIPIcs.ICALP.2023.2},
  annote =	{Keywords: Maximum flow, Minimum cost flow, Data structures, Interior point methods, Convex optimization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.69
Efficient Data Structures for Incremental Exact and Approximate Maximum Flow

Authors: Gramoz Goranci and Monika Henzinger

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
We show an (1+ε)-approximation algorithm for maintaining maximum s-t flow under m edge insertions in m^{1/2+o(1)} ε^{-1/2} amortized update time for directed, unweighted graphs. This constitutes the first sublinear dynamic maximum flow algorithm in general sparse graphs with arbitrarily good approximation guarantee. Furthermore we give an algorithm that maintains an exact maximum s-t flow under m edge insertions in an n-node graph in Õ(n^{5/2}) total update time. For sufficiently dense graphs, this gives to the first exact incremental algorithm with sub-linear amortized update time for maintaining maximum flows.
Cite as

Gramoz Goranci and Monika Henzinger. Efficient Data Structures for Incremental Exact and Approximate Maximum Flow. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 69:1-69:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goranci_et_al:LIPIcs.ICALP.2023.69,
  author =	{Goranci, Gramoz and Henzinger, Monika},
  title =	{{Efficient Data Structures for Incremental Exact and Approximate Maximum Flow}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{69:1--69:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.69},
  URN =		{urn:nbn:de:0030-drops-181212},
  doi =		{10.4230/LIPIcs.ICALP.2023.69},
  annote =	{Keywords: dynamic graph algorithms, maximum flow, data structures}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.69
A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems

Authors: Monika Henzinger, Billy Jin, Richard Peng, and David P. Williamson

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form 𝐋𝐱 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution 𝐱 of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1-ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an Õ(m^{1.5}) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ε}) for any ε > 0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart.
Cite as

Monika Henzinger, Billy Jin, Richard Peng, and David P. Williamson. A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 69:1-69:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{henzinger_et_al:LIPIcs.ITCS.2023.69,
  author =	{Henzinger, Monika and Jin, Billy and Peng, Richard and Williamson, David P.},
  title =	{{A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{69:1--69:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.69},
  URN =		{urn:nbn:de:0030-drops-175720},
  doi =		{10.4230/LIPIcs.ITCS.2023.69},
  annote =	{Keywords: Laplacian solver, electrical flow, data structure}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.83
Vertex Sparsification for Edge Connectivity in Polynomial Time

Authors: Yang P. Liu

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
An important open question in the area of vertex sparsification is whether (1+ε)-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. The work [Parinya Chalermsook et al., 2021] (SODA 2021) introduced a relaxation called connectivity-c mimicking networks, which asks to construct a vertex sparsifier which preserves connectivity among k terminals exactly up to the value of c, and showed applications to dynamic connectivity data structures and survivable network design. We show that connectivity-c mimicking networks with Õ(kc³) edges exist and can be constructed in polynomial time in n and c, improving over the results of [Parinya Chalermsook et al., 2021] for any c ≥ log n, whose runtimes depended exponentially on c.
Cite as

Yang P. Liu. Vertex Sparsification for Edge Connectivity in Polynomial Time. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 83:1-83:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{liu:LIPIcs.ITCS.2023.83,
  author =	{Liu, Yang P.},
  title =	{{Vertex Sparsification for Edge Connectivity in Polynomial Time}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{83:1--83:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.83},
  URN =		{urn:nbn:de:0030-drops-175863},
  doi =		{10.4230/LIPIcs.ITCS.2023.83},
  annote =	{Keywords: Vertex-sparsification, edge-connectivity, Gammoids}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.70
Vertex Sparsifiers for Hyperedge Connectivity

Authors: Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Recently, Chalermsook et al. {[}SODA'21{]} introduces a notion of vertex sparsifiers for c-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for c-edge st-connectivity {[}Jin and Sun FOCS'22{]}. We study a natural extension called vertex sparsifiers for c-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph G = (V,E) with n vertices and m hyperedges with k terminal vertices and a parameter c, there exists a hypergraph H containing only O(kc³) hyperedges that preserves all minimum cuts (up to value c) between all subset of terminals. This matches the best bound of O(kc³) edges for normal graphs by [Liu'20]. Moreover, H can be constructed in almost-linear O(p^{1+o(1)} + n(rclog n)^{O(rc)}log m) time where r = max_{e ∈ E}|e| is the rank of G and p = ∑_{e ∈ E}|e| is the total size of G, or in poly(m, n) time if we slightly relax the size to O(kc³log^{1.5}(kc)) hyperedges.
Cite as

Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang. Vertex Sparsifiers for Hyperedge Connectivity. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 70:1-70:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jiang_et_al:LIPIcs.ESA.2022.70,
  author =	{Jiang, Han and Huang, Shang-En and Saranurak, Thatchaphol and Zhang, Tian},
  title =	{{Vertex Sparsifiers for Hyperedge Connectivity}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{70:1--70:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.70},
  URN =		{urn:nbn:de:0030-drops-170081},
  doi =		{10.4230/LIPIcs.ESA.2022.70},
  annote =	{Keywords: Vertex sparsifier, hypergraph, connectivity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.11
Decremental Matching in General Graphs

Authors: Sepehr Assadi, Aaron Bernstein, and Aditi Dudeja

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

Abstract
We consider the problem of maintaining an approximate maximum integral matching in a dynamic graph G, while the adversary makes changes to the edges of the graph. The goal is to maintain a (1+ε)-approximate maximum matching for constant ε > 0, while minimizing the update time. In the fully dynamic setting, where both edge insertion and deletions are allowed, Gupta and Peng (see [Manoj Gupta and Richard Peng, 2013]) gave an algorithm for this problem with an update time of O(√m/ε²). Motivated by the fact that the O_ε(√m) barrier is hard to overcome (see Henzinger, Krinninger, Nanongkai, and Saranurak [Henzinger et al., 2015]; Kopelowitz, Pettie, and Porat [Kopelowitz et al., 2016]), we study this problem in the decremental model, where the adversary is only allowed to delete edges. Recently, Bernstein, Probst-Gutenberg, and Saranurak (see [Bernstein et al., 2020]) gave an O(poly({log n}/ε)) update time decremental algorithm for this problem in bipartite graphs. However, beating O(√m) update time remained an open problem for general graphs. In this paper, we bridge the gap between bipartite and general graphs, by giving an O_ε(poly(log n)) update time algorithm that maintains a (1+ε)-approximate maximum integral matching under adversarial deletions. Our algorithm is randomized, but works against an adaptive adversary. Together with the work of Grandoni, Leonardi, Sankowski, Schwiegelshohn, and Solomon [Fabrizio Grandoni et al., 2019] who give an O_ε(1) update time algorithm for general graphs in the incremental (insertion-only) model, our result essentially completes the picture for partially dynamic matching.
Cite as

Sepehr Assadi, Aaron Bernstein, and Aditi Dudeja. Decremental Matching in General Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.ICALP.2022.11,
  author =	{Assadi, Sepehr and Bernstein, Aaron and Dudeja, Aditi},
  title =	{{Decremental Matching in General Graphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{11:1--11:19},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.11},
  URN =		{urn:nbn:de:0030-drops-163528},
  doi =		{10.4230/LIPIcs.ICALP.2022.11},
  annote =	{Keywords: Dynamic algorithms, matching, primal-dual algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.33
Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

Authors: Sílvia Casacuberta and Rasmus Kyng

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected O(n^{2.2131}) time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory.
Cite as

Sílvia Casacuberta and Rasmus Kyng. Faster Sparse Matrix Inversion and Rank Computation in Finite Fields. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 33:1-33:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{casacuberta_et_al:LIPIcs.ITCS.2022.33,
  author =	{Casacuberta, S{\'\i}lvia and Kyng, Rasmus},
  title =	{{Faster Sparse Matrix Inversion and Rank Computation in Finite Fields}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{33:1--33:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.33},
  URN =		{urn:nbn:de:0030-drops-156290},
  doi =		{10.4230/LIPIcs.ITCS.2022.33},
  annote =	{Keywords: Matrix inversion, rank computation, displacement operators, numerical linear algebra}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.FSTTCS.2020.3
Convex Optimization and Dynamic Data Structure (Invited Talk)

Authors: Yin Tat Lee

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
In the last three years, there are many breakthroughs in optimization such as nearly quadratic time algorithms for bipartite matching, linear programming algorithms that are as fast as Ax = b. All of these algorithms are based on a careful combination of optimization techniques and dynamic data structures. In this talk, we will explain the framework underlying all the recent breakthroughs. Joint work with Jan van den Brand, Michael B. Cohen, Sally Dong, Haotian Jiang, Tarun Kathuria, Danupon Nanongkai, Swati Padmanabhan, Richard Peng, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, Di Wang, Sam Chiu-wai Wong, Guanghao Ye, Qiuyi Zhang.
Cite as

Yin Tat Lee. Convex Optimization and Dynamic Data Structure (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lee:LIPIcs.FSTTCS.2020.3,
  author =	{Lee, Yin Tat},
  title =	{{Convex Optimization and Dynamic Data Structure}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.3},
  URN =		{urn:nbn:de:0030-drops-132440},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.3},
  annote =	{Keywords: Convex Optimization, Dynamic Data Structure}
}
Document
DOI: 10.4230/DagRep.8.6.19
High-Performance Graph Algorithms (Dagstuhl Seminar 18241)

Authors: Henning Meyerhenke, Richard Peng, and Ilya Safro

Published in: Dagstuhl Reports, Volume 8, Issue 6 (2019)

Abstract
This report documents the program and outcomes of Dagstuhl Seminar 18241 ``High-performance Graph Algorithms''. The seminar reflected the ongoing qualitative change how graph algorithms are used in practice due to (i) the complex structure of graphs in new and emerging applications, (ii) the size of typical inputs, and (iii) the computer systems with which graph problems are solved. This change is having a tremendous impact on the field of graph algorithms in terms of algorithm theory and implementation as well as hardware requirements and application areas. The seminar covered recent advances in all these aspects, trying to balance and mediate between theory and practice. The abstracts included in this report contain and survey recent state-of-the-art results, but also point to promising new directions for high-performance graph algorithms and their applications, both from a theoretical and a practical point of view.
Cite as

Henning Meyerhenke, Richard Peng, and Ilya Safro. High-Performance Graph Algorithms (Dagstuhl Seminar 18241). In Dagstuhl Reports, Volume 8, Issue 6, pp. 19-39, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@Article{meyerhenke_et_al:DagRep.8.6.19,
  author =	{Meyerhenke, Henning and Peng, Richard and Safro, Ilya},
  title =	{{High-Performance Graph Algorithms (Dagstuhl Seminar 18241)}},
  pages =	{19--39},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2018},
  volume =	{8},
  number =	{6},
  editor =	{Meyerhenke, Henning and Peng, Richard and Safro, Ilya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.6.19},
  URN =		{urn:nbn:de:0030-drops-100475},
  doi =		{10.4230/DagRep.8.6.19},
  annote =	{Keywords: algorithm engineering, combinatorial scientific computing, graph algorithms, high-performance computing, theoretical computer science}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.14
Density Independent Algorithms for Sparsifying k-Step Random Walks

Authors: Gorav Jindal, Pavel Kolev, Richard Peng, and Saurabh Sawlani

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

Abstract
We give faster algorithms for producing sparse approximations of the transition matrices of k-step random walks on undirected and weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with n vertices and m edges, our algorithm produces a graph with about nlog(n) edges that approximates the k-step random walk graph in about m + k^2 nlog^4(n) time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices.
Cite as

Gorav Jindal, Pavel Kolev, Richard Peng, and Saurabh Sawlani. Density Independent Algorithms for Sparsifying k-Step Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 14:1-14:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jindal_et_al:LIPIcs.APPROX-RANDOM.2017.14,
  author =	{Jindal, Gorav and Kolev, Pavel and Peng, Richard and Sawlani, Saurabh},
  title =	{{Density Independent Algorithms for Sparsifying k-Step Random Walks}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{14:1--14:17},
  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.2017.14},
  URN =		{urn:nbn:de:0030-drops-75638},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.14},
  annote =	{Keywords: random walks, graph sparsification, spectral graph theory, effective resistances}
}
Document
DOI: 10.4230/LITES-v004-i001-a002
Utility-Based Scheduling of (m,k)-firm Real-Time Tasks - New Empirical Results

Authors: Florian Kluge

Published in: LITES, Volume 4, Issue 1 (2017). Leibniz Transactions on Embedded Systems, Volume 4, Issue 1

Abstract
The concept of a firm real-time task implies the notion of a firm deadline that should not be missed by the jobs of this task. If a deadline miss occurs, the concerned job yields no value to the system. For some applications domains, this restrictive notion can be relaxed. For example, robust control systems can tolerate that single executions of a control loop miss their deadlines, and still yield an acceptable behaviour. Thus, systems can be developed under more optimistic assumptions, e.g. by allowing overloads. However, care must be taken that deadline misses do not accumulate. This restriction can be expressed by the model of (m,k)-firm real-time tasks that require that from any k consecutive jobs at least m are executed successfully. In this article, we extend our prior work on the MKU scheduling heuristic. MKU uses history-cognisant utility functions as means for making decisions in overload situations. We present new theoretical results on MKU and on other schedulers for (m,k)-firm real-time tasks. Based on extensive simulations, we assess the performance of these schedulers. The results allow us to identify task set characteristics that can be used as guidelines for choosing a scheduler for a concrete use case.
Cite as

Florian Kluge. Utility-Based Scheduling of (m,k)-firm Real-Time Tasks - New Empirical Results. In LITES, Volume 4, Issue 1 (2017). Leibniz Transactions on Embedded Systems, Volume 4, Issue 1, pp. 02:1-02:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@Article{kluge:LITES-v004-i001-a002,
  author =	{Kluge, Florian},
  title =	{{Utility-Based Scheduling of (m,k)-firm Real-Time Tasks - New Empirical Results}},
  booktitle =	{LITES, Volume 4, Issue 1 (2017)},
  pages =	{02:1--02:25},
  journal =	{Leibniz Transactions on Embedded Systems},
  ISSN =	{2199-2002},
  year =	{2017},
  volume =	{4},
  number =	{1},
  editor =	{Kluge, Florian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES-v004-i001-a002},
  doi =		{10.4230/LITES-v004-i001-a002},
  annote =	{Keywords: Real-time Scheduling, (m, k)-Firm Real-Time Tasks}
}
Document
DOI: 10.4230/LITES-v002-i002-a001
From Dataflow Specification to Multiprocessor Partitioned Time-triggered Real-time Implementation

Authors: Thomas Carle, Dumitru Potop-Butucaru, Yves Sorel, and David Lesens

Published in: LITES, Volume 2, Issue 2 (2015). Leibniz Transactions on Embedded Systems, Volume 2, Issue 2

Abstract
Our objective is to facilitate the development of complex time-triggered systems by automating the allocation and scheduling steps. We show that full automation is possible while taking into account the elements of complexity needed by a complex embedded control system. More precisely, we consider deterministic functional specifications provided (as often in an industrial setting) by means of synchronous data-flow models with multiple modes and multiple relative periods. We first extend this functional model with an original real-time characterization that takes advantage of our time-triggered framework to provide a simpler representation of complex end-to-end flow requirements. We also extend our specifications with additional non-functional properties specifying partitioning, allocation, and preemptability constraints. Then, we provide novel algorithms for the off-line scheduling of these extended specifications onto partitioned time-triggered architectures à la ARINC 653. The main originality of our work is that it takes into account at the same time multiple complexity elements: various types of non-functional properties (real-time, partitioning, allocation, preemptability) and functional specifications with conditional execution and multiple modes. Allocation of time slots/windows to partitions can be fully or partially provided, or synthesized by our tool. Our algorithms allow the automatic allocation and scheduling onto multi-processor (distributed) systems with a global time base, taking into account communication costs. We demonstrate our technique on a model of space flight software system with strong real-time determinism requirements.
Cite as

Thomas Carle, Dumitru Potop-Butucaru, Yves Sorel, and David Lesens. From Dataflow Specification to Multiprocessor Partitioned Time-triggered Real-time Implementation. In LITES, Volume 2, Issue 2 (2015). Leibniz Transactions on Embedded Systems, Volume 2, Issue 2, pp. 01:1-01:30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@Article{carle_et_al:LITES-v002-i002-a001,
  author =	{Carle, Thomas and Potop-Butucaru, Dumitru and Sorel, Yves and Lesens, David},
  title =	{{From Dataflow Specification to Multiprocessor Partitioned Time-triggered Real-time Implementation}},
  booktitle =	{LITES, Volume 2, Issue 2 (2015)},
  pages =	{01:1--01:30},
  journal =	{Leibniz Transactions on Embedded Systems},
  ISSN =	{2199-2002},
  year =	{2015},
  volume =	{2},
  number =	{2},
  editor =	{Carle, Thomas and Potop-Butucaru, Dumitru and Sorel, Yves and Lesens, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES-v002-i002-a001},
  doi =		{10.4230/LITES-v002-i002-a001},
  annote =	{Keywords: Time-triggered, Off-line real-time scheduling, Temporal partitioning}
}
Document
DOI: 10.4230/LIPIcs.STACS.2012.266
Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices

Authors: Ioannis Koutis, Alex Levin, and Richard Peng

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Abstract
We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/epsilon^2) edges that provides a strong approximation of G. Namely, for all vectors x and any epsilon>0, we have (1-epsilon) x^T L_G x <= x^T L_H x <= (1+epsilon) x^T L_G x, where L_G and L_H are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in tilde{O}(m log^2 n) time, an O(log n) factor faster than before. The second algorithm runs in tilde{O}(m log n) time and generates a sparsifier with tilde{O}(n log^3 n) edges. The third algorithm applies to graphs where m>n log^5 n and runs in tilde{O}(m log_{m/ n log^5 n} n time. In the range where m>n^{1+r} for some constant r this becomes softO(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.
Cite as

Ioannis Koutis, Alex Levin, and Richard Peng. Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 266-277, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{koutis_et_al:LIPIcs.STACS.2012.266,
  author =	{Koutis, Ioannis and Levin, Alex and Peng, Richard},
  title =	{{Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{266--277},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.266},
  URN =		{urn:nbn:de:0030-drops-34348},
  doi =		{10.4230/LIPIcs.STACS.2012.266},
  annote =	{Keywords: Spectral sparsification, linear system solving}
}
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