Derandomized Squaring of Graphs

We introduce a “derandomized ” analogue of graph squaring. This op-eration 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, instead of squaring the degree. One application of this product is an alternative proof of Reingold’s re-cent breakthrough result that S-T Connectivity in Undirected Graphs can be solved in deterministic logspace.

Pseudorandom Walks in Biregular Graphs and the RL vs. L Problem

Motivated by Reingold’s recent deterministic log-space algorithm for UNDIRECTED S-T CONNEC-TIVITY (ECCC TR 04-94), we revisit the general RL vs. L question, obtaining the following results. 1. We exhibit a new complete problem for RL: S-T CONNECTIVITY restricted to directed graphs for which the random walk is promised to have polynomial mixing time. 2. Generalizing Reingold’s techniques, we present a deterministic, log-space algorithm that given a directed graph G that is biregular (i.e., all in-degrees and out-degrees are equal) and two vertices s and t, finds a path between s and t if one exists. 3. Using the same techniques as in Item 2, we give a “pseudorandom generator” for random walks on “consistently labelled” biregular graphs. Roughly speaking, given a random seed of logarithmic length, the generator constructs, in log-space, a “short” pseudorandom walk that ends at an almostuniformly distributed vertex when taken in any consistently labelled biregular graph. 4. We prove that if our pseudorandom generator from Item 3 could be generalized to all biregular graphs (instead of just consistently labelled ones), then our complete problem from Item 1 can be solved in log-space and hence RL = L

Expander Graphs and Coding Theory

Expander graphs are highly connected sparse graphs which lie at the interface of many different fields of study. For example, they play important roles in prime sieves, cryptography, compressive sensing, metric embedding, and coding theory to name a few. This thesis focuses on the connections between sparse graphs and coding theory. It is a major challenge to explicitly construct sparse graphs with good expansion properties, for example Ramanujan graphs. Nevertheless, explicit constructions do exist, and in this thesis, we survey many of these constructions up to this point including a new construction which slightly improves on an earlier edge expansion bound. The edge expansion of a graph is crucial in applications, and it is well-known that computing the edge expansion of an arbitrary graph is NP-hard. We present a simple algo-rithm for approximating the edge expansion of a graph using linear programming techniques. While Andersen and Lang (2008) proved similar results, our analysis attacks the problem from a different vantage point and was discovered independently. The main contribution in the thesis is a new result in fast decoding for expander codes. Current algorithms in the literature can decode a constant fraction of errors in linear time but require that the underlying graphs have vertex expansion at least 1/2. We present a fast decoding algorithm that can decode a constant fraction of errors in linear time given any vertex expansion (even if it is much smaller than 1/2) by using a stronger local code, and the fraction of errors corrected almost doubles that of Viderman (2013)