# An analysis of load-balancing algorithms on edge-Markovian evolving graphs

Analysis of algorithms on time-varying networks (often called evolving graphs) is a modern challenge in theoretical computer science. The edge-Markovian is a relatively simple and comprehensive model of evolving graphs: every pair of vertices which is not a current edge independently becomes an edge with probability p at each time-step, as well as every edge disappears with probability q. Clearly, the edge-Markovian graph changes its shape depending on the current shape, and the dependency refuses some useful techniques for an independent sequence of random graphs which often behaves similarly to a static random graph. It motivates this paper to develop a new technique for analysis of algorithms on edge-Markovian evolving graphs. Specifically speaking, this paper is concerned with load-balancing, which is a popular subject in distributed computing, and we analyze the so-called random matching algorithms, which is a standard scheme for load-balancing. We prove that major random matching algorithms achieve nearly optimal load balance in O(r log (Δ n)) steps on edge-Markovian evolving graphs, where r = max{p/(1-q), (1-q)/p}, n is the number of vertices (i.e., processors) and Δ denotes the initial gap of loads unbalance. We remark that the independent sequences of random graphs correspond to r=1. To avoid the difficulty of an analysis caused by a complex correlation with the history of an execution, we develop a simple proof technique based on history-independent bounds. As far as we know, this is the first theoretical analysis of load-balancing on randomly evolving graphs, not only for the edge-Markovian but also for the independent sequences of random graphs.

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