Streaming Balanced Graph Partitioning for Random Graphs

There has been a recent explosion in the size of stored data, partially due to advances in storage technology, and partially due to the growing popularity of cloud-computing and the vast quantities of data generated. This motivates the need for streaming algorithms that can compute approximate solutions without full random access to all of the data. We model the problem of loading a graph onto a distributed cluster as computing an approximately balanced $k$-partitioning of a graph in a streaming fashion with only one pass over the data. We give lower bounds on this problem, showing that no algorithm can obtain an $o(n)$ approximation with a random or adversarial stream ordering. We analyze two variants of a randomized greedy algorithm, one that prefers the $\arg\max$ and one that is proportional, on random graphs with embedded balanced $k$-cuts and are able to theoretically bound the performance of each algorithms - the $\arg\max$ algorithm is able to recover the embedded $k$-cut, while, surprisingly, the proportional variant can not. This matches the experimental results in [25].