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Derandomized Squaring of Graphs [chapter]

Eyal Rozenman, Salil Vadhan
2005 Lecture Notes in Computer Science  
We introduce a "derandomized" analogue of graph squaring.  ...  This operation increases the connectivity of the graph (as measured by the second eigenvalue) almost as well as squaring the graph does, yet only increases the degree of the graph by a constant factor,  ...  Acknowledgments This work emerged from of our collaborations with Omer Reingold, Luca Trevisan, and Avi Wigderson. We are deeply grateful to them for their insights on this topic  ... 
doi:10.1007/11538462_37 fatcat:wpjiq4l2qnamvhwk57wi2phrna

Deterministic Approximation of Random Walks in Small Space

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs.  ...  Our algorithm combines the derandomized square graph operation [21] , which we recently used for solving Laplacian systems in nearly logarithmic space [16] , with ideas from [5] , which gave an algorithm  ...  Our solution is to generalize the derandomized square to produce sparse approximations to the product of distinct graphs.  ... 
doi:10.4230/lipics.approx-random.2019.42 dblp:conf/approx/MurtaghRSV19 fatcat:qbobogy745ds7b43u23fx3oj4y

Deterministic Approximation of Random Walks in Small Space [article]

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan
2019 arXiv   pre-print
Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs.  ...  Our algorithm combines the derandomized square graph operation (Rozenman and Vadhan, 2005), which we recently used for solving Laplacian systems in nearly logarithmic space (Murtagh, Reingold, Sidford,  ...  Our solution is to generalize the derandomized square to produce sparse approximations to the product of distinct graphs.  ... 
arXiv:1903.06361v2 fatcat:6mm4uijvojg5fggde4zqsmemxa

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan
2017 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)  
We give a deterministicÕ(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and  ...  escape probabilities for undirected graphs.  ...  We obtain a sequence of graphs G 0 , G 1 , . . . , G k , where G 0 = G and G i is the derandomized square of G i−1 , a graph of degree d · c i .  ... 
doi:10.1109/focs.2017.79 dblp:conf/focs/MurtaghRSV17 fatcat:hw3sxqd5wvgnljea3yrdqb54oy

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space [article]

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan
2017 arXiv   pre-print
We give a deterministic Õ( n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape  ...  probabilities for undirected graphs.  ...  Derandomized squaring of graphs.  ... 
arXiv:1708.04634v1 fatcat:mooju4ooy5ezlpnhajkoyghcy4

An Optimal Lower Bound for Monotonicity Testing over Hypergrids [chapter]

Deeparnab Chakrabarty, C. Seshadhri
2013 Lecture Notes in Computer Science  
Approximating Chromatic Number Sangxia Huang 12:14-1:45 Lunch 1:45-3:13 RANDOM (Random structures) 3:13-3:30 Break 3:30-4:36 APPROX Online Square-into-Square Packing Sandor Fekete and Hella-Franziska  ...  Problems on Planar Graphs Feng Pan and Aaron Schild Approximation Algorithms for Movement Repairmen Mohammadtaghi Hajiaghayi, Rohit Khandekar, Reza Khani and Guy Kortsarz The Approximability of the Binary  ... 
doi:10.1007/978-3-642-40328-6_30 fatcat:tdimjhbxkveu5l724kwo37lina

Derandomized graph products

Noga Alon, Uriel Feige, Avi Wigderson, David Zuckerman
1995 Computational Complexity  
We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs [1].  ...  The main technical contribution of this paper is in lower bounding the probability that all steps of a random walk stay within a specified set of vertices of a graph.  ...  Since our technique is a derandomization of the Berman and Schnitger technique, we call it derandomized graph products.  ... 
doi:10.1007/bf01277956 fatcat:xjuibrzzv5eznh5braolfndyki

Small-Bias Sets for Nonabelian Groups: Derandomizing the Alon-Roichman Theorem [article]

Sixia Chen, Cristopher Moore, Alexander Russell
2013 arXiv   pre-print
Our techniques include derandomized squaring (in both the matrix product and tensor product senses) and a Chernoff-like bound on the expected norm of the product of independently random operators that  ...  Equivalently, such sets make G's Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon^2) suffice.  ...  derandomized squaring.  ... 
arXiv:1304.5010v4 fatcat:mv5pjrun5jfv7hyrqhmcytje4m

High-precision Estimation of Random Walks in Small Space

AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, Salil Vadhan
2020 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)  
A running theme throughout these contributions is an analysis of "cycle-lifted graphs," where we take a graph and "lift" it to a new graph whose adjacency matrix is the tensor product of the original adjacency  ...  We also give improvements on the space complexity of both of these previous algorithms for non-Eulerian directed graphs when the error is negligible (ǫ = 1/N ω(1) ), generalizing what Hoza and Zuckerman  ...  Acknowledgements We thank William Hoza for pointing out an error in an earlier version of Theorem 4.8.  ... 
doi:10.1109/focs46700.2020.00123 fatcat:oydcdvq7cjg4leezgeeljnldce

High-precision Estimation of Random Walks in Small Space [article]

AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, Salil Vadhan
2022 arXiv   pre-print
A running theme throughout these contributions is an analysis of "cycle-lifted graphs", where we take a graph and "lift" it to a new graph whose adjacency matrix is the tensor product of the original adjacency  ...  We also give improvements on the space complexity of both of these previous algorithms for non-Eulerian directed graphs when the error is negligible (ϵ=1/N^ω(1)), generalizing what Hoza and Zuckerman (  ...  We thank William Hoza for pointing out an error in an earlier version of Theorem 4.8 and Ori Sberlo and Dean Doron for finding a bug in an earlier version of the proof of Theorem 5.9.  ... 
arXiv:1912.04524v3 fatcat:peiozxcjrff75c53cvzlx2daju

Derandomization of Euclidean Random Walks [chapter]

Ilia Binder, Mark Braverman
2007 Lecture Notes in Computer Science  
We consider the problem of derandomizing random walks in the Euclidean space R k .  ...  As a corollary, we show that the Dirichlet Problem can be deterministically simulated in space O(log n √ log log n), where 1/n is the desired precision of the simulation.  ...  Attempts to solve DirDP, which is at least as hard as derandomizing the class BPSPACE(S), can now be restricted to different classes of graphs.  ... 
doi:10.1007/978-3-540-74208-1_26 fatcat:p56crfbuf5dblkxsguavymlpzy

Spectral Sparsification via Bounded-Independence Sampling

Dean Doron, Jack Murtagh, Salil Vadhan, David Zuckerman, Emanuela Merelli, Artur Czumaj, Anuj Dawar
2020 International Colloquium on Automata, Languages and Programming  
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  ...  2 in G. 39:4 Spectral Sparsification via Bounded-Independence Sampling and Vadhan [41] showed that the derandomized square actually produces a spectral sparsifier of the square of a graph and this  ... 
doi:10.4230/lipics.icalp.2020.39 dblp:conf/icalp/DoronMVZ20 fatcat:i4wyqtosjzhqlpihlef3djthdm

Pseudorandom Bit Generators That Fool Modular Sums [chapter]

Shachar Lovett, Omer Reingold, Luca Trevisan, Salil Vadhan
2009 Lecture Notes in Computer Science  
The most interesting technical component of our second construction is a variant of the derandomized graph squaring operation of [RV].  ...  Our generalization handles a product of two distinct graphs with distinct bounds on their expansion.  ...  Here we study the analogue of derandomized squaring for taking products of two distinct Cayley graphs over an abelian group (namely Z M ).  ... 
doi:10.1007/978-3-642-03685-9_46 fatcat:xafdsn5p2fcadib7fd7ayrd2ey

Randomized Rounding in the Presence of a Cardinality Constraint

Benjamin Doerr, Magnus Wahlström
2015 ACM Journal of Experimental Algorithmics  
Our work results in (a) an improved version of the bitwise derandomization given by the first author, (b) the first derandomization of Srinivasan's treebased randomized approach, together with a proof  ...  When implementing the derandomized tree-based approach, however, the choice of the tree is important.  ...  Our first set of tests are square random matrices with 0/1 entries.  ... 
doi:10.1145/2594409 fatcat:d6zlzyaglrfwzgwnwu3ad5ovc4

Randomized Rounding in the Presence of a Cardinality Constraint [chapter]

Benjamin Doerr, Magnus Wahlström
2009 2009 Proceedings of the Eleventh Workshop on Algorithm Engineering and Experiments (ALENEX)  
Our work results in (a) an improved version of the bitwise derandomization given by the first author, (b) the first derandomization of Srinivasan's treebased randomized approach, together with a proof  ...  When implementing the derandomized tree-based approach, however, the choice of the tree is important.  ...  Our first set of tests are square random matrices with 0/1 entries.  ... 
doi:10.1137/1.9781611972894.16 dblp:conf/alenex/DoerrW09 fatcat:jedu4qbxive73dkpm4cibag3mm
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