We show how to solve all-pairs shortest paths on n nodes in deterministic n 3> /2> Ω ( √ 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 [46] and Abboud, Williams, and Yu [8], 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 deterministically count k -satisfiability ( 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 [24]. A key to our constructions is an efficient way to deterministically simulate certain probabilistic polynomials critical to the algorithms of prior work, carefully applying small-biased sets and modulus-amplifying polynomials.

中文翻译：

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确定性 APSP、正交向量等

我们展示了如何解决所有对最短路径n中的节点确定性的 n 3>/2>Ω( √ 日志n) 时间，以及如何计算其中的正交向量对n0−1 个向量d=C日志n尺寸在确定性的 n 2−1/○（日志C)时间。这些运行时间基本上分别与最知名的 Williams [46] 和 Abboud、Williams 和 Yu [8] 的随机算法相匹配，并且即使对于随机算法，计数的能力也是开放的。通过减少，这两个结果为许多其他问题产生了更快的确定性算法。我们的技术也可用于确定性计数ķ-可满足性（ķ-SAT) 上的作业n2中的变量公式 n-n/○(ķ)时间，大致匹配用于检测可满足性和解决 Santhanam [24] 的开放问题的已知运行时间。我们构建的关键是一种有效的方法来确定性地模拟对先前工作的算法至关重要的某些概率多项式，仔细应用小偏差集和模放大多项式。