Suffix trees and applications. String Algorithms
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1 Suffix trees and applications String Algorithms
2 Tries a trie is a data structure for storing and retrieval of strings.
3 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b x 2 = a b c
4 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b a x 2 = a b c
5 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b x 2 = a b c a b 1
6 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b x 2 = a b c a b 1 a
7 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b x 2 = a b c a b 1 a b
8 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b x 2 = a b c a b 1 2 a b c
9 Tries a trie is a data structure for storing and retrieval of strings. x 1 = a b x 2 = a b c a b 1 2 a b c Observations: shared prefixes implies shared initial paths Often we want each string to correspond to a unique root-to-leaf path, i.e. make sure that no input-string is a prefix of another. How?
10 Tries a trie is a data structure for storing and retrieval of strings. a a x 1 = a b b b x 1 = a b $ x 2 = a b c 1 c x 2 = a b c $ 2 x 3 = $ Observations: shared prefixes implies shared initial paths Often we want each string to correspond to a unique root-to-leaf path, i.e. make sure that no input-string is a prefix of another. How?
11 Tries a trie is a data structure for storing and retrieval of strings. a a a x 1 = a b x 2 = a b c b 1 2 b c x 1 = a b $ x 2 = a b c $ x 3 = $ b $ 1 Observations: shared prefixes implies shared initial paths Often we want each string to correspond to a unique root-to-leaf path, i.e. make sure that no input-string is a prefix of another. How?
12 Tries a trie is a data structure for storing and retrieval of strings. a a a a x 1 = a b x 2 = a b c b 1 2 b c x 1 = a b $ x 2 = a b c $ x 3 = $ b b c $ $ 1 2 Observations: shared prefixes implies shared initial paths Often we want each string to correspond to a unique root-to-leaf path, i.e. make sure that no input-string is a prefix of another. How?
13 Tries a trie is a data structure for storing and retrieval of strings. a a a a $ x 1 = a b x 1 = a b $ b b b b x 2 = a b c 1 x 2 = a b c $ c c $ $ 2 x 3 = $ Observations: shared prefixes implies shared initial paths Often we want each string to correspond to a unique root-to-leaf path, i.e. make sure that no input-string is a prefix of another. How?
14 Tries a trie is a data structure for storing and retrieval of strings. a a a a $ x 1 = a b x 1 = a b $ b b b b x 2 = a b c 1 x 2 = a b c $ c c $ $ 2 x 3 = $ Observations: shared prefixes implies shared initial paths Application: Given a query-string y[1...m], we can determine if y equals one of the input-strings (or a prefix of one) in time O(m)...
15 What is the space complexity?
16 Compacted tries Saving space: Eliminate all internal nodes of degree 2... x 1 = a b $ x 2 = a b c $ x 3 = $ a b a b $ c $ $ a b $ 1 $ c $ 3 2 If we have n input-strings, then the trie has n+1 leaves and at most n internal nodes, i.e space O(n) for the tree. What about the labels?
17 Compacted tries Saving space: Eliminate all internal nodes of degree 2... x 1 = a b $ x 2 = a b c $ x 3 = $ a b a b $ c $ $ a b $ 1 $ c $ 3 2 If we have n input-strings, then the trie has n+1 leaves and at most n internal nodes, i.e space O(n) for the tree. What about the labels? Labels can be represented in space O(1), i.e. ab (1,1,2)
18 If there are n strings... How many leaves are there? How many inner nodes? How many edges?
19 Suffix tree The suffix tree T(x) of string x[1..n] is the compacted trie of all suffixes x[i..n] for i = 1,.., n+1, i.e. including the empty suffix
20 Suffix tree The suffix tree T(x) of string x[1..n] is the compacted trie of all suffixes x[i..n] for i = 1,..,n+1, i.e. including the empty suffix Example for x = tatat tatat$ atat$ tat$ at$ t$ ε$
21 Suffix tree The suffix tree T(x) of string x[1..n] is the compacted trie of all suffixes x[i..n] for i = 1,..,n+1, i.e. including the empty suffix Example for x = tatat tatat$ atat$ tat$ at$ t$ ε$
22 A larger example
23 A larger example Node has path-label ssi and is at depth 3...
24 A larger example Node has path-label ssi and is at depth 3... Path-label of leaf i is suffix i, i.e. x[i..n]$...
25 A larger example Path-label of lowest common ancestor of leaf i and j, is longest common prefix of suffix i and j of x... Node has path-label ssi and is at depth 3... Path-label of leaf i is suffix i, i.e. x[i..n]$...
26 What is the space complexity?
27 Space consumption
28 Space consumption
29 Space consumption
30 Space consumption
31 Constructing suffix trees Constructing T(x) by inserting each suffix one by one takes time O(n 2 ) Can we do better?
32 Constructing suffix trees Constructing T(x) by inserting each suffix one by one takes time O(n 2 ) Can we do better? [Weiner 1973]: T(x) can be constructed in time O(n)... There are two practical algorithms that construct the suffix tree in linear time: McCreight (1976) and Ukkonen (1993)...
33 What about applications?... exact matching, finding repeats, longest common substring...
34 Exact matching Given string x and pattern y, report where y occurs in x If y occurs in x at position i, then y is a prefix of suffix i of x i j y is spelled by an initial part of the path from the root to leaf i in T(x)
35 Exact matching Given string x and pattern y, report where y occurs in x
36 Exact matching Given string x and pattern y, report where y occurs in x
37 Exact matching Given string x and pattern y, report where y occurs in x
38 Exact matching Given string x and pattern y, report where y occurs in x
39 Exact matching Given string x and pattern y, report where y occurs in x Pattern ata occurs at position 2 in tatat Time: O( P ) using the suffix tree T(S)
40 Exact matching Given string x and pattern y, report where y occurs in x
41 Exact matching Given string x and pattern y, report where y occurs in x
42 Exact matching Given string x and pattern y, report where y occurs in x
43 Exact matching Given string x and pattern y, report where y occurs in x
44 Exact matching Given string x and pattern y, report where y occurs in x Pattern tatt does not occur in tatat Time: O( P ) using the suffix tree T(S)
45 Repeats A pair of substrings R=(S[i 1..j 1 ], S[i 2..j 2 ]) is a...
46 Repeats A pair of substrings R=(S[i 1..j 1 ], S[i 2..j 2 ]) is a...
47 Repeats A pair of substrings R=(S[i 1..j 1 ], S[i 2..j 2 ]) is a...
48 Finding exact repeats
49 Finding exact repeats
50 Finding exact repeats
51 Finding exact repeats
52 Finding exact repeats
53 Finding exact repeats
54 Finding exact repeats
55 A larger example
56 Finding maximal exact repeats
57 Finding maximal exact repeats
58 Finding maximal exact repeats
59 Finding maximal exact repeats
60 Finding maximal exact repeats
61 Other repeats Maximal repeats with bounded gap in time O(n log n + z) Tandem repeats in time O(n log n + z) Palindromic repeats in O(n + z)... all using suffix trees...
62 More strings The longest common substring of x[1..n] and y[1..m] is the longest string z which occurs in both x and y... Can this be found efficiently using a suffix tree?
63 More strings The longest common substring of x[1..n] and y[1..m] is the longest string z which occurs in both x and y... Can this be found efficiently using a suffix tree? z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] x z y
64 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε
65 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 a a t a
66 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a a t a
67 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a a t a a 3
68 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a 4 a t a a 3
69 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a 4 a t a 5 a 3
70 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] Idea: Build a compacted trie of all suffixes of x and y, such that each suffix of x and y corresponds to unique root-to-leaf paths... tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a 4 a t a 5 a 3 6
71 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a 4 a t a 5 a 3 6 Observe: z is the path-label of the deepest node with suffixes from both x and y as leaves in its sub-tree...
72 More strings z is the longest common prefix of any pair of suffixes x[i..n] and y[j..m] tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a 4 a t a 5 a 3 6 Observe: z is the path-label of the deepest node with suffixes from both x and y as leaves in its sub-tree... Time: O(n+m)
73 Generalized suffix tree This is the generalized suffix tree of tatat and aataa tatat$ atat$ tat$ at$ t$ ε$ aataa ataa taa aa a ε 1 2 a a 4 a t a 5 a 3 6 Can be constructed by constructing the suffix tree of... tatat$aataa
74 Generalized suffix tree 1 n n+2 n+m+1 $ 1 m... we must argue that we get the same branching structure...
75 Generalized suffix tree 1 n n+2 n+m+1 $ 1 m Case 1: i j $ $
76 Generalized suffix tree 1 n n+2 n+m+1 $ 1 m Case 1: i j $ $ i j $ $ $ $ j i
77 Generalized suffix tree 1 n n+2 n+m+1 $ 1 m Case 1: i j $ $ i j $ $ $ i $ j $ (1, j) $ (1, i)
78 Generalized suffix tree 1 n n+2 n+m+1 $ 1 m Case 2: i j i +(n+1) j +(n+1) j i (2, j ) (2, i )
79 Generalized suffix tree 1 n n+2 n+m+1 $ 1 m Case 3: i j $ i j +(n+1) $ j $ (2, j ) $ (1, i) i
80 Is everything great?
81 Space consumption Fact: T(x) requires O(n) space, where n= x... but how much space does it consume in practice?
82
83 Space consumption Fact: T(x) requires O(n) space, where n= x, but in practice somewhere between 10 and 40 bytes per letter in x... Is this a problem? Depends on n, if then yes...
84 Alphabet size How much time does it take to find the proper edge out from a node when searching in a suffix tree?
85 Alphabet size How much time does it take to find the proper edge out from a node when searching in a suffix tree? Time proportional to the out-degree of the node A search time in pratice is O( A P )... If A is large, e.g. 256, this matters!!
86 Alphabet size How much time does it take to find the proper edge out from a node when searching in a suffix tree? Idea 1: Organising children in a search-tree, reduces search time from A to O(log A )... (requires an ordered alphabet)
87 Alphabet size How much time does it take to find the proper edge out from a node when searching in a suffix tree? Idea 2: Organising children in a vector of size A indexed by letters, reduces search time from A to O(1)... (requires a finite alphabet)
88 Alphabet size How much time does it take to find the proper edge out from a node when searching in a suffix tree? Idea 3: Use some other dictionary for mapping letters to children the alphabet size matters in practice...
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