Expanders via Random Spanning Trees

Abstract

Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree nvertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph G n,p , for p = Ω(log n/n), we give a randomized algorithm for constructing two spanning trees whose union is an expander. This is suggested by the case of the complete graph, where we prove that two random spanning trees give an expander. The construction of the splicer is elementary; each spanning tree can be produced independently using an algorithm by Aldous and Broder: A random walk in the graph with edges leading to previously unvisited vertices included in the tree. Splicers also turn out to have applications to graph cut-sparsification where the goal is to approximate every cut using only a small subgraph of the original graph. For random graphs, splicers provide simple algorithms for sparsifiers of size O(n) that approximate every cut to within a factor of O(log n).

Twice-Ramanujan Sparsifiers

“…We resolve this conjecture. Very recently, partial progress was made towards the conjecture by Goyal, Rademacher and Vempala [GRV08], who show how to find graphs H with only 2n edges that O(log n)-approximate bounded degree graphs G under the cut notion of Benczur and Karger.…”

Fast distributed random walks

“…Broder [9] and Wilson [34] gave algorithms to generate random spanning trees using random walks and Broder's algorithm was later applied to the network setting by BarIlan and Zernik [6]. Recently Goyal et al [19] show how to construct an expander/sparsifier using random spanning trees. If their algorithm is implemented on a distributed network, the techniques presented in this paper would yield an additional speed-up in the random walk constructions.…”

Efficient random walk sampling in distributed networks

“…Broder [13] and Wilson [45] gave algorithms to generate random spanning trees using random walks and Broder's algorithm was later applied to the network setting by Bar-Ilan and Zernik [8]. Recently Goyal et al [30] show how to construct an expander/sparsifier using random spanning trees. The approach of Broder and Wilson's algorithm is essentially performing a random walk from a specified node (root node) until all nodes are visited.…”

“…We note that the work of Bar-Ilan and Zernik constructed random spanning trees only in rings and complete graphs. The improved random spanning tree algorithm can be useful in implementing Goyal et al's [30] approach in a distributed way. A variety of such applications can greatly benefit using the random walk sampling techniques presented in this paper.…”