Practical linear-time O (1)-workspace suffix sorting for constant alphabets

Abstract

This article presents an O ( n )-time algorithm called SACA-K for sorting the suffixes of an input string T [0, n -1] over an alphabet A [0, K -1]. The problem of sorting the suffixes of T is also known as constructing the suffix array (SA) for T . The theoretical memory usage of SACA-K is n log K + n log n + K log n bits. Moreover, we also have a practical implementation for SACA-K that uses n bytes + ( n + 256) words and is suitable for strings over any alphabet up to full ASCII, where a word is log n bits. In our experiment, SACA-K outperforms SA-IS that was previously the most time- and space-efficient linear-time SA construction algorithm (SACA). SACA-K is around 33% faster and uses a smaller deterministic workspace of K words, where the workspace is the space needed beyond the input string and the output SA. Given K = O (1), SACA-K runs in linear time and O (1) workspace. To the best of our knowledge, such a result is the first reported in the literature with a practical source code publicly available.

Induced Sorting Suffixes in External Memory

“…The time complexity of induced sorting is linear as given by the recursive formula T (n) = T (n/2) + O(n) = O(n). The induced sorting method also facilitates time-and space-efficient designs in practice, for example, SA-IS [Nong et al 2011], its optimized implementation [Mori 2008], and a recent improvement of SA-IS called SACA-K, which uses only a σ log n-bit workspace beyond the input string and the output suffix array [Nong 2013]. …”

Inducing the Lyndon Array

“…The type of each suffix can be computed with a right-to-left scanning of T [18], or otherwise it can be computed on-the-fly in constant time during Nong's algorithm [17,Section 3]. By extension, the type of each symbol in T can be classified according to the type of the suffix starting with such symbol.…”

“…Induced sorting was introduced by Itoh and Tanaka [10] and later refined by Ko and Aluru [11] and by Nong et al [19,18]. In 2013, Nong [17] proposed a space efficient linear time algorithm based on induced sorting, called SACA-K, which uses only σ + O(1) words of working space, where σ is the alphabet size and the working space is the space used in addition to the input and the output. Since a small working space is a very desirable feature, there have been many algorithms adapting induced suffix sorting to the computation of data structures related to the suffix array, such as the Burrows-Wheeler transform [21], the Φ-array [8], the LCP array [4,14], and the document array [13].…”

A bioinformatician's guide to the forefront of suffix array construction algorithms

“…We also tested SACA-K [26], a recently published memory-efficient successor of SA-IS and included an implementation of the Deep-Shallow algorithm [40], which is theoretically not a linear-time algorithm, but has been shown to be fast and lightweight in practice [22]. For each program, we used the default parameter settings.…”