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All pairs almost shortest paths

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3 Author(s)
Dor, D. ; Dept. of Comput. Sci., Tel Aviv Univ., Israel ; Halperin, S. ; Zwick, U.

Let G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of D. Aingworth et al. (1996), we describe an O˜(min{n3/2m1/2,n7/3 }) time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k>2, we describe an O˜(min{n2-(2)(k+2)/m(2)(k+2)/, n2+(2) (3k-2)/}) time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an O˜(n2) time algorithm APASP for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in O˜(n2) time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every u, v∈V we have δG(u,v)⩽δF (u,v)⩽δG(u,v)+k. We show that every unweighted graph on n vertices has a 2-emulator with O˜(n3/2 ) edges and a 4-emulator with O˜(n4/3) edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with O˜(n 3/2) edges and that such a 3-spanner can be built in O˜(mn1/2) time. We also describe an O˜(n(m2/3+n)) time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3

Published in:

Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on

Date of Conference:

14-16 Oct 1996