Freivalds算法的非确定性去随机化：后果、途径和算法进展
On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress
摘要：
出于研究随机性的力量，验证算法和细粒度约简的障碍，我们研究了两个$n\times n$矩阵的乘法是否可以在接近最优的非确定性时间$\tilde{O}(n^2)$中执行的问题。由于Freivalds的一个经典算法在时间$O(n^2)$上概率性地证明矩阵乘积的正确性，所以我们的问题是Freivalds算法去随机化公开问题的一个放松.我们讨论了这个问题的积极或消极的解决方案的后果，并提供了解决它的潜在途径。特别是，我们表明，足够快的确定性验证3SUM或单变量多项式身份测试产生更快的确定性验证矩阵乘法。此外，我们提出了部分算法的进展，区分是否整数矩阵产品是正确的或包含1和$n$之间的错误条目可以在时间$\tilde{O}(n^2)$ -有趣的是，确定性矩阵产品验证的困难情况下，不是一个问题“大海捞针”，而是在存在许多错误的抵消效应。我们的主要技术贡献是一个确定性的算法，纠正整数矩阵产品包含在时间$\tilde{O}(\sqrt{t} n^2 + t^2)$最$t$错误。为了得到这个结果，我们展示了如何在相同的运行时间内计算一个至多有$t$非零值的整数矩阵乘积。这改进了已知的确定性输出敏感的整数矩阵乘法算法的$t = Ω(n^{2/3})$非零，这是独立的利益。
英文摘要：
Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two $n\times n$ matrices can be performed in near-optimal nondeterministic time $\tilde{O}(n^2)$. Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time $O(n^2)$, our question is a relaxation of the open problem of derandomizing Freivalds' algorithm. We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and $n$ erroneous entries can be performed in time $\tilde{O}(n^2)$ -- interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors. Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most $t$ errors in time $\tilde{O}(\sqrt{t} n^2 + t^2)$. To obtain this result, we show how to compute an integer matrix product with at most $t$ nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for $t = Ω(n^{2/3})$ nonzeroes, which is of independent interest.
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