Object location using path separators

Abstract

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.

Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus and Minor-Free Graphs

“…Thorup [Tho04] proves that any planar graph can be recursively separated by three shortest paths. Abraham and Gavoille [AG06] extend his result to minor-closed families, proving that any minor-free graph can be recursively separated by O(1) shortest paths. Since bounded-genus graphs exclude minors, we could use their result to obtain a linear-space distance oracle.…”

“…Since bounded-genus graphs exclude minors, we could use their result to obtain a linear-space distance oracle. The constant in [AG06] however depends on the size of the minor in an unspecified way. In the following, we prove that genus g graphs can be recursively separated using at most O(g) shortest paths.…”

“…Restricted graph families The only known constructions that achieve (1 + ) stretch are for restricted families of graphs: planar graphs [Tho04], minor-excluded graphs [AG06], and graphs of low doubling dimension [Tal04,HPM06,Sli07,BGK + 10]. (In Section 2.2, we survey previously known results in this area.…”

Graph Augmentation via Metric Embedding

“…It can be distinguished two main kinds of augmentation processes in the navigable networks literature. One kind of augmentation relies on the graph density and its similarity with a mesh (like augmentations in [7,17,18,22]), while the other kind relies on the existence of good separators in the graph (like augmentations in [4,10]). Augmentation via embedding cannot be directly extended to augmentations using separators because of the difficulty to handle the distortion in the analysis of greedy routing.…”

“…where, for any node u and radius r ≥ 1, the 2r-neighborhood of u is of size at most a constant times its r-neighborhood. Fraigniaud [10] demonstrates that any bounded treewidth graph can also be augmented by one link per node to become navigable, and Abraham and Gavoille [4] showed that, more generally, this is possible for all graphs excluding a fixed minor. The definition of the problem can directly be extended to metric spaces by asking which n-points metric spaces 1 M = (V, δ) can be augmented by O(log n) links such that, in the resulting graph, greedy routing computes polylog(n) routes between any pair with the only knowledge of M .…”