Directed Shortest Paths via Approximate Cost Balancing

We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [31] for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an O(m root n logn) preprocessing step. Using these reduced costs, every shortest path query can be solved in O(m) time using an adaptation of Thorup's component hierarchy method. The balancing result can also be applied to the l(infinity)-matrix balancing problem.

Automated summary

We propose an O(nm) algorithm for computing all-pairs shortest paths in directed graphs with nonnegative integer arc costs. This algorithm matches the complexity bound achieved by Thorup's algorithm for the same problem in undirected graphs. The key observation is that directed shortest paths problems with approximately balanced cost functions can be solved similarly to the undirected case. The algorithm first finds an approximately balanced reduced cost function in an O(m root n logn) preprocessing step, and then uses this reduced cost to solve each shortest path query in O(m) time using an adaptation of Thorup's component hierarchy method. The balancing result can also be applied to the l(infinity)-matrix balancing problem.