Given a simple connected unweighted undirected graph G = (V (G), E(G)) with |V (G)| = n and |E(G)| = m, we present a new algorithm for the all-pairs shortest-path (APSP) problem. The running time of our algorithm is in O(n2 log n). This bound is an improvement over previous best known O(n2.376) time bound of Raimund Seidel (1995) for general graphs. The algorithm presented does not rely on fast matrix multiplication. Our algorithm with slight modifications, enables us to compute the APSP problem for unweighted directed graph in time O(n2 log n), improving a previous best known O(n2.575) time bound of Uri Zwick (2002).<\/p>\r\n","references":"[1] N. Alon, Z. Galil, and O. Margalit, \"On the exponent of the all-pairs\r\nshortest path problem,\" J. Comput. Sys. Sci., vol. 54, pp. 255-262, 1997.\r\n[2] T. M. Chan, \"All-pairs shortest paths for Unweighted Undirected Graphs\r\nin o(mn) Time,\" In Proc. 17th annual ACM-SIAM Sympos. on Discrete\r\nAlgorithms, pp. 514-523, 2006.\r\n[3] F. F. Dragan, \"Estimating all pairs shortest paths in restricted graph\r\nfamilies: a unified approach,\" J. of Algorithms, vol. 57, pp. 1-21, 2005.\r\n[4] T. Feder and R. Motwani, \"Clique patritions, graph compression and\r\nspeeding-up algorithms,\" J. Comput. Sys. Sci., vol. 51, pp. 261-272,\r\n1995.\r\n[5] Z. Galil and O. Margalit, \"All pairs shortest distances for graphs with\r\nsmall integer length edges,\" Inf. Comput., vol. 134, pp. 103-139, 1997.\r\n[6] Z. Galil and O. Margalit, \"All pairs shortest paths for graphs with small\r\ninteger length edges,\" J. Comput. Sys. Sci., vol. 54, pp. 243-254, 1997.\r\n[7] R. Seidel, \"On the all-pairs shortest path problem in unweighted\r\nundirected graphs,\" J. Comput. Sys. Sci., vol. 51, pp. 400-403, 1995.\r\n[8] U. Zwick, \"All-pairs shortest paths using bridging sets and rectangular\r\nmatrix multiplication,\" J. ACM, vol. 49, pp. 289-317, 2002.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 26, 2009"}