Problem
Research questions and friction points this paper is trying to address.
Low-Stretch Spanning Tree
Smoothed Analysis
Dijkstra's Algorithm
Shortest-Path Tree
Low-Stretch Spanning Trees via Smoothed Analysis of Dijkstra's Algorithm
🤖 AI Summary
This study addresses the algorithmic complexity of existing low-stretch spanning tree (LSST) constructions and the lack of theoretical justification for Dijkstra-based heuristics. We introduce the first smoothed analysis framework for this problem, demonstrating that applying slight perturbations to graph edge weights enables the shortest path tree rooted at any arbitrary node to serve as an approximate LSST. By further integrating low-diameter decomposition techniques, we propose a novel and efficient paradigm for LSST construction. Theoretically, we prove that perturbed shortest path trees achieve an $\tilde{O}(1)$ approximation ratio, thereby elucidating the underlying mechanism behind the empirical success of Dijkstra-based heuristics. Practically, our approach enables simple and efficient LSST computation, significantly reducing overall algorithmic complexity compared to prior methods.
📝 Abstract
Given an undirected weighted graph $G$, a $\gamma$-approximate low-stretch spanning tree (LSST) $T \subseteq G$ is a tree that approximates the distance metric of $G$ up to a $\gamma$-factor in expectation. Currently, existing algorithms to find a provably good LSST carefully construct an approximate shortest-path tree from an arbitrary source. The resulting algorithms are intricate. In contrast, practitioners observed that a much simpler heuristic performs surprisingly well: choose an arbitrary root, run Dijkstra's algorithm, and use the resulting shortest-path tree as an LSST. In this paper, we give a smoothed analysis of shortest-path tree algorithms, such as Dijkstra's algorithm, that explains this behavior. We show that adding a small perturbation to the weights of the input graph suffices to turn the shortest path tree rooted at an arbitrary node in the resulting graph into an $\tilde{O}(1)$-approximate LSST. We further show that the set of perturbations can be computed efficiently from few low-diameter decompositions (LDDs). Thus, our proof is also constructive in the sense of giving a novel approach to computing LSSTs.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Smoothed Analysis
Low-Stretch Spanning Trees
Dijkstra's Algorithm
Low-Diameter Decompositions
Edge Perturbation
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