The Zig-Zag Product and Reingold's Theorem Anurag Sahay University of Rochester [email protected] 27th February, 2020 Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 1 / 20 Overview of the Talk 1 Introduction Reingold's Theorem: ustcon 2 L 2 Expander Graphs Diameter of an Expander Graph 3 Initial Ideas Graph Exponentiation Tensor Product 4 The Zig-Zag Product 5 Reingold's Algorithm Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 2 / 20 References Omer Reingold. Undirected connectivity in log-space. Journal of the ACM (JACM), 55(4):1{24, 2008. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of mathematics, 155(1):157{187, 2002. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439{561, 2006. Luca Trevisan. Lecture Notes { U.C. Berkeley, CS278: Computational Complexity, 2004. https://people.eecs.berkeley.edu/~luca/cs278-04/notes/. Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 3 / 20 Definition (LogSpace) L = DSPACE(log n) In other words, L contains all languages L ⊆ f0; 1g∗ for which a Turing machine M decides L using no more that O(log n) space on input of length n. ustcon and L Recall the definition of ustcon and L: Definition (Undirected st-connectivity) ustcon is the following decision problem: Input: an undirected graph G = (V ; E), and two vertices s; t 2 V Output: 1 if s t in G, and 0 otherwise. Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 4 / 20 ustcon and L Recall the definition of ustcon and L: Definition (Undirected st-connectivity) ustcon is the following decision problem: Input: an undirected graph G = (V ; E), and two vertices s; t 2 V Output: 1 if s t in G, and 0 otherwise. Definition (LogSpace) L = DSPACE(log n) In other words, L contains all languages L ⊆ f0; 1g∗ for which a Turing machine M decides L using no more that O(log n) space on input of length n. Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 4 / 20 Reingold's Theorem The goal of this talk is to describe the proof of Reingold's theorem: Theorem (Reingold, 2005) ustcon 2 L Note that the graph is not assumed to be simple (multiedges and loops are both allowed). We will restrict ourselves largely to regular graphs, and then show how the problem in a general graph can be reduced to the regular case. Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 5 / 20 The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 Every set of vertices has many neighbours Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Every cut has many edges across it Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 The morality of expansion In different contexts, an expander graph (or more accurately, an expander graph family) is defined morally in one of the following ways: The graph is a sparse approximation for the complete graph The graph satisfies a strong isoperimetric inequality Every set of vertices has many neighbours Every cut has many edges across it A (uniform) random walk on on the graph quickly converges to the stationary distribution The graph has a large spectral gap Loosely speaking, under the right assumptions on the graphs, all of the above notions are equivalent. Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 6 / 20 Note here that λ1 = d, and that the maximum on the right is 1 if and only if G is either disconnected or bipartite. Definition (Expander Graphs) A family of d-regular graphs fGj gj is called an expander family with spectral gap 1 − λ (or a λ-expander family), if every Gj is an (n; d; λ)-graph for some n = nj . Definition of Expander Graphs We adopt the spectral point of view, restricted to regular graphs: Definition ((n; d; λ)-graph) We say that G is an (n; d; λ)-graph if G is a d-regular graph on n vertices such that jλ j λ ≥ max i i6=1 d where λ1 ≥ λ2 ≥ · · · ≥ λn is the spectrum of the adjacency matrix A of G. Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 7 / 20 Definition of Expander Graphs We adopt the spectral point of view, restricted to regular graphs: Definition ((n; d; λ)-graph) We say that G is an (n; d; λ)-graph if G is a d-regular graph on n vertices such that jλ j λ ≥ max i i6=1 d where λ1 ≥ λ2 ≥ · · · ≥ λn is the spectrum of the adjacency matrix A of G. Note here that λ1 = d, and that the maximum on the right is 1 if and only if G is either disconnected or bipartite. Definition (Expander Graphs) A family of d-regular graphs fGj gj is called an expander family with spectral gap 1 − λ (or a λ-expander family), if every Gj is an (n; d; λ)-graph for some n = nj . Anurag Sahay (UoR) ustcon 2 L 27th February, 2020 7 / 20 The Punchline: Expanders have Logarithmic Diameter The usefulness of expanders in solving ustcon comes from the following lemma about their diameters: Lemma (Diameter of an Expander) Let G = (V ; E) be a connected (n; d; λ)-graph. Then, log n diam(G) = max d(u; v) = Oλ(log n) = O u;v 1 − λ where the implicit constant is effective, and efficiently computable.