**3**Author(s)

# All pairs almost shortest paths

- Already Purchased? View Article
- Subscription Options Learn More

By Topic

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{n^{3/2}m^{1/2},n^{7/3 }}) time algorithm APASP_{2} for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP_{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^{2-(2)}(k+2)/m^{(2)}(k+2)/, n^{2+(2) }(3k-2)/}) time algorithm APASP_{k} 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_{∞} 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^{2}) 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 δ_{G}(u,v)⩽δ_{F }(u,v)⩽δ_{G}(u,v)+k. We show that every unweighted graph on n vertices has a 2-emulator with O˜(n^{3/2 }) edges and a 4-emulator with O˜(n^{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 ^{3/2}) edges and that such a 3-spanner can be built in O˜(mn^{1/2}) time. We also describe an O˜(n(m^{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

- Page(s):
- 452 - 461
- Meeting Date :
- 14 Oct 1996-16 Oct 1996
- ISSN :
- 0272-5428
- Print ISBN:
- 0-8186-7594-2
- INSPEC Accession Number:
- 5450830

- Conference Location :
- Burlington, VT
- DOI:
- 10.1109/SFCS.1996.548504
- Publisher:
- IEEE