An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs
Sayan Bhattacharya, Janardhan Kulkarni

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 with nodes. In this setting, initially the edge-set of the graph is empty. Subsequently, at each time-step an edge gets inserted into . 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 . Let be the total number of edges that get inserted into . We present a randomized algorithm for this problem with total expected update time.
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