Bounded-degree light approximate shortest-path trees in doubling metrics

Abstract

We study the Bounded-Degree Light Approximate Shortest-path Tree problem: Given a metric space (V,d) on n points, a root r∈V and a degree bound b, the goal is to find a tree T spanning V with maximum degree at most b, total weight comparable to the weight of a Minimum Spanning Tree of (V,d), and low stretch of tree distances from r to every vertex compared to input distances.Our main result is an algorithm that given a metric (V,d) with doubling dimension k, returns a tree T rooted at r spanning V with degree b and stretch max{O(k)/logb,O(1)} having weight within a constant factor the weight of a MST of (V,d).On the negative side, we show that the trade-off between degree and stretch is unavoidable (up to constant factors in the Big-O) even when the weight of the tree is not taken into account.