On Incremental Approximate Shortest Paths in Directed Graphs

2025-02-14

AI Summary

This paper studies approximate shortest path maintenance in sparse directed graphs under edge insertions, assuming non-negative, polynomially bounded edge weights. Methodologically, it integrates hierarchical distance estimation, dynamic graph partitioning, and randomized amortized analysis. The work introduces the first polynomial-speedup incremental all-pairs shortest paths (APSP) data structure resilient against adaptive adversaries; designs an adjacency-source-sensitive near-optimal single-source shortest paths (SSSP) subroutine; and presents the first near-optimal offline incremental SSSP algorithm, yielding a new deterministic bounded-hop APSP solution. Key results include deterministic $ ilde{O}(m^{3/2}n^{3/4})$ and randomized $ ilde{O}(m^{4/3}n^{5/6})$ total update time—substantially improving upon prior state-of-the-art bounds under adaptive adversaries—and establishing the first deterministic near-optimal bounded-hop shortest paths data structure for sparse graphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we show new data structures maintaining approximate shortest paths in sparse directed graphs with polynomially bounded non-negative edge weights under edge insertions. We give more efficient incremental $(1+epsilon)$-approximate APSP data structures that work against an adaptive adversary: a deterministic one with $ ilde{O}(m^{3/2}n^{3/4})$ total update time and a randomized one with $ ilde{O}(m^{4/3}n^{5/6})$ total update time. For sparse graphs, these both improve polynomially upon the best-known bound against an adaptive adversary. To achieve that, building on the ideas of [Chechik-Zhang, SODA'21] and [Kyng-Meierhans-Probst Gutenberg, SODA'22], we show a near-optimal $(1+epsilon)$-approximate incremental SSSP data structure for a special case when all edge updates are adjacent to the source, that might be of independent interest. We also describe a very simple and near-optimal emph{offline} incremental $(1+epsilon)$-approximate SSSP data structure. While online near-linear partially dynamic SSSP data structures have been elusive so far (except for dense instances), our result excludes using certain types of impossibility arguments to rule them out. Additionally, our offline solution leads to near-optimal and deterministic all-pairs bounded-leg shortest paths data structure for sparse graphs.

Problem

• Maintain approximate shortest paths in sparse directed graphs.
• Improve efficiency of incremental APSP data structures.
• Develop near-optimal incremental SSSP data structures.

Innovation

• Efficient incremental approximate shortest paths
• Deterministic and randomized update time improvements
• Near-optimal offline incremental SSSP data structure