## Abstract

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 Aingworth et al. 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. 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^{HLF}) 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.

Original language | English |
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Pages (from-to) | 1740-1759 |

Number of pages | 20 |

Journal | SIAM Journal on Computing |

Volume | 29 |

Issue number | 5 |

DOIs | |

State | Published - Mar 2000 |