Slow Links, Fast Links, and the Cost of Gossip

Abstract

Consider the classical problem of information dissemination: one (or more) nodes in a network have some information that they want to distribute to the remainder of the network. In this paper, we study the cost of information dissemination in networks where edges have latencies, i.e., sending a message from one node to another takes some amount of time. We first generalize the idea of conductance to weighted graphs by defining $\\phi _*$φ* to be the “critical weighted conductance” and $\\ell _*$ℓ* to be the “critical latency”. One goal of this paper is to argue that $\\phi _*$φ* characterizes the connectivity of a weighted graph with latencies in much the same way that conductance characterizes the connectivity of unweighted graphs. We give near tight lower and upper bounds on the problem of information dissemination, up to polylogarithmic factors. Specifically, we show that in a graph with (weighted) diameter $D$D (with latencies as weights) and maximum degree $\\Delta$Δ, any information dissemination algorithm requires at least $\\Omega (\\min (D+\\Delta, \\ell _*/\\phi _*))$Ω(min(D+Δ,ℓ*/φ*)) time in the worst case. We show several variants of the lower bound (e.g., for graphs with small diameter, graphs with small max-degree, etc.) by reduction to a simple combinatorial game. We then give nearly matching algorithms, showing that information dissemination can be solved in $O(\\min ((D+\\Delta)\\log ^3{n}, (\\ell _*/\\phi _*)\\log n)$O(min((D+Δ)log3n,(ℓ*/φ*)logn) time. This is achieved by combining two cases. We show that the classical push-pull algorithm is (near) optimal when the diameter or the maximum degree is large. For the case where the diameter and the maximum degree are small, we give an alternative strategy in which we first discover the latencies and then use an algorithm for known latencies based on a weighted spanner construction. (Our algorithms are within polylogarithmic factors of being tight both for known and unknown latencies.) While it is easiest to express our bounds in terms of $\\phi _*$φ* and $\\ell _*$ℓ*, in some cases they do not provide the most convenient definition of conductance in weighted graphs. Therefore we give a second (nearly) equivalent characterization, namely the average weighted conductance $\\phi _{avg}$φavg.