Suffix automaton

In computer science, a suffix automaton is an efficient data structure for representing the substring index of a given string which allows the storage, processing, and retrieval of compressed information about all its substrings. The suffix automaton of a string is the smallest directed acyclic graph with a dedicated initial vertex and a set of "final" vertices, such that paths from the initial vertex to final vertices represent the suffixes of the string. Formally speaking, a suffix automaton is defined by the following set of properties:

  1. Its arcs are tagged with letters;
  2. none of its nodes have two outgoing arcs tagged with the same letter;
  3. for every suffix of there exists a path from initial vertex to some final vertex such that the concatenation of letters on the path forms this suffix;
  4. it has the least number of vertices among all graphs defined by the properties above.
Suffix automaton
Suffix automaton bold.svg
TypeSubstring index
Invented1983
Invented byAnselm Blumer; Janet Blumer; Andrzej Ehrenfeucht; David Haussler; Ross McConnell
Time complexity in big O notation
Algorithm Average Worst case
Space

Speaking in the terms of automata theory, a suffix automaton is the minimal partial deterministic finite automaton that recognizes the set of suffixes of a given string . The state graph of a suffix automaton is called a directed acyclic word graph (DAWG), a term that is also sometimes used for any deterministic acyclic finite state automaton.

Suffix automata were introduced in 1983 by a group of scientists from the University of Denver and the University of Colorado Boulder. They suggested a linear time online algorithm for its construction and showed that the suffix automaton of a string having length at least two characters has at most states and at most transitions. Further works have shown a close connection between suffix automata and suffix trees, and have outlined several generalizations of suffix automata, such as compacted suffix automaton obtained by compression of nodes with a single outgoing arc.

Suffix automata provide efficient solutions to problems such as substring search and computation of the largest common substring of two and more strings.

HistoryEdit

 
Anselm Blumer with a drawing of generalized CDAWG for strings ababc and abcab

The concept of suffix automaton was introduced in 1983[1] by a group of scientists from University of Denver and University of Colorado Boulder consisting of Anselm Blumer, Janet Blumer, Andrzej Ehrenfeucht, David Haussler and Ross McConnell, although similar concepts had earlier been studied alongside suffix trees in the works of Peter Weiner,[2] Vaughan Pratt[3] and Anatol Slissenko.[4] In their initial work, Blumer et al. showed a suffix automaton built for the string   of length greater than   has at most   states and at most   transitions, and suggested a linear algorithm for automaton construction.[5]

In 1983, Mu-Tian Chen and Joel Seiferas independently showed that Weiner's 1973 suffix-tree construction algorithm[2] while building a suffix tree of the string   constructs a suffix automaton of the reversed string   as an auxiliary structure.[6] In 1987, Blumer et al. applied the compressing technique used in suffix trees to a suffix automaton and invented the compacted suffix automaton, which is also called the compacted directed acyclic word graph (CDAWG).[7] In 1997, Maxime Crochemore and Renaud Vérin developed a linear algorithm for direct CDAWG construction.[1] In 2001, Shunsuke Inenaga et al. developed an algorithm for construction of CDAWG for a set of words given by a trie.[8]

DefinitionsEdit

Usually when speaking about suffix automata and related concepts, some notions from formal language theory and automata theory are used, in particular:[9]

  • "Alphabet" is a finite set   that is used to construct words. Its elements are called "characters";
  • "Word" is a finite sequence of characters  . "Length" of the word   is denoted as  ;
  • "Formal language" is a set of words over given alphabet;
  • "Language of all words" is denoted as  (where the "*" character stands for Kleene star), "empty word" (the word of zero length) is denoted by the character  ;
  • "Concatenation of words"   and   is denoted as   or   and corresponds to the word obtained by writing   to the right of  , that is,  ;
  • "Concatenation of languages"   and   is denoted as   or   and corresponds to the set of pairwise concatenations  ;
  • If the word   may be represented as  , where  , then words  ,   and   are called "prefix", "suffix" and "subword" (substring) of the word   correspondingly;
  • If   then   is said to "occur" in   as a subword. Here   and   are called left and right positions of occurrence of   in   correspondingly.

Automaton structureEdit

Formally, deterministic finite automaton is determined by 5-tuple  , where:[10]

  •   is an "alphabet" that is used to construct words,
  •   is a set of automaton "states",
  •   is an "initial" state of automaton,
  •   is a set of "final" states of automaton,
  •   is a partial "transition" function of automaton, such that   for   and   is either undefined or defines a transition from   over character  .

Most commonly, deterministic finite automaton is represented as a directed graph ("diagram") such that:[10]

  • Set of graph vertices corresponds to the state of states  ,
  • Graph has a specific marked vertex corresponding to initial state  ,
  • Graph has several marked vertices corresponding to the set of final states  ,
  • Set of graph arcs corresponds to the set of transitions  ,
  • Specifically, every transition   is represented by an arc from   to   marked with the character  . This transition also may be denoted as  .

In terms of its diagram, the automaton recognizes the word   only if there is a path from the initial vertex   to some final vertex   such that concatenation of characters on this path forms  . The set of words recognized by an automaton forms a language that is set to be recognized by the automaton. In these terms, the language recognized by a suffix automaton of   is the language of its (possibly empty) suffixes.[9]

Automaton statesEdit

"Right context" of the word   with respect to language   is a set   that is a set of words   such that their concatenation with   forms a word from  . Right contexts induce a natural equivalence relation   on the set of all words. If language   is recognized by some deterministic finite automaton, there exists unique up to isomorphism automaton that recognizes the same language and has the minimum possible number of states. Such an automaton is called a minimal automaton for the given language  . Myhill–Nerode theorem allows it to define it explicitly in terms of right contexts:[11][12]

Theorem — Minimal automaton recognizing language   over the alphabet   may be explicitly defined in the following way:

  • Alphabet   stays the same,
  • States   correspond to right contexts   of all possible words  ,
  • Initial state   corresponds to the right context of the empty word  ,
  • Final states   correspond to right contexts   of words from  ,
  • Transitions   are given by  , where   and  .

In these terms, "suffix automaton" is the minimal deterministic finite automaton recognizing the language of suffixes of the word  . Right context of the word   with respect to this language constitutes of words  , such that   is a suffix of  . It allows one to formulate the following lemma defining bijection between right context of the word and the set of right positions of its occurrences in  :[13][14]

Theorem — Let   be the set of right positions of occurrences of   in  .

There is a following bijection between   and  :

  • If  , then  ;
  • If  , then  .

For example, for the word   and its subword  , it holds   and  . Informally,   is formed by words that follow occurrences of   to the end of   and   is formed by right positions of those occurrences. In this example, the element   corresponds with the word   while the word   corresponds with the element  .

It implies several structure properties of suffix automaton states. Let  , then:[14]

  • If   and   have at least one common element  , then   and   have a common element as well. It implies   is a suffix of   and therefore   and  . In aforementioned example,  , so   is a suffix of   and thus   and  ;
  • If  , then  , thus   occurs in   only as a suffix of  . For example, for   and   it holds that   and  ;
  • If   and   is a suffix of   such that  , then  . In the example above   and it holds for "intermediate" suffix   that  .

Any state   of the suffix automaton recognizes some continuous chain of nested suffixes of the longest word recognized by this state.[14]

"Left extension"   of the string   is the longest string   that has the same right context as  . Length   of the longest string recognized by   is denoted by  . It holds:[15]

Theorem — Left extension of   may be represented as  , where   is the longest word such that any occurrence of   in   is preceded by  .

"Suffix link"   of the state   is the pointer to the state   that contains the largest suffix of   that is not recognized by  .

In this terms it can be said   recognizes exactly all suffixes of   that is longer than   and not longer than  . It also holds:[15]

Theorem — Suffix links form a tree   that may be defined explicitly in the following way:

  1. Vertices   of the tree correspond to left extensions   of all   substrings,
  2. Edges   of the tree connect pairs of vertices  , such that   and  .

Connection with suffix treesEdit

 
Relationship of the suffix trie, suffix tree, DAWG and CDAWG

A "prefix tree" (or "trie") is a rooted directed tree in which arcs are marked by characters in such a way no vertex   of such tree has two out-going arcs marked with the same character. Some vertices in trie are marked as final. Trie is said to recognize a set of words defined by paths from its root to final vertices. In this way prefix trees are a special kind of deterministic finite automata if you perceive its root as an initial vertex.[16] The "suffix trie" of the word   is a prefix tree recognizing a set of its suffixes. "A suffix tree" is a tree obtained from a suffix trie via the compaction procedure, during which consequent edges are merged if the degree of the vertex between them is equal to two.[15]

By its definition, a suffix automaton can be obtained via minimization of the suffix trie. It may be shown that a compacted suffix automaton is obtained by both minimization of the suffix tree (if one assumes each string on the edge of the suffix tree is a solid character from the alphabet) and compaction of the suffix automaton.[17] Besides this connection between the suffix tree and the suffix automaton of the same string there is as well a connection between the suffix automaton of the string   and the suffix tree of the reversed string  .[18]

Similarly to right contexts one may introduce "left contexts"  , "right extensions"   corresponding to the longest string having same left context as   and the equivalence relation  . If one considers right extensions with respect to the language   of "prefixes" of the string   it may be obtained:[15]

Theorem — Suffix tree of the string   may be defined explicitly in the following way:

  • Vertices   of the tree correspond to right extensions   of all   substrings,
  • Edges   correspond to triplets   such that   and  .

Here triplet   means there is an edge from   to   with the string   written on it

, which implies the suffix link tree of the string   and the suffix tree of the string   are isomorphic:[18]

Suffix structures of words "abbcbc" and "cbcbba" 

Similarly to the case of left extensions, the following lemma holds for right extensions:[15]

Theorem — Right extension of the string   may be represented as  , where   is the longest word such that every occurrence of   in   is succeeded by  .

SizeEdit

A suffix automaton of the string   of length   has at most   states and at most   transitions. These bounds are reached on strings   and   correspondingly.[13] This may be formulated in a stricter way as   where   and   are the numbers of transitions and states in automaton correspondingly.[14]

Maximal suffix automata

ConstructionEdit

Initially the automaton only consists of a single state corresponding to the empty word, then characters of the string are added one by one and the automaton is rebuilt on each step incrementally.[19]

State updatesEdit

After a new character is appended to the string, some equivalence classes are altered. Let   be the right context of   with respect to the language of   suffixes. Then the transition from   to   after   is appended to   is defined by lemma:[14]

Theorem — Let   be some words over   and   be some character from this alphabet. Then there is a following correspondence between   and  :

  •   if   is a suffix of  ;
  •   otherwise.

After adding   to the current word   the right context of   may change significantly only if   is a suffix of  . It implies equivalence relation   is a refinement of  . In other words, if  , then  . After the addition of a new character at most two equivalence classes of   will be split and each of them may split in at most two new classes. First, equivalence class corresponding to empty right context is always split into two equivalence classes, one of them corresponding to   itself and having   as a right context. This new equivalence class contains exactly   and all its suffixes that did not occur in  , as the right context of such words was empty before and contains only empty word now.[14]

Given the correspondence between states of the suffix automaton and vertices of the suffix tree, it is possible to find out the second state that may possibly split after a new character is appended. The transition from   to   corresponds to the transition from   to   in the reversed string. In terms of suffix trees it corresponds to the insertion of the new longest suffix   into the suffix tree of  . At most two new vertices may be formed after this insertion: one of them corresponding to  , while the other one corresponds to its direct ancestor if there was a branching. Returning to suffix automata, it means the first new state recognizes   and the second one (if there is a second new state) is its suffix link. It may be stated as a lemma:[14]

Theorem — Let  ,   be some word and character over  . Also let   be the longest suffix of  , which occurs in  , and let  . Then for any substrings   of   it holds:

  • If   and  , then  ;
  • If   and  , then  ;
  • If   and  , then  .

It implies that if   (for example, when   didn't occur in   at all and  ), then only the equivalence class corresponding to the empty right context is split.[14]

Besides suffix links it is also needed to define final states of the automaton. It follows from structure properties that all suffixes of a word   recognized by   are recognized by some vertex on suffix path   of  . Namely, suffixes with length greater than   lie in  , suffixes with length greater than   but not greater than   lie in   and so on. Thus if the state recognizing   is denoted by  , then all final states (that is, recognizing suffixes of  ) form up the sequence  .[19]

Transitions and suffix links updatesEdit

After the character   is appended to   possible new states of suffix automaton are   and  . Suffix link from   goes to   and from   it goes to  . Words from   occur in   only as its suffixes therefore there should be no transitions at all from   while transitions to it should go from suffixes of   having length at least   and be marked with the character  . State   is formed by subset of  , thus transitions from   should be same as from  . Meanwhile, transitions leading to   should go from suffixes of   having length less than   and at least  , as such transitions have lead to   before and corresponded to seceded part of this state. States corresponding to these suffixes may be determined via traversal of suffix link path for  .[19]

Construction of the suffix automaton for the word abbcbc 
∅ → a
   
After first character is appended, only one state is created in suffix automaton. Similarly, only one leaf is added to the suffix tree.
a → ab
   
New transitions are drawn from all previous final states as b didn't appear before. For the same reason another leaf is added to the root of the suffix tree.
ab → abb
   
The state 2 recognizes words ab and b, but only b is the new suffix, therefore this word is separated into state 4. In the suffix tree it corresponds to the split of the edge leading to the vertex 2.
abb → abbc
   
Character c occurs for the first time, so transitions are drawn from all previous final states. Suffix tree of reverse string has another leaf added to the root.
abbc → abbcb
   
State 4 consists of the only word b, which is suffix, thus the state is not split. Correspondingly, new leaf is hanged on the vertex 4 in the suffix tree.
abbcb → abbcbc
   
The state 5 recognizes words abbc, bbc, bc and c, but only last two are suffixes of new word, so they're separated into new state 8. Correspondingly, edge leading to the vertex 5 is split and vertex 8 is put in the middle of the edge.

Construction algorithmEdit

Theoretical results above lead to the following algorithm that takes character   and rebuilds the suffix automaton of   into the suffix automaton of  :[19]

  1. The state corresponding to the word   is kept as  ;
  2. After   is appended, previous value of   is stored in the variable   and   itself is reassigned to the new state corresponding to  ;
  3. States corresponding to suffixes of   are updated with transitions to  . To do this one should go through  , until there is a state that already has a transition by  ;
  4. Once the aforementioned loop is over, there are 3 cases:
    1. If none of states on the suffix path had a transition by  , then   never occurred in   before and the suffix link from   should lead to  ;
    2. If the transition by   is found and leads from the state   to the state  , such that  , then   does not have to be split and it is a suffix link of  ;
    3. If the transition is found but  , then words from   having length at most   should be segregated into new "clone" state  ;
  5. If the previous step was concluded with the creation of  , transitions from it and its suffix link should copy those of  , at the same time   is assigned to be common suffix link of both   and  ;
  6. Transitions that have lead to   before but corresponded to words of the length at most   are redirected to  . To do this, one continues going through the suffix path of   until the state is found such that transition by   from it doesn't lead to  .

The whole procedure is described by the following pseudo-code:[19]

function add_letter(x):
    define p = last
    assign last = new_state()
    assign len(last) = len(p) + 1
    while δ(p, x) is undefined:
        assign δ(p, x) = last, p = link(p)
    define q = δ(p, x)
    if q = last:
        assign link(last) = q0
    else if len(q) = len(p) + 1:
        assign link(last) = q
    else:
        define cl = new_state()
        assign len(cl) = len(p) + 1
        assign δ(cl) = δ(q), link(cl) = link(q)
        assign link(last) = link(q) = cl
        while δ(p, x) = q:
            assign δ(p, x) = cl, p = link(p)

Here   is the initial state of the automaton and   is a function creating new state for it. It is assumed  ,  ,   and   are stored as global variables.[19]

ComplexityEdit

Complexity of the algorithm may vary depending on the underlying structure used to store transitions of the automaton. It may be implemented in   with   memory overhead or in   with   memory overhead if one assumes that memory allocation is done in  . To obtain such complexity, one has to use the methods of amortized analysis. The value of   strictly reduces with each iteration of the cycle while it may only increase by as much as one after the first iteration of the cycle on the next add_letter call. Overall value of   never exceeds   and it is only increased by one between iterations of appending new letters that suggest total complexity is at most linear as well. The linearity of the second cycle is shown in a similar way.[19]

GeneralizationsEdit

The suffix automaton is closely related to other suffix structures and substring indices. Given a suffix automaton of a specific string one may construct its suffix tree via compacting and recursive traversal in linear time.[20] Similar transforms are possible in both directions to switch between the suffix automaton of   and the suffix tree of reversed string  .[18] Other than this several generalizations were developed to construct an automaton for the set of strings given by trie,[8] compacted suffix automation (CDAWG),[7] to maintain the structure of the automaton on the sliding window,[21] and to construct it in a bidirectional way, supporting the insertion of a characters to both the beginning and the end of the string.[22]

Compacted suffix automatonEdit

As was already mentioned above, a compacted suffix automaton is obtained via both compaction of a regular suffix automaton (by removing states which are non-final and have exactly one out-going arc) and the minimization of a suffix tree. Similarly to the regular suffix automaton, states of compacted suffix automaton may be defined in explicit manner. A two-way extension   of a word   is the longest word  , such that every occurrence of   in   is preceded by   and succeeded by  . In terms of left and right extensions it means that two-way extension is the left extension of the right extension or, which is equivalent, the right extension of the left extension, that is  . In terms of two-way extensions compacted automaton is defined as follows:[15]

Theorem — Compacted suffix automaton of the word   is defined by a pair  , where:

  •   is a set of automaton states;
  •   is a set of automaton transitions.

Two-way extensions induce an equivalence relation   which defines the set of words recognized by the same state of compacted automaton. This equivalence relation is a transitive closure of the relation defined by  , which highlights the fact that a compacted automaton may be obtained by both gluing suffix tree vertices equivalent via   relation (minimization of the suffix tree) and gluing suffix automaton states equivalent via   relation (compaction of suffix automaton).[23] If words   and   have same right extensions, and words   and   have same left extensions, then cumulatively all strings  ,   and   have same two-way extensions. At the same time it may happen that neither left nor right extensions of   and   coincide. As an example one may take  ,   and  , for which left and right extensions are as follows:  , but   and  . That being said, while equivalence relations of one-way extensions were formed by some continuous chain of nested prefixes or suffixes, bidirectional extensions equivalence relations are more complex and the only thing one may conclude for sure is that strings with the same two-way extension are substrings of the longest string having the same two-way extension, but it may even happen that they don't have any non-empty substring in common. The total number of equivalence classes for this relation does not exceed   which implies that compacted suffix automaton of the string having length   has at most   states. The amount of transitions in such automaton is at most  .[15]

Suffix automaton of several stringsEdit

Consider a set of words  . It is possible to construct a generalization of suffix automaton that would recognize the language formed up by suffixes of all words from the set. Constraints for the number of states and transitions in such automaton would stay the same as for a single-word automaton if you put  .[23] The algorithm is similar to the construction of single-word automaton except instead of   state, function add_letter would work with the state corresponding to the word   assuming the transition from the set of words   to the set  .[24][25]

This idea is further generalized to the case when   is not given explicitly but instead is given by a prefix tree with   vertices. Mohri et al. showed such an automaton would have at most   and may be constructed in linear time from its size. At the same time, the number of transitions in such automaton may reach  , for example for the set of words   over the alphabet   the total length of workds is equal to  , the number of vertices in corresponding suffix trie is equal to   and corresponding suffix automaton is formed of   states and   transitions. Algorithm suggested by Mohri mainly repeats the generic algorithm for building automaton of several strings but instead of growing words one by one, it traverses the trie in a breadth-first search order and append new characters as it meet them in the traversal, which guarantees amortized linear complexity.[26]

Sliding windowEdit

Some compression algorithms, such as LZ77 and RLE may benefit from storing suffix automaton or similar structure not for the whole string but for only last   its characters while the string is updated. This is because compressing data is usually expressively large and using   memory is undesirable. In 1985, Janet Blumer developed an algorithm to maintain a suffix automaton on a sliding window of size   in   worst-case and   on average, assuming characters are distributed independently and uniformly. She also showed   complexity cannot be improved: if one considers words construed as a concatenation of several   words, where  , then the number of states for the window of size   would frequently change with jumps of order  , which renders even theoretical improvement of   for regular suffix automata impossible.[27]

The same should be true for the suffix tree because its vertices correspond to states of the suffix automaton of the reversed string but this problem may be resolved by not explicitly storing every vertex corresponding to the suffix of the whole string, thus only storing vertices with at least two out-going edges. A variation of McCreight's suffix tree construction algorithm for this task was suggested in 1989 by Edward Fiala and Daniel Greene;[28] several years later a similar result was obtained with the variation of Ukkonen's algorithm by Jesper Larsson.[29][30] The existence of such an algorithm, for compacted suffix automaton that absorbs some properties of both suffix trees and suffix automata, was an open question for a long time until it was discovered by Martin Senft and Tomasz Dvorak in 2008, that it is impossible if the alphabet's size is at least two.[31]

One way to overcome this obstacle is to allow window width to vary a bit while staying  . It may be achieved by an approximate algorithm suggested by Inenaga et al. in 2004. The window for which suffix automaton is built in this algorithm is not guaranteed to be of length   but it is guaranteed to be at least   and at most   while providing linear overall complexity of the algorithm.[32]

ApplicationsEdit

Suffix automaton of the string   may be used to solve such problems as:[33][34]

  • Counting the number of distinct substrings of   in   on-line,
  • Finding the longest substring of   occurring at least twice in  ,
  • Finding the longest common substring of   and   in  ,
  • Counting the number of occurrences of   in   in  ,
  • Finding all occurrences of   in   in  , where   is the number of occurrences.

It is assumed here that   is given on the input after suffix automaton of   is constructed.[33]

Suffix automata are also used in data compression,[35] music retrieval[36][37] and matching on genome sequences.[38]

ReferencesEdit

BibliographyEdit

External linksEdit