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

We present improved bounds on the cover time of the coalescing-branching 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 \ge 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-\lambda)) \log n) for r-regular connected graphs with the second eigenvalue \lambda. Our bounds improve the O(n^{11/4}\log n) and ((r^4/\phi^2)\log^2 n) bounds shown in [Mitzenmacher et al., SPAA 2016], where \phi is the conductance of the graph, and complement the O((1/(1-\l))^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.