Summary: External Memory Algorithms for Diameter and
AllPairs ShortestPaths on Sparse Graphs
Lars Arge 1,# , Ulrich Meyer 2,## , and Laura Toma 3,# # #
1 Department of Computer Science, Duke University, Durham, NC 27708, USA.
2 MaxPlanckInstitut f˜ur Informatik, 66123 Saarbr˜ucken, Germany.
3 Department of Computer Science, Bowdoin College, Brunswick, ME 04011, USA.
Abstract. We develop I/Oe#cient algorithms for diameter and all
pairs shortestpaths (APSP). For general undirected graphs G(V, E) with
nonnegative edge weights and E/V = o(B/ log V ) our approaches are
the first to achieve o(V 2 ) I/Os. We also show that for unweighted undi
rected graphs, APSP can be solved with just O(V · sort(E)) I/Os. Both
our weighted and unweighted approaches require O(V 2 ) space. For di
ameter computations we provide I/Ospace tradeo#s. Finally, we provide
improved results for both diameter and APSP computation on directed
Computing shortest paths and diameter of a graph are fundamental problems in
algorithmic graph theory. For example, research in web modeling uses shortest
path and diameter computations as primitive routines for investigating the struc
ture of the web. Further applications often appear in Geographic Information