Distributed Scheduling in Finite Capacity Networks

In this paper, we show that a simple distributed algorithm for network scheduling in arbitrary m processor networks with unit capacity links is an O(logm)-approximation algorithm if the optimal schedule length is su ciently large. We will assume that there are m machines or processors labeled p1; p2; : : : ; pm, such that processor pi has ji jobs and P i ji = n. Let d be the maximum degree in the network and let L be the length of the optimal schedule. We also assume that each processor knows the current number of jobs on its neighboring processors. In one step of the algorithm, which we will refer to as Eager-scheduler, a processor with ji jobs will pass one job to each of its neighbors with less than ji 2d jobs. A precise statement of Eager-scheduler appears in Figure 1. This algorithm was analyzed for the load balancing problem by Aiello et al. [1] and Ghosh and Muthukrishnan [2]. The di erence between the load balancing problem and network scheduling is that in the load balancing problem there is no overlap of communication and computation. The objective is to balance (either perfectly or approximately, depending on the variation of the problem) the number of jobs on each machine. The bounds of [1, 2] are not in terms of an actual lower bound but instead an existential lower bound. They can prove that their algorithm balances the number of jobs on each processor so that no two neighbors are more than d apart. Let be the average number of jobs in a network with edge expansion . Their algorithm will perform this approximate balancing in O( = ) steps where = maxi(ji ) and is referred to as the imbalance in the network. Further, they show that there exists a network with imbalance and edge expansion that requires O( = ) steps to balance. Thus, their algorithm is not an approximation algorithm by the usual de nition since the lower bound is not necessarily for the instance at hand but instead an existential lower bound. While their algorithms may produce a short schedule by load balancing, our algorithm gives a guarantee that the schedule produced is no more than an O(logm) factor longer than the shortest possible schedule for the instance given.