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

Abstract

The suffix array and its variants are text-indexing data structures that have become indispensable in the field of bioinformatics. With the uninitiated in mind, we provide an accessible exposition of the SA-IS algorithm, which is the state of the art in suffix array construction. We also describe DisLex, a technique that allows standard suffix array construction algorithms to create modified suffix arrays designed to enable a simple form of inexact matching needed to support 'spaced seeds' and 'subset seeds' used in many biological applications.

Keywords: linear-time algorithm; spaced seeds; subset seeds; suffix array construction; text index.

Figure 1:

A string (above) and its suffix array (shown vertically) along with the position index on the left and the corresponding suffixes to the right.

Figure 2:

Divide phase of the algorithm by Kim et al. Here, accca$. The set of sampled suffixes {accca$,cca$,a$}. ac; cc; a$. Since a$ < ac < cc, RANK()=1, RANK()=2, and RANK()=0. Therefore, 120.

Figure 3:

Divide Phase. (a) String T with its suffixes classified as , , . (b) Construction of reduced instance by lexical naming.

Figure 4:

Buckets of a DNA-string suffix array of length n. Gray indicates -type positions. The bucket for T does not have a subbucket for because there cannot be any suffix starting with the lexically greatest character of the alphabet.

Figure 5:

Array A at the end of Step 0 in which the suffixes have been placed in their buckets in sorted order. Gray indicates -type positions. This order of suffixes is obtained from recursion.

Figure 6:

Animation of Step 1 of the combine phase as the sweep proceeds from left to right. The original text T is also shown for reference. The symbols point to the current heads of -type subbuckets, the ∙ symbol shows the current position of the sweep and cells with thick boundaries indicate changes. For example, in the topmost row, suffix index 13 is encountered; and as is -type, 12 is inserted at A₅, the current head of the bucket for -type suffixes starting with g. The sweep proceeds accordingly. Whenever a pointer reaches the edge of its bucket, we change its representation to a dashed arrow. From sweep position 10 onwards, the array does not change and so this animation excludes those steps.

Figure 7:

Animation of Step 2 of the combine phase as the sweep proceeds from right to left. The original text T is also shown for reference. The symbols point to the current tails of -type subbuckets, the ∙ symbol shows the current position of the sweep, and cells with thick boundaries indicate changes. For example, in the topmost row, suffix index 0 is encountered, and therefore no action needs to be taken. Next, suffix index 2 is encountered; and as is -type, 1 is inserted at A₉, the current tail of the bucket for -type suffixes starting with g. Whenever a pointer reaches the edge of its bucket, we change its representation to a dashed arrow. From sweep position 2 onwards, the array does not change and so this animation excludes those steps.

Figure 8:

Time and memory performance of implementations of select suffix sorting algorithms. The shapes of the markers distinguish the different programs, and the fill-styles distinguish the data sets. The nonfilled markers connected by lines correspond to the performance for the four data sets constructed from increasingly many polymorphic copies of human chromosome 22.

Figure 9:

Contrasting the ordinary suffix array (left) of cagctat$ with its spaced suffix array under mask 101 (right). The characters at don’t-care positions have been replaced by *.

Figure 10:

A demonstration of how DisLex constructs a spaced suffix array using an example string atggacgacac$ and mask 101. The characters at the 0 positions of the mask have been mapped to the character . (a) The input string with extra padding. (b) Lexically sorting all the length-3 distinct substrings of T. The mapping RANK is defined using this ordering. (c), (d), (e) Constructing , and , respectively. (f) Constructing by concatenating , and . (g) The suffix array of (above) is transformed to the spaced suffix array of T (below).