We present a deterministic distributed algorithm to compute all-pairs shortest paths(APSP) in an edge-weighted directed or undirected graph. Our algorithm runs inÕ(n 3/2 ) rounds in the Congest model, where n is the number of nodes in the graph. This is the first o(n 2 ) rounds deterministic distributed algorithm for the weighted APSP problem. Our algorithm is fairly simple and incorporates a deterministic distributed algorithm we develop for computing a 'blocker set' [14], which has been used earlier in sequential dynamic computation of APSP.

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We consider the fine-grained complexity of sparse graph problems that currently haveÕ(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (minimum weight cycle), and Eccentricities, which is the problem of computing, for each vertex in the graph, the length of a longest shortest path starting at that vertex.We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in thẽ O(mn) class. Surprisingly, very few of the known nontrivial reductions between problems in thẽ O(mn) class are sparse reductions. In the directed case, our results give a partial order on a large collection of problems in theÕ(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to a natural variant of APSP. The latter reduction also gives an improved algorithm for ANSC (for dense graphs).We propose the MWC Conjecture, a new conditional hardness conjecture that the weight of a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in theÕ(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, and Eccentricities, are MWCC hard. We also identify Eccentricities as a key problem in theÕ(mn) class which is simultaneously MWCC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than n k .Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which theÕ(mn) time algorithms are the ones typically used in practice, and not theÕ(n 3 ) time algorithms.1Õ andΘ can hide sub-polynomial factors; in our new results they only hide polylog factors.

We present a new pipelined approach to compute all pairs shortest paths (APSP) in a directed graph with nonnegative integer edge weights (including zero weights) in the Congest model in the distributed setting. Our deterministic distributed algorithm computes shortest paths of distance at most ∆ for all pairs of vertices in at most 2n √ ∆ + 2n rounds, and more generally, it computes h-hop shortest paths for k sources in 2 √ nkh + n + k rounds. The algorithm is simple, and it has some novel features and a nontrivial analysis. It uses only the directed edges in the graph for communication. This algorithm can be used as a base within asymptotically faster algorithms that match or improve on the current best deterministic bound ofÕ(n 3/2 ) rounds for this problem when edge weights are O(n) or shortest path distances areÕ(n 3/2 ). These latter results are presented in a companion paper [1].

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