All pairs almost shortest paths
- Tel Aviv Univ. (Israel)
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 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe an O(min n{sup 3/2}m{sup 1/2}, n{sup 7/3}) time algorithm APASP{sub 2} for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP{sub 2} 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 n{sup 2} 2/k +2 m 2/2+k, n{sup 2+} 2/3k-2) time algorithm APASP{sub {infinity}} for computing all distances in G with an additive one-sided error of at most k. We also give an O(n 2) time algorithm APASP{sub {infinity}}. for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in O(n{sup 2}) time. We say that a weighted graph F = (V,E`) k-emulates an unweighted graph G = (V,E) if for every u, v {element_of} V we have {delta}{sub G} (u, {nu}) {le} {delta}{sub F} (u, {nu}) {le} {delta}{sub G} (u, {nu}) + k. We show that every unweighted graph on n vertices has a 2-emulator with O(n{sup 3/2}) edges and a 4-emulator with O(n{sup 4/3}) edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with O(n{sup 1/2}) edges and that such a 3-spanner can be built in O(mn{sup 1/2}) time. We also describe an O(n (m{sup 2/3} + n)) time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3.
- OSTI ID:
- 457675
- Report Number(s):
- CONF-961004--
- Country of Publication:
- United States
- Language:
- English
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