Vertex Ordering Problems in Directed Graph Streams

TL;DR

This paper investigates the complexity of vertex ordering problems in directed graph streams, revealing high space requirements for adversarial streams and proposing efficient algorithms for random streams, especially in tournaments.

Contribution

It provides new lower bounds for directed graph problems in streaming models and introduces sublinear algorithms for specific cases like tournaments and random graphs.

Findings

High space complexity for adversarial streams in many problems.

Random ordering can significantly reduce space requirements.

Sublinear algorithms are effective for tournaments and random graphs.

Abstract

We consider directed graph algorithms in a streaming setting, focusing on problems concerning orderings of the vertices. This includes such fundamental problems as topological sorting and acyclicity testing. We also study the related problems of finding a minimum feedback arc set (edges whose removal yields an acyclic graph), and finding a sink vertex. We are interested in both adversarially-ordered and randomly-ordered streams. For arbitrary input graphs with edges ordered adversarially, we show that most of these problems have high space complexity, precluding sublinear-space solutions. Some lower bounds also apply when the stream is randomly ordered: e.g., in our most technical result we show that testing acyclicity in the $p$-pass random-order model requires roughly $n^{1+1/p}$ space. For other problems, random ordering can make a dramatic difference: e.g., it is possible to find a sink in an acyclic tournament in the one-pass random-order model using polylog$(n)$ space whereas under adversarial ordering roughly $n^{1/p}$ space is necessary and sufficient given $\Theta(p)$ passes. We also design sublinear algorithms for the feedback arc set problem in tournament graphs; for random graphs; and for randomly ordered streams. In some cases, we give lower bounds establishing that our algorithms are essentially space-optimal. Together, our results complement the much maturer body of work on algorithms for undirected graph streams.