后缀树及模式匹配

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Tree(hattivatti)
hattivatti attivatti ttivatti tivatti ivatti vatti atti tti ti i
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vatti t ti atti hattivatti i vatti ti vatti atti hattivatti attivatti ttivatti vatti vatti tti tivatti ivatti i i vatti
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Tree(T) is of linear size
• only the internal branching nodes and the leaves represented explicitly • edges labeled by substrings of T • v = node(α) if the path from root to v spells α • one-to-one correspondence of leaves and suffixes • |T| leaves, hence < |T| internal nodes • |Tree(T)| = O(|T| + size(edge labels))
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Algorithms for combinatorial string matching?
• • • • • deep beauty? +shallow beauty? + applications? ++ intensive algorithmic miniatures sources of new problems: text processing, DNA, music,…
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Suffix trie and suffix tree
abaab baab aab ab b
Trie(abaab)
a b a a b b a b b a
Tree(abaab)
a baab ab baab a
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Trie(T) can be large
• |Trie(T)| = O(|T|2) • bad example: T = anbn • Trie(T) can be seen as a DFA: language accepted = the suffixes of T • minimize the DFA => directed cyclic word graph (’DAWG’)
– static text, any given pattern P
• Pattern synthesis problem: Learn from T new patterns that occur surprisingly often • What is a pattern? Exact substring, approximate 5 substring, with generalized symbols, with gaps, …
Suffix tree and suffix array techniques for pattern analysis in strings
Esko Ukkonen Univ Helsinki Erice School 30 Oct 2005
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ttttttttttttttgagacggagtctcgctctg tcgcccaggctggagtgcagtggcggg atctcggctcactgcaagctccgcctcc cgggttcacgccattctcctgcctcagcc tcccaagtagctgggactacaggcgcc cgccactacgcccggctaattttttgtattt ttagtagagacggggtttcaccgttttagc cgggatggtctcgatctcctgacctcgtg atccgcccgcctcggcctcccaaagtgc tgggattacaggcgt
1 2 6,10 7 3 6,10 vatti 8 4 hattivatti 2,5 3,3 4,5 i vatti 9 5 i 10 vatti 6
hattivatti
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Tree(T) is full text index
Tree(T)
P
P occurs in T at locations 8, 31, …
a b a a a b b a b a a a b a )
Trie(abaa)
a b a a a b b a b a a a b a a b a
Add next symbol = b
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Trie(abaab)
Trie(abaa)
a b a a a b b a b a a a b a a b a
• time: O(size of Trie(T)) • suffix links: slink(node(aα)) = node(α)
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On-line procedure for suffix trie
1. Create Trie(t1): nodes root and v, an arc son(root, t1) = v, and suffix links slink(v) := root and slink(root) := root 2. for i := 2 to n do begin 3. 4. 5. 6. 7. 8. 9. vi-1 := leaf of Trie(t1…ti-1) for string t1…ti-1 (i.e., the deepest leaf) v := vi-1; v´ := 0 while node v has no outgoing arc for ti do begin Create a new node v´´ and an arc son(v,ti) = v´´ if v´ ≠ 0 then slink(v) := v´´ v := slink(v); v´ := v´´ end for the node v´´ such that v´´= son(v,ti) do if v´´ = v´ then slink(v’) := root else slink(v´) := v´´
hattivatti
substring labels of edges represented as pairs of pointers 12
Tree(hattivatti)
hattivatti attivatti ttivatti tivatti ivatti vatti atti tti ti i
Before
ai
From here on the ai-arc exists already => stop updating here
After
ai ai ai
ai
ai
New suffix links New nodes
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What happens in Trie(Pi) => Trie(Pi+1) ?
hattivatti attivatti ttivatti tivatti ivatti vatti atti tti ti i
hattivatti attivatti vatti atti vatti vatti tti tivatti ttivatti hattivatti ti t i i vatti ti ivatti vatti vatti
’on-line’ algorithm (Ukkonen 1992)
i
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On-line construction of Trie(T)
• • • • T = t1t2 … tn$ Pi = t1t2 … ti i:th prefix of T on-line idea: update Trie(Pi) to Trie(Pi+1) => very simple construction
Add next symbol = b From here on b-arc already exists
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Trie(abaab)
a b a a a b b a b a a a b a a b a
Trie(abaab)
b a a
a a
b a b a b
b
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What happens in Trie(Pi) => Trie(Pi+1) ?
1.
Suffix tree
2. Suffix array 3. Some applications 4. Finding motifs
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The suffix tree Tree(T) of T
• data structure suffix tree, Tree(T), is compacted trie that represents all the suffixes of string T • linear size: |Tree(T)| = O(|T|) • can be constructed in linear time O(|T|) • has myriad virtues (A. Apostolico) • is well-known: 366 000 Google hits
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P occurs in T P is a prefix of some suffix of T Path for P exists in Tree(T) All occurrences of P in time O(|P| + #occ)
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Find att from Tree(hattivatti)
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Trie(abaab)
Trie(a)
a b a
Trie(ab)