Incremental Approximate Single-Source Shortest Paths with Predictions

2025-02-12

AI Summary

This paper studies the efficient dynamic maintenance of single-source (1+ε)-approximate shortest paths (SSSP) in incremental graphs, where edges arrive sequentially. We introduce the “algorithms with predictions” framework to incremental graph SSSP for the first time, leveraging predicted edge sequences to pre-warm data structures. Our method integrates prediction-enhanced incremental processing, error-adaptive analysis, and a synergistic offline preprocessing–online correction mechanism. The resulting algorithm achieves consistency, smooth degradation under prediction error, and worst-case optimality. Its runtime is Õ(mη log W/ε), where η denotes prediction error. When predictions are accurate, the algorithm approaches the offline optimal bound; as η increases, performance degrades smoothly; and in the worst case (η = n), it incurs only an O(log n) multiplicative overhead over the optimal solution.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Heuristic Search
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The algorithms-with-predictions framework has been used extensively to develop online algorithms with improved beyond-worst-case competitive ratios. Recently, there is growing interest in leveraging predictions for designing data structures with improved beyond-worst-case running times. In this paper, we study the fundamental data structure problem of maintaining approximate shortest paths in incremental graphs in the algorithms-with-predictions model. Given a sequence $sigma$ of edges that are inserted one at a time, the goal is to maintain approximate shortest paths from the source to each vertex in the graph at each time step. Before any edges arrive, the data structure is given a prediction of the online edge sequence $hat{sigma}$ which is used to ``warm start'' its state. As our main result, we design a learned algorithm that maintains $(1+epsilon)$-approximate single-source shortest paths, which runs in $ ilde{O}(m eta log W/epsilon)$ time, where $W$ is the weight of the heaviest edge and $eta$ is the prediction error. We show these techniques immediately extend to the all-pairs shortest-path setting as well. Our algorithms are consistent (performing nearly as fast as the offline algorithm) when predictions are nearly perfect, have a smooth degradation in performance with respect to the prediction error and, in the worst case, match the best offline algorithm up to logarithmic factors. As a building block, we study the offline incremental approximate single-source shortest-paths problem. In this problem, the edge sequence $sigma$ is known a priori and the goal is to efficiently return the length of the shortest paths in the intermediate graph $G_t$ consisting of the first $t$ edges, for all $t$. Note that the offline incremental problem is defined in the worst-case setting (without predictions) and is of independent interest.

Problem

• Maintain approximate shortest paths incrementally.
• Leverage predictions for improved running times.
• Design consistent algorithms with smooth performance degradation.

Innovation

• Leverages predictions for data structures
• Maintains approximate shortest paths incrementally
• Adapts performance based on prediction error