Improved Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

Abstract

We present improved bounds on the cover time of the coalescingbranching random walk process COBRA. The COBRA process, introduced in [Dutta et al., SPAA 2013], can be viewed as spreading a single item of information throughout an undirected graph in synchronised rounds. In each round, each vertex which has received the information in the previous round (possibly simultaneously from more than one neighbour and possibly not for the first time), 'pushes' the information to b randomly selected neighbours. The COBRA process is typically studied for integer branching rates b ≥ 2 (with the case b = 1 corresponding to a random walk). The aim of the process is to propagate the information quickly, but with a limited number of transmissions per vertex per round.The cover time of COBRA is defined as the expected number of rounds until each vertex has received the information at least once. Our main results are a bound of O(m + (d max ) 2 log n) = O(n 2 log n) on the COBRA cover time for an arbitrary connected graph with n vertices, m edges and the maximum vertex degree d max , and a bound of O((r 2 +r /(1−λ)) log n) for r -regular connected graphs with the second eigenvalue λ. Our bounds improve the O(n 11/4 log n) and O((r 4 /ϕ 2 ) log 2 n) bounds shown in [Mitzenmacher et al., SPAA 2016], where ϕ is the conductance of the graph, and complement the O((1/(1 − λ)) 3 log n) bound shown in [Cooper et al., PODC 2016]. We obtain our bounds by analysing the process called Biased Infection with Persistent Source (BIPS), which was introduced in [Cooper et al., PODC 2016] as a dual process for COBRA .

Tight Bounds for Coalescing-Branching Random Walks on Regular Graphs

“…This result improves the previous bounds of O n 2−1/d log n for d-regular graphs [27], and the recent O n 2 log n bound which however applies also to non-regular graphs [9]. Our bound is tight, as demonstrated by the example of an n-node graph obtained by two cliques of size n/2, by removing one of the edges in each clique, and adding two new edges between the cliques such that the graph remains regular.…”

“…Cooper et al [8] showed that the cover time on d-regular graphs is bounded by O (1 − λ) −3 · log n w.h.p., where 1 − λ is the absolute spectral gap of the graph's random walk matrix. Very recently, Cooper et al [9] complemented this result by a…”

“…bound in [27], the O (1 − λ) −3 · log n bound in [8], [9]. Note that φ is an upper bound on 1 − λ, but the latter can be much smaller than φ.…”

“…for arbitrary graphs with m edges and maximum degree ∆. Both [8,9] base their analysis on the duality between the k-CoBra process and the k-Bips process, while earlier papers use techniques from the analysis of parallel random walks.…”

Better Bounds for Coalescing-Branching Random Walks

“…The bound achieved by Reference [12] for expander graphs is optimal; for general graphs, our bounds and theirs are incomparable. In more recent results obtained subsequent to this article, Cooper et al [13] have obtained an improved O (n 2 log n) bound for cobra walks for general graphs and an improved bound for regular graphs in terms of the second eigenvalue of the graph.…”