@MISC{_fasteralgorithms, author = {}, title = {Faster Algorithms for Approximate Distance Oracles and All-Pairs Small StretchPaths}, year = {} }

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Abstract

ffi(u, v) < = ^ffi(u, v) < = t * ffi(u, v). The most efficient al-gorithms known for computing small stretch distances in Gare the approximate distance oracles of [16] and the three algorithms in [9] to compute all-pairs stretch t distancesfor t = 2, 7/3, and 3. We present faster algorithms forthese problems. For any integer k> = 1, Thorup and Zwick in [16] gavean O(kmn1/k) algorithm to construct a data structure ofsize O(kn1+1/k) which, given a query (u, v) 2 V * V,returns in O(k) time, a 2k- 1 stretch estimate of ffi(u, v).But for small values of k, the time to construct the oracle israther high. Here we present an O(n2 log n) algorithm toconstruct such a data structure of size O(kn1+1/k) for allintegers k> = 2. Our query answering time is O(k) for k>2 and \Theta (log n) for k = 2. We use a new generic scheme forall-pairs approximate shortest paths for these results. This scheme also enables us to design faster algorithms for all-pairs t-stretch distances for t = 2 and 7/3, and computeall-pairs almost stretch 2 distances in O(n2 log n) time.