Expansion and the cover time of parallel random walks

Abstract

We study the cover time of parallel random walks which was recently introduced by Alon et al. [2]. We consider k parallel (independent) random walks starting from arbitrary vertices. The expected number of steps until these k walks have visited all n vertices is called cover time of G. In this paper we present a lower bound on the cover time of Ω( √n/k • √(1/Φ(G))}, where Φ(G) is the geometric expansion (a.k.a. as edge expansion or conductance). This bound is matched for any 1 ≤ k ≤ n by binary trees up to logarithmic factors. Our lower bound combined with previous results also implies a new characterization of expanders. Roughly speaking, the edge expansion Φ(G) satisfies 1/Φ(G) = O(polylog(n)) if and only if G has a cover time of O(n/k • polylog (n)) for all 1 ≤ k ≤ n. We also present new upper bounds on the cover time with sublinear dependence on the (algebraic) expansion.