A faster all-pairs shortest path algorithm for real-weighted sparse graphs

We present a faster all-pairs shortest paths algorithm for arbitrary real-weighted directed graphs. The algorithm works in the fundamental comparison-addition model and runs in O(mn + n(2) log log n) time, where m and n are the number of edges & vertices, respectively. This is strictly faster than Johnson's algorithm (for arbitrary edge-weights) and Dijkstra's algorithm (for positive edge-weights) when m = o(n log n) and matches the running time of Hagerup's APSP algorithm, which assumes integer edge-weights and a more powerful model of computation.