Light Tree Covers, Routing, and Path-Reporting Oracles via Spanning Tree Covers in Doubling Graphs
Abstract
A (1+)-stretch tree cover of an edge-weighted n-vertex graph G is a collection of trees, where every pair of vertices has a (1+)-stretch path in one of the trees. The celebrated Dumbbell Theorem by Arya et. al. [STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+)-stretch tree cover with a constant number of trees, where the constant depends on and the dimension d. This result was generalized for arbitrary doubling metrics by Bartal et. al. [ICALP'19]. While the total number of edges in the tree covers of Arya et. al. and Bartal et. al. is O(n), all known tree cover constructions incur a total lightness of ( n); whether one can get a tree cover of constant lightness has remained a longstanding open question, even for 2-dimensional point sets. In this work we resolve this fundamental question in the affirmative, as a direct corollary of a new construction of (1+)-stretch spanning tree cover for doubling graphs; in a spanning tree cover, every tree may only use edges of the input graph rather than the corresponding metric. To the best of our knowledge, this is the first constant-stretch spanning tree cover construction (let alone for (1+)-stretch) with a constant number of trees, for any nontrivial family of graphs. Concrete applications of our spanning tree cover include a (1+)-stretch light tree cover, a compact (1+)-stretch routing scheme in the labeled model, and a (1+)-stretch path-reporting distance oracle, for doubling graphs. [...]
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