Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
Algorithms for two bottleneck optimization problems
Journal of Algorithms
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Witnesses for Boolean matrix multiplication and for transitive closure
Journal of Complexity
On the all-pairs-shortest-path problem in unweighted undirected graphs
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
Rectangular matrix multiplication revisited
Journal of Complexity
Fast rectangular matrix multiplication and applications
Journal of Complexity
All pairs lightest shortest paths
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Lowest common ancestors in trees and directed acyclic graphs
Journal of Algorithms
Finding a maximum weight triangle in n3-Δ time, with applications
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Faster algorithms for finding lowest common ancestors in directed acyclic graphs
Theoretical Computer Science
LCA queries in directed acyclic graphs
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Finding the smallest H-Subgraph in real weighted graphs and related problems
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
All-pairs shortest paths with real weights in O(n3/ log n) time
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
All-pairs bottleneck paths for general graphs in truly sub-cubic time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Nondecreasing paths in a weighted graph or: how to optimally read a train schedule
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Bottleneck flows in unit capacity networks
Information Processing Letters
On Cartesian Trees and Range Minimum Queries
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Unique lowest common ancestors in dags are almost as easy as matrix multiplication
ESA'07 Proceedings of the 15th annual European conference on Algorithms
ESWC'12 Proceedings of the 9th international conference on The Semantic Web: research and applications
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Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights. The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem. A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).