Suffix tree

Suffix tree for the string BANANA. Each substring is terminated with special character $. The six paths from the root to a leaf (shown as boxes) correspond to the six suffixes A$, NA$, ANA$, NANA$, ANANA$ and BANANA$. The numbers in the boxes give the start position of the corresponding suffix. Suffix links drawn dashed.

In computer science, a suffix tree (also called PAT tree or, in an earlier form, position tree) is a data structure that presents the suffixes of a given string in a way that allows for a particularly fast implementation of many important string operations.

The suffix tree for a string $S$ is a tree whose edges are labeled with strings, such that each suffix of $S$ corresponds to exactly one path from the tree's root to a leaf. It is thus a radix tree (more specifically, a Patricia tree) for the suffixes of $S$.

Constructing such a tree for the string $S$ takes time and space linear in the length of $S$. Once constructed, several operations can be performed quickly, for instance locating a substring in $S$, locating a substring if a certain number of mistakes are allowed, locating matches for a regular expression pattern etc. Suffix trees also provided one of the first linear-time solutions for the longest common substring problem. These speedups come at a cost: storing a string's suffix tree typically requires significantly more space than storing the string itself.

History

The concept was first introduced as a position tree by Weiner in 1973, which Donald Knuth subsequently characterized as "Algorithm of the Year 1973". The construction was greatly simplified by McCreight in 1976 , and also by Ukkonen in 1995. Ukkonen provided the first linear-time online-construction of suffix trees, now known as Ukkonen's algorithm. These algorithms are all linear-time for constant-size alphabet, and have worst-case running time of $O(n\log n)$ in general.

In 1997, Farach gave the first suffix tree construction algorithm that is optimal for all alphabets. In particular, this is the first linear-time algorithm for strings drawn from an alphabet of integers in a polynomial range. This latter algorithm has become the basis for new algorithms for constructing both suffix trees and suffix arrays, for example, in external memory, compressed, succinct, etc.

Definition

The suffix tree for the string $S$ of length $n$ is defined as a tree such that ( page 90):

• the paths from the root to the leaves have a one-to-one relationship with the suffixes of $S$,
• edges spell non-empty strings,
• and all internal nodes (except perhaps the root) have at least two children.

Since such a tree does not exist for all strings, $S$ is padded with a terminal symbol not seen in the string (usually denoted $). This ensures that no suffix is a prefix of another, and that there will be $n$ leaf nodes, one for each of the $n$ suffixes of $S$. Since all internal non-root nodes are branching, there can be at most n −  1 such nodes, and n + (n − 1) + 1 = 2n nodes in total (n leaves, n − 1 internal nodes, 1 root).

Suffix links are a key feature for older linear-time construction algorithms, although most newer algorithms, which are based on Farach's algorithm, dispense with suffix links. In a complete suffix tree, all internal non-root nodes have a suffix link to another internal node. If the path from the root to a node spells the string $\chi \alpha$, where $\chi$ is a single character and $\alpha$ is a string (possibly empty), it has a suffix link to the internal node representing $\alpha$. See for example the suffix link from the node for ANA to the node for NA in the figure above. Suffix links are also used in some algorithms running on the tree.

Functionality

A suffix tree for a string $S$ of length $n$ can be built in $\Theta (n)$ time, if the letters come from an alphabet of integers in a polynomial range (in particular, this is true for constant-sized alphabets). For larger alphabets, the running time is dominated by first sorting the letters to bring them into a range of size $O(n)$; in general, this takes $O(n\log n)$ time. The costs below are given under the assumption that the alphabet is constant.

Assume that a suffix tree has been built for the string $S$ of length $n$, or that a generalised suffix tree has been built for the set of strings $D=\{S_{1},S_{2},\dots ,S_{K}\}$ of total length $n=|n_{1}|+|n_{2}|+\cdots +|n_{K}|$. You can:

• Search for strings:
  • Check if a string $P$ of length $m$ is a substring in $O(m)$ time ( page 92).
  • Find the first occurrence of the patterns $P_{1},\dots ,P_{q}$ of total length $m$ as substrings in $O(m)$ time.
  • Find all $z$ occurrences of the patterns $P_{1},\dots ,P_{q}$ of total length $m$ as substrings in $O(m+z)$ time ( page 123).
  • Search for a regular expression P in time expected sublinear in $n$ ().
  • Find for each suffix of a pattern $P$, the length of the longest match between a prefix of $P[i\dots m]$ and a substring in $D$ in $\Theta (m)$ time ( page 132). This is termed the matching statistics for $P$.
• Find properties of the strings:
  • Find the longest common substrings of the string $S_{i}$ and $S_{j}$ in $\Theta (n_{i}+n_{j})$ time ( page 125).
  • Find all maximal pairs, maximal repeats or supermaximal repeats in $\Theta (n+z)$ time ( page 144).
  • Find the Lempel–Ziv decomposition in $\Theta (n)$ time ( page 166).
  • Find the longest repeated substrings in $\Theta (n)$ time.
  • Find the most frequently occurring substrings of a minimum length in $\Theta (n)$ time.
  • Find the shortest strings from $\Sigma$ that do not occur in $D$, in $O(n+z)$ time, if there are $z$ such strings.
  • Find the shortest substrings occurring only once in $\Theta (n)$ time.
  • Find, for each $i$, the shortest substrings of $S_{i}$ not occurring elsewhere in $D$ in $\Theta (n)$ time.

The suffix tree can be prepared for constant time lowest common ancestor retrieval between nodes in $\Theta (n)$ time ( chapter 8). You can then also:

• Find the longest common prefix between the suffixes $S_{i}[p..n_{i}]$ and $S_{j}[q..n_{j}]$ in $\Theta (1)$ ( page 196).
• Search for a pattern P of length m with at most k mismatches in $O(kn+z)$ time, where z is the number of hits ( page 200).
• Find all $z$ maximal palindromes in $\Theta (n)$( page 198), or $\Theta (gn)$ time if gaps of length $g$ are allowed, or $\Theta (kn)$ if $k$ mismatches are allowed ( page 201).
• Find all $z$ tandem repeats in $O(n\log n+z)$, and k-mismatch tandem repeats in $O(kn\log(n/k)+z)$ ( page 204).
• Find the longest substrings common to at least $k$ strings in $D$ for $k=2,\dots ,K$ in $\Theta (n)$ time ( page 205).

Applications

Suffix trees can be used to solve a large number of string problems that occur in text-editing, free-text search, computational biology and other application areas. Primary applications include:

• String search, in O(m) complexity, where m is the length of the sub-string (but with initial O(n) time required to build the suffix tree for the string)
• Finding the longest repeated substring
• Finding the longest common substring
• Finding the longest palindrome in a string

Suffix trees are often used in bioinformatics applications, searching for patterns in DNA or protein sequences (which can be viewed as long strings of characters). The ability to search efficiently with mismatches might be considered their greatest strength. Suffix trees are also used in data compression; they can be used to find repeated data, and can be used for the sorting stage of the Burrows–Wheeler transform. Variants of the LZW compression schemes use suffix trees (LZSS). A suffix tree is also used in suffix tree clustering, a data clustering algorithm used in some search engines (first introduced in ).

Implementation

If each node and edge can be represented in $\Theta (1)$ space, the entire tree can be represented in $\Theta (n)$ space. The total length of all the strings on all of the edges in the tree is $O(n^{2})$, but each edge can be stored as the position and length of a substring of S, giving a total space usage of $\Theta (n)$ computer words. The worst-case space usage of a suffix tree is seen with a fibonacci word, giving the full $2n$ nodes.

An important choice when making a suffix tree implementation is the parent-child relationships between nodes. The most common is using linked lists called sibling lists. Each node has a pointer to its first child, and to the next node in the child list it is a part of. Hash maps, sorted/unsorted arrays (with array doubling), and balanced search trees may also be used, giving different running time properties. We are interested in:

• The cost of finding the child on a given character.
• The cost of inserting a child.
• The cost of enlisting all children of a node (divided by the number of children in the table below).

Let $\sigma$ be the size of the alphabet. Then you have the following costs:

	Lookup	Insertion	Traversal
Sibling lists / unsorted arrays	$O(\sigma )$	$\Theta (1)$	$\Theta (1)$
Hash maps	$\Theta (1)$	$\Theta (1)$	$O(\sigma )$
Balanced search tree	$O(\log \sigma )$	$O(\log \sigma )$	$O(1)$
Sorted arrays	$O(\log \sigma )$	$O(\sigma )$	$O(1)$
Hash maps + sibling lists	$O(1)$	$O(1)$	$O(1)$

Note that the insertion cost is amortised, and that the costs for hashing are given perfect hashing.

The large amount of information in each edge and node makes the suffix tree very expensive, consuming about ten to twenty times the memory size of the source text in good implementations. The suffix array reduces this requirement to a factor of four, and researchers have continued to find smaller indexing structures.

External construction

Suffix trees quickly outgrow the main memory on standard machines for sequence collections in the order of gigabytes. As such, their construction calls for external memory approaches.

There are theoretical results for constructing suffix trees in external memory. The algorithm by Farach et al. is theoretically optimal, with an I/O complexity equal to that of sorting. However, as discussed for example in , the overall intricacy of this algorithm has prevented, so far, its practical implementation.

On the other hand, there have been practical works for constructing disk-based suffix trees which scale to (few) GB/hours. The state of the art methods are TDD , TRELLIS , DiGeST , and B²ST .

TDD and TRELLIS scale up to the entire human genome – approximately 3GB – resulting in a disk-based suffix tree of a size in the tens of gigabytes ,. However, these methods cannot handle efficiently collections of sequences exceeding 3GB . DiGeST performs significantly better and is able to handle collections of sequences in the order of 6GB in about 6 hours . The source code and documentation for the latter is available from . All these methods can efficiently build suffix trees for the case when the tree does not fit in main memory, but the input does. The most recent method, B²ST , scales to handle inputs that do not fit in main memory.

See also

• Suffix array
• Generalised suffix tree

External links

• Suffix Trees by Dr. Sartaj Sahni (CISE Department Chair at University of Florida)
• Suffix Trees by Lloyd Allison
• NIST's Dictionary of Algorithms and Data Structures: Suffix Tree
• suffix_tree ANSI C implementation of a Suffix Tree
• libstree, a generic suffix tree library written in C
• Tree::Suffix, a Perl binding to libstree
• Strmat a faster generic suffix tree library written in C (uses arrays instead of linked lists)
• SuffixTree a Python binding to Strmat
• Universal Data Compression Based on the Burrows-Wheeler Transformation: Theory and Practice, application of suffix trees in the BWT
• Theory and Practice of Succinct Data Structures, C++ implementation of a compressed suffix tree]
• Practical Algorithm Template Library, a C++ library with suffix tree implementation on PATRICIA trie, by Roman S. Klyujkov
• A Java implementation