# A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

• Andrzej Lingas
• Dzmitry Sledneu
Conference paper

DOI: 10.1007/978-3-642-27660-6_31

Volume 7147 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Lingas A., Sledneu D. (2012) A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs. In: Bieliková M., Friedrich G., Gottlob G., Katzenbeisser S., Turán G. (eds) SOFSEM 2012: Theory and Practice of Computer Science. SOFSEM 2012. Lecture Notes in Computer Science, vol 7147. Springer, Berlin, Heidelberg

## Abstract

We consider the problem of computing all-pairs shortest paths in a directed graph with non-negative real weights assigned to vertices.

For an n×n 0 − 1 matrix C, let KC be the complete weighted graph on the rows of C where the weight of an edge between two rows is equal to their Hamming distance. Let MWT(C) be the weight of a minimum weight spanning tree of KC.

We show that the all-pairs shortest path problem for a directed graph G on n vertices with non-negative real weights and adjacency matrix AG can be solved by a combinatorial randomized algorithm in time
$$\widetilde{O}(n^{2}\sqrt{n + \min\{MWT(A_G), MWT(A_G^t)\}})$$
As a corollary, we conclude that the transitive closure of a directed graph G can be computed by a combinatorial randomized algorithm in the aforementioned time.

We also conclude that the all-pairs shortest path problem for vertex-weighted uniform disk graphs induced by point sets of bounded density within a unit square can be solved in time $$\widetilde{O}(n^{2.75})$$.