We show how to solve all-pairs shortest paths on n nodes in deterministic n^3 / 2^{Omega(sqrt{log n})} time, and how to count the pairs of orthogonal vectors among n 0-1 vectors in d = c log n dimensions in deterministic n^{2 - 1/O(log c)} time. These running times essentially match the best known randomized algorithms of (Williams, STOC'14) and (Abboud, Williams, and Yu, SODA 2015) respectively, and the ability to count was open even for randomized algorithms. By reductions, these two results yield faster deterministic algorithms for many other problems. Our techniques can also be used to count k-SAT assignments on n variable formulas in 2^{n - n/O(k)} time, roughly matching the best known running times for detecting satisfiability and resolving an open problem of Santhanam (2013).
A key to our constructions is an efficient way to deterministically simulate certain probabilistic polynomials critical to the algorithms of prior work, applying epsilon-biased sets and modulus-amplifying polynomials.
We present a new combinatorial algorithm for Boolean matrix multiplication that runs in O(n^3 (loglog n)^3 / log^3 n) time. This improves the previous combinatorial algorithm by Bansal and Williams [FOCS'09] that runs in O(n^3 (loglog n)^2 / log^{9/4} n) time. Whereas Bansal and Williams' algorithm uses regularity lemmas for graphs, the new algorithm is simple and uses entirely elementary techniques: table lookup, word operations, plus a deceptively straightforward divide-and-conquer.
Our algorithm is in part inspired by a recent result of Impagliazzo, Lovett, Paturi, and Schneider (2014) on a different geometric problem, offline dominance range reporting; we improve their analysis for that problem as well.
In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time approaching O(n^3 log^3log n / log^2 n), which improves all known algorithms for general real-weighted dense graphs.
In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of "geometrically weighted" graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n^{3-(3-w)/(2d+4)}), where w < 2.376; in two dimensions, this is O(n^{2.922}). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n^{3-(3-w)/4}) = O(n^{2.844}) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n^{(3+w)/2}) = O(n^{2.688}) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs.
We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the l_p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p=2, and p=infty. For p=2, we reduce the problem to standard convolution, while for p=infty and p=1, we reduce the problem to (min,+) convolution and (median,+) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X+Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X+Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in Theta(n^2) time.
We revisit the all-pairs-shortest-paths problem for an unweighted undirected graph with n vertices and m edges. We present new algorithms with the following running times:
We describe an O(n^3 / log n)-time algorithm for the all-pairs-shortest-paths problem for a real-weighted directed graph with n vertices. This slightly improves a series of previous, slightly subcubic algorithms by Fredman (1976), Takaoka (1992), Dobosiewicz (1990), Han (2004), Takaoka (2004), and Zwick (2004). The new algorithm is surprisingly simple and different from previous ones.
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