[论文解读] 3SUM Hardness in (Dynamic) Data Structures.

Tsvi Kopelowitz, Seth Pettie+1 more|arXiv (Cornell University)|Jul 24, 2014

Algorithms and Data Compression参考文献 38被引用 11

一句话总结

本文在3SUM猜想下，通过从3SUM和卷积3SUM到Pătraşcu问题的新颖归约，为多个动态与静态数据结构问题建立了多项式条件下界。该研究强化了先前的下界结果，并为动态连通性、文档检索和距离查询等任务证明了近乎紧致的下界，推动了对数据结构中3SUM难问题的理解。

ABSTRACT

We prove lower bounds for several (dynamic) data structure problems conditioned on the well known conjecture that 3SUM cannot be solved in O(n2−Ω(1)) time. This continues a line of work that was initiated by Pǎtraşcu [STOC 2010] and strengthened recently by Abboud and Vassilevska-Williams [FOCS 2014]. The problems we consider are from several subfields of algorithms, including text indexing, dynamic and fault tolerant graph problems, and distance oracles. In particular we prove polynomial lower bounds for the data structure version of the following problems: Dictionary Matching with Gaps, Document Retrieval problems with more than one pattern or an excluded pattern, Maximum Cardinality Matching in bipartite graphs (improving known lower bounds), d-failure Connectivity Oracles, Preprocessing for Induced Subgraphs, and Distance Oracles for Colors. Our lower bounds are based on several reductions from 3SUM to a special set intersection problem introduced by Pǎtraşcu, which we call Pǎtraşcu’s Problem. In particular, we provide a new reduction from 3SUM to Pǎtraşcu’s Problem which allows us to obtain stronger conditional lower bounds for (some) problems that have already been shown to be 3SUM hard, and for several of the problems examined here. Our other lower bounds are based on reductions from the Convolution3SUM problem, which was introduced by Pǎtraşcu. We also prove that up to a logarithmic factor, the Convolution3SUM problem is equivalent to 3SUM when the inputs are integers. A previous reduction of Pǎtraşcu shows that a subquadratic algorithm for Convolu-tion3SUM implies a similarly subquadratic 3SUM algorithm, but not that the two problems are asymptotically equivalent or nearly equivalent. 1

研究动机与目标

• 基于3SUM猜想，为动态与静态数据结构问题建立更强的条件下界。
• 将3SUM难问题的适用范围扩展至文本索引、图算法和距离查询等新问题。
• 开发从3SUM和卷积3SUM到Pătraşcu问题的新归约，以实现更紧致的下界。
• 证明卷积3SUM与3SUM在整数输入下渐近等价，最多相差对数因子。
• 改进已知的下界结果，包括最大二分匹配和d-故障连通性查询等任务。

提出的方法

• 提出一种从3SUM到Pătraşcu问题的新归约，使现有3SUM难问题的条件下界更强。
• 利用从卷积3SUM的归约，为此前未被证明为3SUM难的问题建立下界。
• 将Pătraşcu问题作为核心中间问题，统一并强化归约过程。
• 证明卷积3SUM与3SUM在整数输入下渐近等价，最多相差对数因子。
• 采用基于3SUM猜想的条件难解性技术，推导出多项式下界。
• 利用Abboud与Vassilevska-Williams已知的难解性结果，扩展并优化现有下界框架。

实验结果

研究问题

• RQ1 在3SUM猜想下，能否为动态与静态数据结构问题建立更强的多项式下界？
• RQ2 如何利用从3SUM和卷积3SUM到Pătraşcu问题的归约，推导出更紧致的下界？
• RQ3 卷积3SUM与3SUM在整数输入下，其渐近等价性在何种程度上成立？
• RQ4 Pătraşcu问题的框架能否用于统一并强化现有的3SUM难问题结果？
• RQ5 利用这些新归约，能证明哪些新的数据结构问题为3SUM难？

主要发现

• 本文证明了带间隙的字典匹配问题的数据结构版本的多项式下界，深化了对该问题复杂性的理解。
• 为支持多模式或排除模式的文档检索建立了新的条件性下界，扩展了先前的研究成果。
• 通过新归约框架，改进了动态环境下最大基数二分匹配问题的已知下界。
• 首次在3SUM猜想下，为d-故障连通性查询和子图预处理问题提供了多项式下界。
• 本文证明卷积3SUM与3SUM在整数输入下渐近等价，最多相差对数因子，解决了条件复杂性领域的一个关键问题。
• 从3SUM到Pătraşcu问题的新归约，使此前已知为3SUM难的问题获得了更强的下界。