Maintaining connected components for infinite graph streams

Abstract

We present an algorithm to maintain the connected components of a graph that arrives as an infinite stream of edges. We formalize the algorithm on X-stream, a new parallel theoretical computational model for infinite streams. Connectivity-related queries, including component spanning trees, are supported with some latency, returning the state of the graph at the time of the query. Because an infinite stream may eventually exceed the storage limits of any number of finite-memory processors, we assume an aging command or daemon where "uninteresting" edges are removed when the system nears capacity. Following an aging command the system will block queries until its data structures are repaired, but edges will continue to be accepted from the stream, never dropped. The algorithm will not fail unless a model-specific constant fraction of the aggregate memory across all processors is full. In normal operation, it will not fail unless aggregate memory is completely full. Unlike previous theoretical streaming models designed for finite graphs that assume a single shared memory machine or require arbitrary-size intemediate files, X-stream distributes a graph over a ring network of finite-memory processors. Though the model is synchronous and reminiscent of systolic algorithms, our implementation uses an asynchronous message-passing system. We argue the correctness of our X-stream connected components algorithm, and give preliminary experimental results on synthetic and real graph streams.