Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models

We present a method for solving the transshipment problem—also known as uncapacitated minimum cost flow—up to a multiplicative error of 1 + in undirected graphs with polynomially bounded integer edge weights using a tailored gradient descent algorithm. An important special case of the transshipment problem is the single-source shortest paths (SSSP) problem. Our gradient descent algorithm takes O( −3 polylogn) iterations and in each iteration it needs to solve a variant of the transshipment problem up to a multiplicative error of polylogn. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. As a consequence, we improve prior work by obtaining the following results: 1. RAM model: (1+ )-approximate transshipment in Õ( −3(m+n1+o(1))) computational steps1 (leveraging a recent O(m1+o(1))-step O(1)-approximation due to Sherman [2016]). 2. Multipass Streaming model: (1 + )-approximate transshipment and SSSP using Õ(n) space and Õ( −O(1)) passes. 3. Broadcast Congested Clique model: (1 + )-approximate transshipment and SSSP using Õ( −O(1)) rounds. 4. Broadcast Congest model: (1+ )-approximate SSSP using Õ( −O(1)( √ n+D)) rounds, where D is the (hop) diameter of the network. The previous fastest algorithms for the last three models above leverage sparse hop sets. We bypass the hop set computation; using a spanner is sufficient in our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. 1We use Õ(·) to hide polylogarithmic factors in n. 1 ar X iv :1 60 7. 05 12 7v 1 [ cs .D S] 1 8 Ju l 2 01 6