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

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.

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} }

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

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.

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} }

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

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.

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} }

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

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.

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} }

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

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.

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} }