Suffix Tree

SYNONYMS Compact suffix trie DEFINITION The suffix tree S(y) of a non-empty string y of length n is a compact trie representing all the suffixes of the string. The suffix tree of y is defined by the following properties: All branches of S(y) are labeled by all suffixes of y. • • Edges of S(y) are labeled by strings. • Internal nodes of S(y) have at least two children. • Edges outgoing an internal node are labeled by segments starting with different letters. • The segments are represented by their starting position on y and their lengths. Moreover, it is assumed that y ends with a symbol occurring nowhere else in it (the space sign is used in the examples of the present entry). This avoids marking nodes, and implies that S(y) has exactly n leaves (number of non-empty suffixes). All the properties then imply that the total size of S(y) is O(n), which makes it possible to design a linear-time construction of the suffix tree. HISTORICAL BACKGROUND The first linear time algorithm for building a suffix tree in from Weiner [15] but it requires quadratic space: O(n × σ) where σ is the size of the alphabet. The first linear time and space algorithm for building a suffix tree is from McCreight [13]. It works " off-line " ie it inserts the suffixes from the longest one to the shortest one. A strictly sequential version of the construction of the suffix tree was described by Ukkonen [14]. When the alphabet is potentially infinite, the construction algorithms of the suffix tree can be implemented to run in time O(n log σ) and then are optimal since they imply an ordering on the letters of the alphabet. On particular integer alphabets, Farach [5] showed that the construction can be done in linear time. The minimization of the suffix trie gives the suffix automaton. The suffix automaton of a string is also known under the name of DAWG, for Directed Acyclic Word Graph. Its linearity was discovered by Blumer et al. (see [1]), who gave a linear construction (on a fixed alphabet). The minimality of the structure as an automaton is from Crochemore [2] who showed how to build with the same complexity the factor automaton of a text The compaction of the suffix automaton gives the compact suffix automaton (see [1]), A direct construction algorithm of the compact suffix automaton was …