关于置乱索引的困难性
On Hardness of Jumbled Indexing
摘要：
混乱索引是针对询问是否存在与表示为Parikh向量的模式匹配的$T$的子串的查询对文本$T$进行索引的问题，即，频率向量对每个字符计数。在过去的四年里，混乱的索引已经引起了人们的兴趣。有一种朴素算法在$O(n^2|Σ|)$时间内对所有答案进行预处理，允许之后进行快速查询，还有一种朴素算法不需要预处理，但有$O(n\log|Σ|)$查询时间。尽管付出了巨大的努力，但在这些运行时间上几乎没有改善。在本文中，我们提供了很好的理由。我们表明，在一个3SUM硬度的假设下，混乱索引的大小$ω(1)$字母需要$Ω(n^{2-ε})$预处理时间或$Ω(n^{1-δ})$查询时间的任何$ε,δ>0$。事实上，在更强的3SUM-硬度假设下，对于任何恒定字母大小$r\ge 3$，存在可描述的固定常数$ε_r$和$δ_r$，使得混乱索引需要$Ω(n^{2-ε_r})$预处理时间或$Ω(n^{1-δ_r})$查询时间。
英文摘要：
Jumbled indexing is the problem of indexing a text $T$ for queries that ask whether there is a substring of $T$ matching a pattern represented as a Parikh vector, i.e., the vector of frequency counts for each character. Jumbled indexing has garnered a lot of interest in the last four years. There is a naive algorithm that preprocesses all answers in $O(n^2|Σ|)$ time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has $O(n\log|Σ|)$ query time. Despite a tremendous amount of effort there has been little improvement over these running times. In this paper we provide good reason for this. We show that, under a 3SUM-hardness assumption, jumbled indexing for alphabets of size $ω(1)$ requires $Ω(n^{2-ε})$ preprocessing time or $Ω(n^{1-δ})$ query time for any $ε,δ>0$. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size $r\ge 3$ there exist describable fixed constant $ε_r$ and $δ_r$ such that jumbled indexing requires $Ω(n^{2-ε_r})$ preprocessing time or $Ω(n^{1-δ_r})$ query time.
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