An expectedcase subcubic solution to the allpairs shortest path problem in R
Abstract
It has been shown by Alon et al. that the socalled 'allpairs shortestpath' problem can be solved in O((MV)^2.688 * log^3(V)) for graphs with V vertices, with integer distances bounded by M. We solve the more general problem for graphs in R (assuming no negative cycles), with expectedcase running time O(V^2.5 * log(V)). While our result appears to violate the Omega(V^3) requirement of "Funny Matrix Multiplication" (due to Kerr), we find that it has a subcubic expected time solution subject to reasonable conditions on the data distribution. The expected time solution arises when certain subproblems are uncorrelated, though we can do better/worse than the expectedcase under positive/negative correlation (respectively). Whether we observe positive/negative correlation depends on the statistics of the graph in question. In practice, our algorithm is significantly faster than FloydWarshall, even for dense graphs.
 Publication:

arXiv eprints
 Pub Date:
 December 2009
 arXiv:
 arXiv:0912.0975
 Bibcode:
 2009arXiv0912.0975M
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 9 pages, 5 figures