真正次二次的3SUM和真正次三次的APSP：基于稀疏偏斜图中的三角形

中文摘要 AI 辅助

我们给出了对3SUM和全对最短路径（APSP）的教科书算法的首个多项式改进：我们展示了如何在O(n^{1.9992})时间内确定性求解n个多项式大小整数的3SUM问题，以及在O(n^{2.9995})时间内求解具有多项式有界整数权重的有向n顶点图的APSP问题。这反驳了3SUM和APSP假设。利用已知的归约，我们还反驳了3SUM和APSP假设的实数值版本、精确三角形假设、零权重k-团假设，以及van den Brand、Nanongkai和Saranurak的三个矩形提示在线矩阵-向量猜想，并为各种其他问题给出了多项式加速。所有这些结果都源于一个用于薄矩阵乘积的新算法。设X为N×D整数矩阵，Y为D×N整数矩阵，其中D≤N^{1/18}，并设W为至多N^2/√D个位置的任意集合。我们计算条目(XY)[I,J]，(I,J)∈W，在O(N^2/D^{0.063})次操作内完成，这比写出XY或逐个计算N^2/√D个内积所需的时间多项式地更少。我们通过修改Coppersmith矩形矩阵乘法算法的一个变体来设计此算法，该变体基于Schönhage的十乘法恒等式构建，仅执行W中条目所需的操作，并证明所需操作很少。作为图算法解释，这以真正次二次时间解决了稀疏偏斜三部图上的全边稀疏三角形问题，其中两个部分各有n个顶点，但一个部分有n^ε个顶点，ε<0.12。通过已知归约，精确三角形，进而3SUM和APSP，归约到该问题。我们还给出了一个数据结构版本，用于回答XY单个条目的查询，这些查询事先未知。

英文摘要

We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n^{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N^{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N^2/D^{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Schönhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n^{\varepsilon}$ vertices for $\varepsilon<0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

发表机构

• Columbia University（哥伦比亚大学）
• MIT（麻省理工学院）

机构由 AI 辅助整理，请以论文原文为准。

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76 pages