An output-sensitive algorithm for all-pairs shortest paths in directed acyclic graphs
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A straightforward dynamic programming method for the single-source shortest paths problem (SSSP) in an edge-weighted directed acyclic graph (DAG) processes the vertices in a topologically sorted order. First, we similarly iterate this method alternatively in a breadth-first search sorted order and the reverse order on an input directed graph with both positive and negative real edge weights, n vertices and m edges. For a positive integer t, after O(t) iterations in O(tm) time, we obtain for each vertex v a path distance from the source to v not exceeding that yielded by the shortest path from the source to v among the so called t+light paths. A directed path between two vertices is t+light if it contains at most t more edges than the minimum edge-cardinality directed path between these vertices. After O(n) iterations, we obtain an O(nm)-time solution to SSSP in directed graphs with real edge weights matching that of Bellman and Ford. Our main result is an output-sensitive algorithm for the all-pairs shortest paths problem (APSP) in DAGs with positive and negative real edge weights. It runs in time O(min{n^ω, nm+n^2log n}+∑_v∈ Vindeg(v)|leaf(T_v)|), where n is the number of vertices, m is the number of edges, ω is the exponent of fast matrix multiplication, indeg(v) stands for the indegree of v, T_v is a tree of lexicographically-first shortest directed paths from all ancestors of v to v, and leaf(T_v) is the set of leaves in T_v. Finally, we discuss an extension of hypothetical improved upper time-bounds for APSP in non-negatively edge-weighted DAGs to include directed graphs with a polynomial number of large directed cycles.
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