Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths

• Authors: Surender Baswana;Telikepalli Kavitha
• Venue: FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2006

Let G = (V,E) be a weighted undirected graph with |V | = n and |E| = m. An estimate \hat \delta \left( {u,v} \right) of the distance\delta \left( {u,v} \right) in G between u, v \in V is said to be of stretch t iff \delta \left( {u,v} \right) \leqslant \hat \delta \left( {u,v} \right) \leqslant t · \delta \left( {u,v} \right). The most efficient algorithms known for computing small stretch distances in G are the approximate distance oracles of [16] and the three algorithms in [9] to compute all-pairs stretch t distances for t = 2, 7/3, and 3. We present faster algorithms for these problems. For any integer k \geqslant 1, Thorup and Zwick in [16] gave an O(kmn^{1/k}) algorithm to construct a data structure of size O(kn^{1+1/k}) which, given a query (u, v) \in V 脳 V , returns in O(k) time, a 2k - 1 stretch estimate of \delta \left( {u,v} \right). But for small values of k, the time to construct the oracle is rather high. Here we present an O(n^2 log n) algorithm to construct such a data structure of size O(kn^{1+1/k}) for all integers k \geqslant 2. Our query answering time is O(k) for k \ge 2 and \Theta (log n) for k = 2. We use a new generic scheme for all-pairs approximate shortest paths for these results. This scheme also enables us to design faster algorithms for allpairs t-stretch distances for t = 2 and 7/3, and compute all-pairs almost stretch 2 distances in O(n^2 log n) time.