Distributed Balanced Partitioning via Linear Embedding

Abstract

Balanced partitioning is often a crucial first step in solving large-scale graph optimization problems, e.g., in some cases, a big graph can be chopped into pieces that fit on one machine to be processed independently before stitching the results together. In other cases, links between different parts may show up in the running time and/or network communications cost. We study a distributed balanced partitioning problem where the goal is to partition the vertices of a given graph into k pieces so as to minimize the total cut size. Our algorithm is composed of a few steps that are easily implementable in distributed computation frameworks. The algorithm first embeds nodes of the graph onto a line, and then processes nodes in a distributed manner guided by the linear embedding order. We examine various ways to find the first embedding, e.g., via a hierarchical clustering or Hilbert curves. Then we apply four different techniques including local swaps, minimum cuts on the boundaries of partitions, as well as contraction and dynamic programming. As our empirical study, we compare the above techniques with each other, and also to previous work in distributed graph algorithms, e.g., a label propagation method, FENNEL and Spinner. We report our results both on a private map graph and several public social networks, and show that our results beat previous distributed algorithms: e.g., compared to the label propagation algorithm, we report an improvement of 15-25% in the cut value. We also observe that our algorithms allow for scalable distributed implementation for any number of partitions. Finally, we apply our techniques for the Google Maps Driving Directions to minimize the number of multi-shard queries with the goal of saving in CPU usage. During live experiments, we observe an ~40% drop in the number of multi-shard queries when comparing our method with a standard geography-based method.