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Fast algorithms for (max,min)matrix multiplication and bottleneck shortest paths
 In Proc. 19th SODA
, 2009
"... Given a directed graph with a capacity on each edge, the allpairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)transitive closure of a realv ..."
Abstract

Cited by 6 (0 self)
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Given a directed graph with a capacity on each edge, the allpairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)transitive closure of a realvalued matrix. In this paper, we give a (max, min)matrix multiplication algorithm running in time O(n (3+ω)/2) ≤ O(n 2.688), where ω is the exponent of binary matrix multiplication. Our algorithm improves on a recent O(n 2+ω/3) ≤ O(n 2.792)time algorithm of Vassilevska, Williams, and Yuster. Although our algorithm is slower than the best APBP algorithm on vertex capacitated graphs, running in O(n 2.575) time, it is just as efficient as the best algorithm for computing the dominance product, a problem closely related to (max, min)matrix multiplication. Our techniques can be extended to give subcubic algorithms for related bottleneck problems. The allpairs bottleneck shortest paths problem (APBSP) asks for the maximum flow that can be routed along a shortest path. We give an APBSP algorithm for edgecapacitated graphs running in O(n (3+ω)/2) time and a slightly faster O(n 2.657)time algorithm for vertexcapactitated graphs. The second algorithm significantly improves on an O(n2.859)time APBSP algorithm of Shapira, Yuster, and Zwick. Our APBSP algorithms make use of new hybrid products we call the distancemaxmin product and dominancedistance product. 1
Exact Complexity Results and PolynomialTime Algorithms for Derivative Accumulation
"... We discuss the complexity of evaluating certain collections of monic multilinear polynomials using operations in {×, +}. The functions we consider, which are defined on paths in directed acyclic graphs (DAGs), represent derivative computations that are based on the chain rule and have direct applica ..."
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We discuss the complexity of evaluating certain collections of monic multilinear polynomials using operations in {×, +}. The functions we consider, which are defined on paths in directed acyclic graphs (DAGs), represent derivative computations that are based on the chain rule and have direct applications in highperformance scientific computing. Our main results concern functions derived from singlesource, singlesink DAGs whose maximal paths all have length three. We derive tight, exact lower bounds for the numbers of multiplications, additions, and total arithmetic operations needed. Moreover, we show that, given such a DAG, an arithmetic circuit (or straightline program) of minimum size that evaluates J (G) can be constructed in polynomial time. In contrast, we show the (perhaps surprising) result that the problem of finding a circuit of minimum size for a given DAG becomes NPhard, even for the restricted class of DAGs considered in this paper, when some subset of the arcs may be labeled with the multiplicative identity 1 rather than an indeterminate. 1