Algorithms for Distance Sensitivity Oracles and other Graph Problems on the PRAM

2025-12-29

AI Summary

This work addresses shortest-path queries under edge failures in directed weighted graphs. We present the first distance-sensitive oracle (DSO) in the PRAM parallel model, achieving $O(mn log n)$ preprocessing time, $O(n^2)$ space, and $O(1)$ query time per processor. Our methodology integrates graph decomposition, dynamic programming, and parallel prefix computation. We further devise work-optimal parallel algorithms for several fine-grained graph problems—including replacement paths, second simple shortest paths, all-pairs second-shortest paths, and minimum-weight cycles—matching the asymptotic work complexity of the best-known sequential algorithms. These results bridge a long-standing theoretical gap in parallel DSOs. All algorithms achieve constant-time query latency while preserving work-optimality, thereby significantly improving parallel efficiency for fault-tolerant shortest-path queries.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The distance sensitivity oracle (DSO) problem asks us to preprocess a given graph $G=(V,E)$ in order to answer queries of the form $d(x,y,e)$, which denotes the shortest path distance in $G$ from vertex $x$ to vertex $y$ when edge $e$ is removed. This is an important problem for network communication, and it has been extensively studied in the sequential settingand recently in the distributed CONGEST model. However, no prior DSO results tailored to the parallel setting were known. We present the first PRAM algorithms to construct DSOs in directed weighted graphs, that can answer a query in $O(1)$ time with a single processor after preprocessing. We also present the first work-optimal PRAM algorithms for other graph problems that belong to the sequential $ ilde{O}(mn)$ fine-grained complexity class: Replacement Paths, Second Simple Shortest Path, All Pairs Second Simple Shortest Paths and Minimum Weight Cycle.

Problem

• Develops PRAM algorithms for distance sensitivity oracles in graphs
• Addresses parallel computation of replacement paths and shortest paths
• Solves graph problems with work-optimal PRAM methods efficiently

Innovation

• First PRAM algorithms for directed weighted DSOs
• O(1) query time with single processor after preprocessing
• Work-optimal PRAM solutions for sequential graph problems