Low-Stretch Spanning Trees via Smoothed Analysis of Dijkstra's Algorithm

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.