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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N, an integer k ≤ log n and an error parameter ε > 0, our algorithm runs in space Õ(k log(N w_max/w_min)) where w_max and w_min are the maximum and minimum edge weights in G, and produces a weighted graph H with Õ(n^(1+2/k)/ε²) edges that spectrally approximates G, in the sense of Spielmen and Teng [Spielman and Teng, 2004], up to an error of ε.
Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava’s effective resistance based edge sampling algorithm [Spielman and Srivastava, 2011] and uses results from recent work on space-bounded Laplacian solvers [Jack Murtagh et al., 2017]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by k above, and the resulting sparsity that can be achieved.

Dean Doron, Jack Murtagh, Salil Vadhan, and David Zuckerman. Spectral Sparsification via Bounded-Independence Sampling. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{doron_et_al:LIPIcs.ICALP.2020.39, author = {Doron, Dean and Murtagh, Jack and Vadhan, Salil and Zuckerman, David}, title = {{Spectral Sparsification via Bounded-Independence Sampling}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {39:1--39:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.39}, URN = {urn:nbn:de:0030-drops-124462}, doi = {10.4230/LIPIcs.ICALP.2020.39}, annote = {Keywords: Spectral sparsification, Derandomization, Space complexity} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk.
Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size.

Jack Murtagh, Omer Reingold, Aaron Sidford, and Salil Vadhan. Deterministic Approximation of Random Walks in Small Space. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 42:1-42:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{murtagh_et_al:LIPIcs.APPROX-RANDOM.2019.42, author = {Murtagh, Jack and Reingold, Omer and Sidford, Aaron and Vadhan, Salil}, title = {{Deterministic Approximation of Random Walks in Small Space}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {42:1--42:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.42}, URN = {urn:nbn:de:0030-drops-112577}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.42}, annote = {Keywords: random walks, space complexity, derandomization, spectral approximation, expander graphs} }

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