For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected dregular graph with eO( 1 1−λ √ n) runtime per query, where λ is the secondlargest eigenvalue of the random walk matrix of the graph in absolute value. Since random dregular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input dregular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lowermore »
Triangledegrees in graphs and tetrahedron coverings in 3graphs
Abstract We investigate a covering problem in 3uniform hypergraphs (3graphs): Given a 3graph F , what is c 1 ( n , F ), the least integer d such that if G is an n vertex 3graph with minimum vertexdegree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G ? We asymptotically determine c 1 ( n , F ) when F is the generalized triangle $K_4^{(3)}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.
 Award ID(s):
 1700622
 Publication Date:
 NSFPAR ID:
 10287722
 Journal Name:
 Combinatorics, Probability and Computing
 Volume:
 30
 Issue:
 2
 Page Range or eLocationID:
 175 to 199
 ISSN:
 09635483
 Sponsoring Org:
 National Science Foundation
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