Constructing Long Paths in Graph Streams

TL;DR

This paper investigates the complexity of constructing long paths in graph streams, providing algorithms and lower bounds for approximating the longest path in both undirected and directed graphs under streaming constraints.

Contribution

It introduces semi-streaming algorithms for undirected graphs achieving certain approximation ratios and establishes space lower bounds, highlighting fundamental differences between directed and undirected graph streaming.

Findings

Semi-streaming algorithms achieve a path length of at least d/3 in undirected graphs.

No similar approximation is possible for directed graphs with subquadratic space.

Lower bounds show space requirements grow with inverse powers of the approximation factor.

Abstract

In the graph stream model of computation, an algorithm processes the edges of an input graph in one or more sequential passes while using a memory sublinear in the input size. This model poses significant challenges for constructing long paths. Many known algorithms tasked with extending an existing path as a subroutine require an entire pass to add a single additional edge. This raises a fundamental question: Are multiple passes inherently necessary to construct paths of non-trivial lengths, or can a single pass suffice? To address this question, we study the Longest Path problem in the one-pass streaming model. In this problem, given a desired approximation factor $\alpha$, the objective is to compute a path of length at least $\lp(G) / \alpha$, where $\lp(G)$ is the length of a longest path in the input graph. We give algorithms as well as space lower bounds for both undirected and directed graphs. Our results include: We show that for undirected graphs, in both the insertion-only and the insertion-deletion models, there are semi-streaming algorithms, that compute a path of length at least $d /3$ with high probability, where $d$ is the average degree of the graph. These algorithms can also yield an $\alpha$-approximation to Longest Path using space $\tilde{O}(n^2 / \alpha)$. Next, we show that such a result cannot be achieved for directed graphs, even in the insertion-only model. We show that computing a $(n^{1 - o(1)})$-approximation to Longest Path in directed graphs in the insertion-only model requires space $\Omega(n^2)$. We further show two additional lower bounds. First, we show that semi-streaming space is insufficient for small constant factor approximations to Longest Path for undirected graphs in the insertion-only model. Last, in undirected graphs in the insertion-deletion model, we show that computing an $\alpha$-approximation requires space $\Omega(n^2 / \alpha^3)$.