Near-Optimal Algorithms for Reachability, Strongly-Connected Components and Shortest Paths in Partially Dynamic Digraphs

TL;DR

This paper introduces near-optimal randomized and deterministic algorithms for dynamic graph problems like reachability, strongly-connected components, and shortest paths, significantly improving efficiency in partially dynamic directed graphs.

Contribution

It presents the first near-optimal randomized data structures for SSR, SCCs, and SSSP in partially dynamic graphs, and deterministic improvements over longstanding barriers.

Findings

Achieved near-linear total update time for SSR and SCCs in edge-deletion scenarios.

Developed near-optimal total update time algorithms for SSSP in dense graphs.

Provided deterministic algorithms surpassing classical $O(mn)$ bounds.

Abstract

In this thesis, we present new techniques to deal with fundamental algorithmic graph problems where graphs are directed and partially dynamic, i.e. undergo either a sequence of edge insertions or deletions: - Single-Source Reachability (SSR), - Strongly-Connected Components (SCCs), and - Single-Source Shortest Paths (SSSP). These problems have recently received an extraordinary amount of attention due to their role as subproblems in various more complex and notoriously hard graph problems, especially to compute flows, bipartite matchings and cuts. Our techniques lead to the first near-optimal data structures for these problems in various different settings. Letting $n$ denote the number of vertices in the graph and by $m$ the maximum number of edges in any version of the graph, we obtain - the first randomized data structure to maintain SSR and SCCs in near-optimal total update time $\tilde{O}(m)$ in a graph undergoing edge deletions. - the first randomized data structure to maintain SSSP in partially dynamic graphs in total update time $\tilde{O}(n^2)$ which is near-optimal in dense graphs. - the first deterministic data structures for SSR and SCC for graphs undergoing edge deletions, and for SSSP in partially dynamic graphs that improve upon the $O(mn)$ total update time by Even and Shiloach from 1981 that is often considered to be a fundamental barrier.