An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs

TL;DR

This paper introduces a randomized algorithm that efficiently maintains cycle detection and topological orderings in a dynamically growing directed graph, significantly improving update times for sparse graphs.

Contribution

It presents a novel randomized algorithm achieving $ ilde{O}(m^{4/3})$ total expected update time for incremental cycle detection and topological ordering.

Findings

Achieves $ ilde{O}(m^{4/3})$ total expected update time

Efficiently maintains topological orderings in sparse graphs

Provides probabilistic guarantees for incremental updates

Abstract

We consider the problem of incremental cycle detection and topological ordering in a directed graph $G = (V, E)$ with $|V| = n$ nodes. In this setting, initially the edge-set $E$ of the graph is empty. Subsequently, at each time-step an edge gets inserted into $G$. After every edge-insertion, we have to report if the current graph contains a cycle, and as long as the graph remains acyclic, we have to maintain a topological ordering of the node-set $V$. Let $m$ be the total number of edges that get inserted into $G$. We present a randomized algorithm for this problem with $\tilde{O}(m^{4/3})$ total expected update time.