# Suffix Tries and Suffix Trees¶

class sage.combinat.words.suffix_trees.ImplicitSuffixTree(word)

Construct the implicit suffix tree of a word w.

The suffix tree of a word w is a compactification of the suffix trie for w. The compactification removes all nodes that have exactly one incoming edge and exactly one outgoing edge. It consists of two components: a tree and a word. Thus, instead of labelling the edges by factors of w, we can labelled them by indices of the occurrence of the factors in w.

The following is a straightforward implementation of Ukkonen’s on-line algorithm for constructing the implicit suffix tree [Ukko1995]. It constructs the suffix tree for w[:i] from that of w[:i-1].

GENERAL IDEA. The suffix tree of w[:i+1] can be obtained from that of w[:i] by visiting each node corresponding to a suffix of w[:i] and modifying the tree by applying one of two rules (either append a new node to the tree, or split an edge into two). The “active state” is the node where the algorithm begins and the “suffix link” carries us to the next node that needs to be dealt with.

TREE. The tree is modelled as an automaton, which is stored as a dictionary of dictionaries: it is keyed by the nodes of the tree, and the corresponding dictionary is keyed by pairs $$(i,j)$$ of integers representing the word w[i-1:j]. This makes it faster to look up a particular transition beginning at a specific node.

STATES/NODES. The states will always be -1, 0, 1, …, n. The state -1 is special and is only used for the purposes of the algorithm. All transitions map -1 to 0, so this information is not explicitly stored in the transition function.

EXPLICIT/IMPLICIT NODES. By definition, some of the nodes will not be states, but merely locations along an edge; these are called implicit nodes. A node r (implicit or explicit) is referenced as a pair (s,(k,p)) where s is an ancestor of r and w[k-1:p] is the word read by transitioning from s to r in the suffix trie. A reference pair is canonical if s is the closest ancestor of r.

SUFFIX LINK. The algorithm makes use of a map from (some) nodes to other nodes, called the suffix link. This is stored as a dictionary.

ACTIVE STATE. We store as ._active_state the active state of the tree, the state where the algorithm will begin when processing the next letter.

RUNNING TIME. The running time and storage space of the algorithm is linear in the length of the word w (whereas for a suffix tree it is quadratic).

REFERENCES:

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: w = Words("aco")("cacao")
sage: t = ImplicitSuffixTree(w); t
Implicit Suffix Tree of the word: cacao
sage: ababb = Words([0,1])([0,1,0,1,1])
sage: s = ImplicitSuffixTree(ababb); s
Implicit Suffix Tree of the word: 01011

LZ_decomposition()

Return a list of index of the beginning of the block of the Lempel-Ziv decomposition of self.word

The Lempel-Ziv decomposition is the factorisation $$u_1...u_k$$ of a word $$w=x_1...x_n$$ such that $$u_i$$ is the longest prefix of $$u_i...u_k$$ that has an occurrence starting before $$u_i$$ or a letter if this prefix is empty.

OUTPUT:

Return a list iB of index such that the blocks of the decomposition are self.word()[iB[k]:iB[k+1]]

EXAMPLES:

sage: w = Word('abababb')
sage: T = w.suffix_tree()
sage: T.LZ_decomposition()
[0, 1, 2, 6, 7]
sage: w = Word('abaababacabba')
sage: T = w.suffix_tree()
sage: T.LZ_decomposition()
[0, 1, 2, 3, 6, 8, 9, 11, 13]
sage: w = Word([0, 0, 0, 1, 1, 0, 1])
sage: T = w.suffix_tree()
sage: T.LZ_decomposition()
[0, 1, 3, 4, 5, 7]
sage: w = Word('0000100101')
sage: T = w.suffix_tree()
sage: T.LZ_decomposition()
[0, 1, 4, 5, 9, 10]

active_state()

Returns the active state of the suffix tree.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: W = Words([0,1,2])
sage: t = ImplicitSuffixTree(W([0,1,0,1,2]))
sage: t.active_state()
(0, (6, 6))

edge_iterator()

Returns an iterator over the edges of the suffix tree. The edge from $$u$$ to $$v$$ labelled by $$(i,j)$$ is returned as the tuple $$(u,v,(i,j))$$.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: sorted( ImplicitSuffixTree(Word("aaaaa")).edge_iterator() )
[(0, 1, (0, None))]
sage: sorted( ImplicitSuffixTree(Word([0,1,0,1])).edge_iterator() )
[(0, 1, (0, None)), (0, 2, (1, None))]
sage: sorted( ImplicitSuffixTree(Word()).edge_iterator() )
[]

factor_iterator(n=None)

Generate distinct factors of self.

INPUT:

• n - an integer, or None.

OUTPUT:

• If n is an integer, returns an iterator over all distinct factors of length n. If n is None, returns an iterator generating all distinct factors.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: sorted( ImplicitSuffixTree(Word("cacao")).factor_iterator() )
[word: , word: a, word: ac, word: aca, word: acao, word: ao, word: c, word: ca, word: cac, word: caca, word: cacao, word: cao, word: o]
sage: sorted( ImplicitSuffixTree(Word("cacao")).factor_iterator(1) )
[word: a, word: c, word: o]
sage: sorted( ImplicitSuffixTree(Word("cacao")).factor_iterator(2) )
[word: ac, word: ao, word: ca]
sage: sorted( ImplicitSuffixTree(Word([0,0,0])).factor_iterator() )
[word: , word: 0, word: 00, word: 000]
sage: sorted( ImplicitSuffixTree(Word([0,0,0])).factor_iterator(2) )
[word: 00]
sage: sorted( ImplicitSuffixTree(Word([0,0,0])).factor_iterator(0) )
[word: ]
sage: sorted( ImplicitSuffixTree(Word()).factor_iterator() )
[word: ]
sage: sorted( ImplicitSuffixTree(Word()).factor_iterator(2) )
[]

number_of_factors(n=None)

Count the number of distinct factors of self.word().

INPUT:

• n - an integer, or None.

OUTPUT:

• If n is an integer, returns the number of distinct factors of length n. If n is None, returns the total number of distinct factors.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: t = ImplicitSuffixTree(Word([1,2,1,3,1,2,1]))
sage: t.number_of_factors()
22
sage: t.number_of_factors(1)
3
sage: t.number_of_factors(9)
0
sage: t.number_of_factors(0)
1

sage: t = ImplicitSuffixTree(Word("cacao"))
sage: t.number_of_factors()
13
sage: list(map(t.number_of_factors, range(10)))
[1, 3, 3, 3, 2, 1, 0, 0, 0, 0]

sage: t = ImplicitSuffixTree(Word("c"*1000))
sage: t.number_of_factors()
1001
sage: t.number_of_factors(17)
1
sage: t.number_of_factors(0)
1

sage: ImplicitSuffixTree(Word()).number_of_factors()
1

sage: blueberry = ImplicitSuffixTree(Word("blueberry"))
sage: blueberry.number_of_factors()
43
sage: list(map(blueberry.number_of_factors, range(10)))
[1, 6, 8, 7, 6, 5, 4, 3, 2, 1]

plot(word_labels=False, layout='tree', tree_root=0, tree_orientation='up', vertex_colors=None, edge_labels=True, *args, **kwds)

Returns a Graphics object corresponding to the transition graph of the suffix tree.

INPUT:

• word_labels - boolean (default: False) if False, labels the edges by pairs $$(i, j)$$; if True, labels the edges by word[i:j].
• layout - (default: 'tree')
• tree_root - (default: 0)
• tree_orientation - (default: 'up')
• vertex_colors - (default: None)
• edge_labels - (default: True)

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: ImplicitSuffixTree(Word('cacao')).plot(word_labels=True)
Graphics object consisting of 23 graphics primitives
sage: ImplicitSuffixTree(Word('cacao')).plot(word_labels=False)
Graphics object consisting of 23 graphics primitives

process_letter(letter)

Modifies the current implicit suffix tree producing the implicit suffix tree for self.word() + letter.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: w = Words("aco")("cacao")
sage: t = ImplicitSuffixTree(w[:-1]); t
Implicit Suffix Tree of the word: caca
sage: t.process_letter(w[-1]); t
Implicit Suffix Tree of the word: cacao

sage: W = Words([0,1])
sage: s = ImplicitSuffixTree(W([0,1,0,1])); s
Implicit Suffix Tree of the word: 0101
sage: s.process_letter(W([1])[0]); s
Implicit Suffix Tree of the word: 01011

show(word_labels=None, *args, **kwds)

Displays the output of self.plot().

INPUT:

• word_labels - (default: None) if False, labels the edges by pairs $$(i, j)$$; if True, labels the edges by word[i:j].

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: w = Words("cao")("cacao")
sage: t = ImplicitSuffixTree(w)
sage: t.show(word_labels=True)
sage: t.show(word_labels=False)

states()

Returns the states (explicit nodes) of the suffix tree.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: W = Words([0,1,2])
sage: t = ImplicitSuffixTree(W([0,1,0,1,2]))
sage: t.states()
[0, 1, 2, 3, 4, 5, 6, 7]


Evaluates the suffix link map of the implicit suffix tree on state. Note that the suffix link is not defined for all states.

The suffix link of a state $$x'$$ that corresponds to the suffix $$x$$ is defined to be -1 is $$x'$$ is the root (0) and $$y'$$ otherwise, where $$y'$$ is the state corresponding to the suffix x[1:].

INPUT:

• state - a state

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: W = Words([0,1,2])
sage: t = ImplicitSuffixTree(W([0,1,0,1,2]))
5
0
-1
Traceback (most recent call last):
...
TypeError: there is no suffix link from -1

suffix_walk(edge, l)

Return the state of “w” if the input state is “aw”.

If the input state (edge, l) is path labeled “aw” with “a” a letter, the output is the state which is path labeled “w”.

INPUT:

• edge – the edge containing the state
• l – the string-depth of the state on edge (l>0)

OUTPUT:

Return ("explicit", end_node) if the state of w is an explicit state and ("implicit", edge, d) is obtained by reading d letters on edge.

EXAMPLES:

sage: T = Word('00110111011').suffix_tree()
sage: T.suffix_walk((0, 5), 1)
('explicit', 0)
sage: T.suffix_walk((7, 3), 1)
('implicit', (9, 4), 1)

to_digraph(word_labels=False)

Returns a DiGraph object of the transition graph of the suffix tree.

INPUT:

• word_labels - boolean (default: False) if False, labels the edges by pairs $$(i, j)$$; if True, labels the edges by word[i:j].

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: W = Words([0,1,2])
sage: t = ImplicitSuffixTree(W([0,1,0,1,2]))
sage: t.to_digraph()
Digraph on 8 vertices

to_explicit_suffix_tree()

Converts self to an explicit suffix tree. It is obtained by processing an end of string letter as if it were a regular letter, except that no new leaf nodes are created (thus, the only thing that happens is that some implicit nodes become explicit).

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: w = Words("aco")("cacao")
sage: t = ImplicitSuffixTree(w)
sage: t.to_explicit_suffix_tree()

sage: W = Words([0,1])
sage: s = ImplicitSuffixTree(W([0,1,0,1,1]))
sage: s.to_explicit_suffix_tree()

transition_function(word, node=0)

Returns the node obtained by starting from node and following the edges labelled by the letters of word. Returns ("explicit", end_node) if we end at end_node, or ("implicit", edge, d) if we end $$d$$ spots along an edge.

INPUT:

• word - a word
• node - (default: 0) starting node

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: W = Words([0,1,2])
sage: t = ImplicitSuffixTree(W([0,1,0,1,2]))
sage: t.transition_function(W([0,1,0]))
('implicit', (3, 1), 1)
sage: t.transition_function(W([0,1,2]))
('explicit', 4)
sage: t.transition_function(W([0,1,2]), 5)
('explicit', 2)
sage: t.transition_function(W([0,1]), 5)
('implicit', (5, 2), 2)

transition_function_dictionary()

Returns the transition function as a dictionary of dictionaries. The format is consistent with the input format for DiGraph.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree
sage: W = Words("aco")
sage: t = ImplicitSuffixTree(W("cac"))
sage: t.transition_function_dictionary()
{0: {1: (0, None), 2: (1, None)}}

sage: W = Words([0,1])
sage: t = ImplicitSuffixTree(W([0,1,0]))
sage: t.transition_function_dictionary()
{0: {1: (0, None), 2: (1, None)}}

trie_type_dict()

Returns a dictionary in a format compatible with that of the suffix trie transition function.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree, SuffixTrie
sage: W = Words("ab")
sage: t = ImplicitSuffixTree(W("aba"))
sage: d = t.trie_type_dict()
sage: len(d)
5
sage: d                     # random
{(4, word: b): 5, (0, word: a): 4, (0, word: b): 3, (5, word: a): 1, (3, word: a): 2}

uncompactify()

Returns the tree obtained from self by splitting edges so that they are labelled by exactly one letter. The resulting tree is isomorphic to the suffix trie.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import ImplicitSuffixTree, SuffixTrie
sage: abbab = Words("ab")("abbab")
sage: s = SuffixTrie(abbab)
sage: t = ImplicitSuffixTree(abbab)
sage: t.uncompactify().is_isomorphic(s.to_digraph())
True

word()

Returns the word whose implicit suffix tree this is.

class sage.combinat.words.suffix_trees.SuffixTrie(word)

Construct the suffix trie of the word w.

The suffix trie of a finite word w is a data structure representing the factors of w. It is a tree whose edges are labelled with letters of w, and whose leafs correspond to suffixes of w.

This is a straightforward implementation of Algorithm 1 from [Ukko1995]. It constructs the suffix trie of w[:i] from that of w[:i-1].

A suffix trie is modelled as a deterministic finite-state automaton together with the suffix_link map. The set of states corresponds to factors of the word (below we write x’ for the state corresponding to x); these are always 0, 1, …. The state 0 is the initial state, and it corresponds to the empty word. For the purposes of the algorithm, there is also an auxiliary state -1. The transition function t is defined as:

t(-1,a) = 0 for all letters a; and
t(x',a) = y' for all x',y' \in Q such that y = xa,


and the suffix link function is defined as:

suffix_link(0) = -1;
suffix_link(x') = y', if x = ay for some letter a.


REFERENCES:

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cacao")
sage: t = SuffixTrie(w); t
Suffix Trie of the word: cacao

sage: e = Words("ab")()
sage: t = SuffixTrie(e); t
Suffix Trie of the word:
sage: t.process_letter("a"); t
Suffix Trie of the word: a
sage: t.process_letter("b"); t
Suffix Trie of the word: ab
sage: t.process_letter("a"); t
Suffix Trie of the word: aba

active_state()

Returns the active state of the suffix trie. This is the state corresponding to the word as a suffix of itself.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cacao")
sage: t = SuffixTrie(w)
sage: t.active_state()
8

sage: u = Words([0,1])([0,1,1,0,1,0,0,1])
sage: s = SuffixTrie(u)
sage: s.active_state()
22

final_states()

Returns the set of final states of the suffix trie. These are the states corresponding to the suffixes of self.word(). They are obtained be repeatedly following the suffix link from the active state until we reach 0.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cacao")
sage: t = SuffixTrie(w)
sage: t.final_states() == Set([8, 9, 10, 11, 12, 0])
True

has_suffix(word)

Return True if and only if word is a suffix of self.word().

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cacao")
sage: t = SuffixTrie(w)
sage: [t.has_suffix(w[i:]) for i in range(w.length()+1)]
[True, True, True, True, True, True]
sage: [t.has_suffix(w[:i]) for i in range(w.length()+1)]
[True, False, False, False, False, True]

node_to_word(state=0)

Returns the word obtained by reading the edge labels from 0 to state.

INPUT:

• state - (default: 0) a state

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("abc")("abcba")
sage: t = SuffixTrie(w)
sage: t.node_to_word(10)
word: abcba
sage: t.node_to_word(7)
word: abcb

plot(layout='tree', tree_root=0, tree_orientation='up', vertex_colors=None, edge_labels=True, *args, **kwds)

Returns a Graphics object corresponding to the transition graph of the suffix trie.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: SuffixTrie(Word("cacao")).plot()
Graphics object consisting of 38 graphics primitives

process_letter(letter)

Modify self to produce the suffix trie for self.word() + letter.

Note

letter must occur within the alphabet of the word.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("ab")("ababba")
sage: t = SuffixTrie(w); t
Suffix Trie of the word: ababba
sage: t.process_letter("a"); t
Suffix Trie of the word: ababbaa

show(*args, **kwds)

Displays the output of self.plot().

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cac")
sage: t = SuffixTrie(w)
sage: t.show()

states()

Returns the states of the automaton defined by the suffix trie.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words([0,1])([0,1,1])
sage: t = SuffixTrie(w)
sage: t.states()
[0, 1, 2, 3, 4]

sage: u = Words("aco")("cacao")
sage: s = SuffixTrie(u)
sage: s.states()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]


Evaluates the suffix link map of the suffix trie on state. Note that the suffix link map is not defined on -1.

INPUT:

• state - a state

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cacao")
sage: t = SuffixTrie(w)
[-1, 0, 3, 0, 5, 1, 7, 2, 9, 10, 11, 12, 0]
-1

to_digraph()

Returns a DiGraph object of the transition graph of the suffix trie.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("cao")("cac")
sage: t = SuffixTrie(w)
sage: d = t.to_digraph(); d
Digraph on 6 vertices
[0 1 0 1 0 0]
[0 0 1 0 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
[0 0 0 0 0 0]
[0 0 0 0 0 0]

transition_function(node, word)

Returns the state reached by beginning at node and following the arrows in the transition graph labelled by the letters of word.

INPUT:

• node - a node
• word - a word

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words([0,1])([0,1,0,1,1])
sage: t = SuffixTrie(w)
sage: all(t.transition_function(u, letter) == v
....:     for ((u, letter), v) in t._transition_function.items())
True

word()

Returns the word whose suffix tree this is.

EXAMPLES:

sage: from sage.combinat.words.suffix_trees import SuffixTrie
sage: w = Words("abc")("abcba")
sage: t = SuffixTrie(w)
sage: t.word()
word: abcba
sage: t.word() == w
True