Lossless Derandomization for Undirected Single-Source Shortest Paths and Approximate Distance Oracles

2025-10-14

AI Summary

This paper addresses the long-standing derandomization challenge for undirected single-source shortest paths (SSSP) and Thorup–Zwick (TZ) approximate distance oracles. We propose the first deterministic framework that avoids extraneous logarithmic factors. Our method introduces an adaptive ball-growing strategy, an improved deterministic center selection scheme, and a structured hitting set optimization—collectively overcoming the inherent trade-off between accuracy and efficiency in classical set-cover approaches. Theoretical contributions include: (1) the first deterministic SSSP algorithm running in $O(msqrt{log n log log n})$ time, improving upon all prior deterministic bounds; and (2) a fully derandomized TZ oracle achieving deterministic construction while preserving the original $O(kn^{1/k})$ space bound and $O(k)$ query time. Our approach applies to arbitrary polynomial cost functions and ensures average-case optimal control.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Reasoning under Uncertainty: Stochastic 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A common step in algorithms related to shortest paths in undirected graphs is that, we select a subset of vertices as centers, then grow a ball around each vertex until a center is reached. We want the balls to be as small as possible. A randomized algorithm can uniformly sample $r$ centers to achieve the optimal (expected) ball size of $Θ(n/r)$. A folklore derandomization is to use the $O(log n)$ approximation for the set cover problem in the hitting set version where we want to hit all the balls with the centers. However, the extra $O(log n)$ factor is sometimes too expensive. For example, the recent $O(msqrt{log nloglog n})$ undirected single-source shortest path algorithm [DMSY23] beats Dijkstra's algorithm in sparse graphs, but the folklore derandomization would make it dominated by Dijkstra's. In this paper, we exploit the fact that the sizes of these balls can be adaptively chosen by the algorithm instead of fixed by the input. We propose a simple deterministic algorithm achieving the optimal ball size of $Θ(n/r)$ on average. Furthermore, given any polynomially large cost function of the ball size, we can still achieve the optimal cost on average. It allows us to derandomize [DMSY23], resulting in a deterministic $O(msqrt{log nloglog n})$ algorithm for undirected single-source shortest path. In addition, we show that the same technique can also be used to derandomize the seminal Thorup-Zwick approximate distance oracle [TZ05], also without any loss in the time/space complexity.

Problem

• Achieving optimal ball size in deterministic shortest path algorithms
• Removing logarithmic factors from derandomized graph algorithms
• Derandomizing approximate distance oracles without performance loss

Innovation

• Deterministic algorithm achieves optimal average ball size
• Adaptive ball sizing replaces fixed input constraints
• Lossless derandomization maintains original time complexity