An Algorithm for the 2-Median Problem on Two-Dimensional Meshes

Abstract

The mesh (and its variant, the torus) is a popular topology for processor interconnection in parallel computers. It has practical advantages such as low degree and perfectly compact layout when compared to other well-known topologies, for example the hypercube. A notable example of parallel computers based on the mesh topology is the iWarp system [1]. Dally has shown that low-dimensional networks have lower latency and higher hot-spot throughput than high-dimensional networks [2]. In this paper, we study the problem of finding a 2-median set in a two-dimensional mesh. The p-median problem is a well-known problem in location theory, which is to locate p identical facilities in a network so that the sum of the distance between every client in the network and its nearest facility is minimized. This has practical application in real mesh-connected parallel computers, which is to locate servers in selected nodes so that service requests from client nodes can be directed to different servers to achieve load balancing, and the traffic so generated can be more spread out over the links. The problem has been proved to be NP-hard for general networks. For a survey on the NP-hardness and algorithms developed in recent years, one can refer to [3]. Mirchandani [4] gives a strong mathematical background for the problem, which also gathers many of the results before 1989. An early survey [5, 6] by Tansel et al. is still very useful, especially for beginners. Shmoys et al. [7] includes a short summary of some of the results. Recently, Guha and Khuller used the greedy approach to handle the problem [8]. Hamacher et al. dealt with a variation of the problem with multicriteria, instead of single criteria [9]. Lai and Chang proposed a