We present an O(n2) time-optimal algorithm for solving the unweighted all-pair shortest path problem on interval graphs, an important subclass of perfect graphs. An interesting structure called the neighborhood tree is studied and used in the algorithm. This tree is formed by identifying

Abstract We obtain three new dynamic algorithms for the approx-imate all-pairs shortest paths problem in unweighted undirected graphs: 1. For any fixed " ? 0, a decremental algorithm withan expected total running time of ~O(mn), where m is the number of edges and n is the number of ver

We show that for every 0 ≤ p ≤ 1 there is an O(n 2.575−p/(7.4−2.3p) ) time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u, v in the graph to within an additive error δ p (u, v), where δ(u, v

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

), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n 2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix

-tiplicative approximation factor of (1 + ) with high probability, where ω is the exponent of ordinary matrix multiplication of n dimensional square matrices. It also computes the edit script. We further solve the local alignment problem; for all substrings of s, we can estimate their language edit distance

by the algorithm is not longer/greater than t · δ(u, v). This paper presents two randomized algorithms for computing all-pairs nearly 2-approximate distances. The first algorithm takes expected O(m 2/3 n log n+n²) time, and for any u, v ∈ V reports distance no greater than 2δ(u, v) + 1. Our second algorithm

The upper bound on the exponent, ω, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even

We design a faster algorithm for the all-pairs shortest path problem under the conventional RAM model, based on distance matrix multiplication (DMM). Specifically we improve the best known time complexity of O(n 3 (log log n) 2 / log n) to O(n 3 log log n / log n). As an application, we show the k

,min)-product of two arbitrary matrices over R ∪ {∞, −∞} in O(n 2+ω/3) ≤ O(n 2.792) time, where n is the number of vertices and ω is the exponent for matrix multiplication over rings. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length