Computing driving directions has motivated many shortest path heuristics that answer queries on continental scale networks, with tens of millions of intersections, literally instantly, and with very low storage overhead. In this paper we complement the experimental evidence with the first rigorous proofs of efficiency for many of the heuristics suggested over the past decade. We introduce the notion of highway dimension and show how low highway dimension gives a unified explanation for several seemingly different algorithms. 782

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We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET ∼T [dT (u, v)/dG(u, v)] is bounded byÕ(log n). Our result is obtained via a new approach of building "highways" between portals and a new strong diameter probabilistic decomposition theorem.

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We study a novel separator property called k-path separable. Roughly speaking, a k-path separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is k-path separable. We then show that k-path separable graphs can be used to solve several object location problems: (1) a small-worldization with an average poly-logarithmic number of hops; (2) an (1 + ε)-approximate distance labeling scheme with O(log n) space labels; (3) a stretch-(1 + ε) compact routing scheme with tables of poly-logarithmic space; (4) an (1 + ε)-approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to bounded-dimension isometric sparable graphs.

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