Local Access to Random Walks
Amartya Shankha Biswas, Edward Pyne, Ronitt Rubinfeld

TL;DR
This paper develops algorithms for efficiently accessing the position of a random walk on large graphs with near-uniform distribution, providing both upper and lower bounds, and explores special cases with improved performance.
Contribution
It introduces the first sublinear-time local access algorithms for random walks on regular graphs, establishes nearly tight lower bounds, and extends results to graphs with algebraic structure and product constructions.
Findings
Efficient $ ilde{O}(rac{1}{1-eta} abla ext{sqrt}(n))$ algorithms for regular graphs.
Lower bounds of $ ilde{ ext{O}}(rac{ ext{sqrt}(n)}{ ext{log}n})$ and $ ext{O}(n^{1/4})$ for general and adversarial cases.
Polylogarithmic-time local access algorithms for abelian Cayley graphs and certain product graphs.
Abstract
For a graph on vertices, naively sampling the position of a random walk of at time requires work . We desire local access algorithms supporting queries, which return the position of a random walk from some start vertex at time , where the joint distribution of returned positions is close to the uniform distribution over such walks in distance. We first give an algorithm for local access to walks on undirected regular graphs with runtime per query, where is the second-largest eigenvalue in absolute value. Since random -regular graphs are expanders with high probability, this gives an algorithm for , which improves on the naive method for small numbers of queries. We then prove that no that algorithm with…
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Videos
Local Access to Random Walks· youtube
Taxonomy
TopicsComplexity and Algorithms in Graphs · Markov Chains and Monte Carlo Methods · Optimization and Search Problems
