This paper studies distance oracles and distance labeling scheme with local stretch. Informally we would like to provide stretch guarantees for the r closest nodes of each node. A distance oracle has local stretch k for r neighborhoods if for any u, v such that v = M(u, r′) and r′ ≤ r: dist(u, v) ≤ dist(u, v) ≤ kdist(u, v), where M(u,r′) is the r′ closest node to u and dist(u, v) is the estimated distance returned by the distance oracle. For parameters r ＞ 1, k ＞ 1, we obtain labels of size O(r~(1/k) ln~(1-1/k) r+ ln k), with local stretch of 2k - 1 for r neighborhoods in O(k) time, significantly improving the query time and stretch constant of [ABN09]. Moreover, our stretch guarantee of 2k - 1 matches the best known (and conjectured optimal) guarantees of standard distance oracles.
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