All pairs almost shortest paths

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/spl tilde/(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/spl tilde/(min{n/sup 2-(2)/(k+2)/m/sup (2)/(k+2)/, n/sup 2+(2)/(3k-2)/}) time algorithm APASP/sub k/ for computing all distances in G with an additive one-sided error of at most k. We also give an O/spl tilde/(n/sup 2/) time algorithm APASP/sub /spl infin// for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in O/spl tilde/(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/spl isin/V we have /spl delta//sub G/(u,v)/spl les//spl delta//sub F/(u,v)/spl les//spl delta//sub G/(u,v)+k. We show that every unweighted graph on n vertices has a 2-emulator with O/spl tilde/(n/sup 3/2/) edges and a 4-emulator with O/spl tilde/(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/spl tilde/(n/sup 3/2/) edges and that such a 3-spanner can be built in O/spl tilde/(mn/sup 1/2/) time. We also describe an O/spl tilde/(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.

[1]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[2]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[3]  Raimund Seidel,et al.  On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs , 1995, J. Comput. Syst. Sci..

[4]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[5]  Edith Cohen.  Fast algorithms for constructing t-spanners and paths with stretch t , 1993, FOCS.

[6]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[7]  Zvi Galil,et al.  All Pairs Shortest Distances for Graphs with Small Integer Length Edges , 1997, Inf. Comput..

[8]  Zvi Galil,et al.  Witnesses for Boolean Matrix Multiplication and for Transitive Closure , 1993, J. Complex..

[9]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[10]  Giri Narasimhan,et al.  New sparseness results on graph spanners , 1992, SCG '92.

[11]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

[12]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[13]  Noga Alon,et al.  Witnesses for Boolean matrix multiplication and for shortest paths , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[14]  Rephael Wenger,et al.  Extremal graphs with no C4's, C6's, or C10's , 1991, J. Comb. Theory, Ser. B.

[15]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[16]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[17]  Rajeev Motwani,et al.  Clique partitions, graph compression and speeding-up algorithms , 1991, STOC '91.

[18]  Mihalis Yannakakis,et al.  High-Probability Parallel Transitive-Closure Algorithms , 1991, SIAM J. Comput..

[19]  Noga Alon,et al.  Finding and Counting Given Length Cycles (Extended Abstract) , 1994, ESA.

[20]  Michael L. Fredman,et al.  New Bounds on the Complexity of the Shortest Path Problem , 1976, SIAM J. Comput..

[21]  Mihalis Yannakakis,et al.  High-probability parallel transitive closure algorithms , 1990, SPAA '90.

[22]  Tadao Takaoka,et al.  A New Upper Bound on the Complexity of the All Pairs Shortest Path Problem , 1991, Inf. Process. Lett..

[23]  Lenore Cowen,et al.  Near-linear cost sequential and distributed constructions of sparse neighborhood covers , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[24]  Zvi Galil,et al.  All Pairs Shortest Paths for Graphs with Small Integer Length Edges , 1997, J. Comput. Syst. Sci..

[25]  David P. Dobkin,et al.  On sparse spanners of weighted graphs , 1993, Discret. Comput. Geom..

[26]  Raimund Seidel,et al.  On the all-pairs-shortest-path problem , 1992, STOC '92.

[27]  Arthur L. Liestman,et al.  Additive graph spanners , 1993, Networks.

[28]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[29]  Noga Alon,et al.  On the Exponent of the All Pairs Shortest Path Problem , 1991, J. Comput. Syst. Sci..

[30]  R. Motwani,et al.  On Diameter Verification and Boolean Matrix Multiplication. , 1995 .

[31]  Jose Augusto Ramos Soares,et al.  Graph Spanners: a Survey , 1992 .

[32]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[33]  Edith Cohen,et al.  All-pairs small-stretch paths , 1997, SODA '97.

[34]  Piotr Indyk,et al.  Fast estimation of diameter and shortest paths (without matrix multiplication) , 1996, SODA '96.

[35]  Edith Cohen,et al.  Polylog-time and near-linear work approximation scheme for undirected shortest paths , 1994, STOC '94.

[36]  David R. Karger,et al.  Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths , 1993, SIAM J. Comput..

[37]  Mikkel Thorup,et al.  Undirected single source shortest paths in linear time , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.