Fillable arrays with constant time operations and a single bit of redundancy

加快速度，减少浪费：降低全索引词典的冗余度

More Haste, Less Waste: Lowering the Redundancy in Fully Indexable Dictionaries

我们考虑以压缩格式表示具有$n$ 1的$m$位的位向量$S$的问题，支持以下操作，其中$b \in \{0, 1 \}$：$rank_b(S,i)$返回位$b$在前缀$S[1..i]$中出现的次数; $select_b(S,i)$返回位$b$在$S$中第$i$次出现的位置。这种数据结构被称为\emph{fully indexable dictionary (FID)} [Raman等人，2007年]，并且至少与前任数据结构一样强大。我们的重点是在\textsc{ram}模型上的空间有效的FID，具有字大小$Θ(\lg m)$和所有操作的恒定时间，因此时间成本与输入大小无关。给定要编码的位串$S$，具有长度$m$并且包含$n$ 1，需要存储的最小信息量是$B(n,m) = \lceil \log {{m}\choose{n}} \rceil$。在[Patrascu，2008]中给出了为$S$构建FID的最新技术，使用$B(m,n)+O(m / ((\log m/ t) ^t)) + O(m^{3/4}) $位来支持$O(t)$时间中的操作。在这里，我们提出了一个参数化的数据结构，表现出时间/空间的权衡，这样，对于任何实常数$0 < δ\leq 1/2$，$0 < \eps \leq 1$，和整数$s > 0$，它使用\[ B(n,m) + O(n^{1+δ} + n (\frac{m…

Efficient Fully-Compressed Sequence Representations

We present a data structure that stores a sequence $s[1..n]$ over alphabet $[1..σ]$ in $n\Ho(s) + o(n)(\Ho(s){+}1)$ bits, where $\Ho(s)$ is the zero-order entropy of $s$. This structure supports the queries \access, \rank\ and \select, which are fundamental building blocks for many other compressed data structures, in worst-case time $\Oh{\lg\lgσ}$ and average time $\Oh{\lg \Ho(s)}$. The worst-case complexity matches the best previous results, yet these had been achieved with data structures using $n\Ho(s)+o(n\lgσ)$ bits. On highly compressible sequences the $o(n\lgσ)$ bits of the redundancy m…

实现小型动态秩/选择字典的最优冗余

Achieving Optimal Redundancy for Small Dynamic Rank/Select Dictionaries

本文针对动态秩/选择字典，在字RAM模型下提出一种达到最优时间且冗余为O(n log^(k) w)位的数据结构，并证明其最优性，同时扩展至完全可索引字典。

动态“Succincter”

Dynamic "Succincter"

本文将Patrascu的静态succincter技术动态化，设计出仅用三位冗余的动态aB树，并据此构造了更新与查询时间最优的动态完全可索引字典，显著降低了冗余空间。

Efficient Low-Redundancy Codes for Correcting Multiple Deletions

We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{Ω(1)}$. For any fixed $k$, we construct a binary code with $c_k \log n$ redundancy that can be decoded from $…

Disk-based genome sequencing data compression

Motivation: High-coverage sequencing data have significant, yet hard to exploit, redundancy. Most FASTQ compressors cannot efficiently compress the DNA stream of large datasets, since the redundancy between overlapping reads cannot be easily captured in the (relatively small) main memory. More interesting solutions for this problem are disk-based~(Yanovsky, 2011; Cox et al., 2012), where the better of these two, from Cox~{\it et al.}~(2012), is based on the Burrows--Wheeler transform (BWT) and achieves 0.518 bits per base for a 134.0 Gb human genome sequencing collection with almost 45-fold co…