Constructing Long Paths in Graph Streams

2025-08-21

AI Summary

This paper investigates the feasibility of constructing long paths in graph streams via a single edge pass, focusing on approximating the longest path: can a path of length at least $l_p(G)/alpha$ be output using sublinear space and one pass over the edge stream? For undirected graphs, we propose a semi-streaming algorithm based on average degree, achieving a path of length $geq ar{d}/3$ in $ ilde{O}(n)$ space. For directed graphs, we establish the first $Omega(n^2)$ space lower bound, proving that no sublinear-space single-pass algorithm can nontrivially approximate the longest path. We further extend this lower bound to dynamic graphs under the insert-delete model. Technically, our approach integrates probabilistic sampling, hashing, and information-theoretic analysis. This work is the first to systematically reveal the inherent necessity of multiple passes for path construction and to delineate a fundamental complexity dichotomy between undirected and directed graphs in the streaming setting.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $α$, the objective is to compute a path of length at least $lp(G) / α$, 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 $α$-approximation to Longest Path using space $ ilde{O}(n^2 / α)$. 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 $Ω(n^2)$. 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 $α$-approximation requires space $Ω(n^2 / α^3)$.

Problem

• Constructing long paths in graph streams with limited passes
• Determining if single-pass suffices for non-trivial path lengths
• Providing algorithms and lower bounds for undirected and directed graphs

Innovation

• Semi-streaming algorithms for undirected graph paths
• Directed graphs require high space for approximation
• Insertion-deletion model needs Ω(n²/α³) space