Object location using path separators

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

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