Fully Dynamic Shortest Paths in Sparse Digraphs

TL;DR

This work delivers the first non-trivial, deterministic fully dynamic shortest-path data structure for sparse weighted directed graphs, achieving worst-case update time $\tilde{O}(m n^{4/5})$ and query time $\tilde{O}(n^{4/5})$ under the no-negative-cycle assumption, with path reporting in time proportional to path length. The method fuses phase-based preprocessing, a hitting-set framework, and a Dijkstra-like recomputation for short-hop paths, complemented by a deterministic derandomization strategy and space-reduction techniques. It also addresses negative weights and cycles and offers a DAG-specific algebraic approach for fully dynamic reachability. Collectively, these ideas close a gap between dense algebraic results and sparse combinatorial methods, providing robust, worst-case guarantees for dynamic shortest paths in practical sparse graphs with potential negative weights. The work further refines the landscape by clarifying prior errors and outlining deterministic alternatives, extending applicability to dynamic reachability in sparse DAGs as well.

Abstract

We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with $\tilde{O}(mn^{4/5})$ worst-case update time processing arbitrary $s,t$-distance queries in $\tilde{O}(n^{4/5})$ time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Remark 3.4
• Lemma 3.5
• proof