On Graph Problems in a Semi-streaming Model

Abstract

We formalize a potentially rich new streaming model, the semi-streaming model, that we believe is necessary for the fruitful study of efficient algorithms for solving problems on massive graphs whose edge sets cannot be stored in memory. In this model, the input graph, G = (V,E), is presented as a stream of edges (in adversarial order), and the storage space of an algorithm is bounded by O(n · polylog n), where n = |V |. We are particularly interested in algorithms that use only one pass over the input, but, for problems where this is provably insufficient, we also look at algorithms using constant or, in some cases, logarithmically many passes. In the course of this general study, we give semi-streaming constant approximation algorithms for the unweighted and weighted matching problems, along with a further algorithm improvement for the bipartite case. We also exhibit log n/ log log n semistreaming approximations to the diameter and the problem of computing the distance between specified vertices in a weighted graph. These are complemented by Ω(log (1−ε) n) lower bounds. Abstract. We formalize a potentially rich new streaming model, the semi-streaming model, that we believe is necessary for the fruitful study of efficient algorithms for solving problems on massive graphs whose edge sets cannot be stored in memory. In this model, the input graph, G = (V, E), is presented as a stream of edges (in adversarial order), and the storage space of an algorithm is bounded by O(n · polylog n), where n = |V |. We are particularly interested in algorithms that use only one pass over the input, but, for problems where this is provably insufficient, we also look at algorithms using constant or, in some cases, logarithmically many passes. In the course of this general study, we give semi-streaming constant approximation algorithms for the unweighted and weighted matching problems, along with a further algorithm improvement for the bipartite case. We also exhibit log n/ log log n semistreaming approximations to the diameter and the problem of computing the distance between specified vertices in a weighted graph. These are complemented by Ω(log (1− ) n) lower bounds.

Bipartite Matching in the Semi-streaming Model

“…For example, it is impossible to decide if there is a path of length 2 between two given vertices x and y [5]. This shows that even easy graph problems are not solvable in this model.…”

“…One is interested in a good approximation of the optimum, and there the number of passes can and does depend on the approximation parameter, e.g., [5,8].…”

“…Feigenbaum et al [5] showed that finding an exact solution of the maximum matching problem in one pass requires Ω(m) bits of space. Therefore, with the given memory restriction, we aim for an approximation of a maximum matching.…”

“…It is desirable to achieve an arbitrarily good, say a 1 + ε approximation, where ε > 0, called approximation parameter, can be arbitrarily small. 1 An algorithm for finding a ( 2 3 − ε) −1 approximation of a maximum matching in bipartite graphs was given by Feigenbaum et al [5]. This algorithm requires O( 1 ε log 1 ε ) passes over the input stream.…”

Near-Optimal Massively Parallel Graph Connectivity

“…Identifying the connected components of a graph is a fundamental problem that has been studied in a variety of settings (see e.g. [2,33,28,61,65,57] and the references therein). This problem is also of great practical importance [60] with a wide range of applications, e.g.…”