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Suffix tree and Suffix array
Karatsuba
CS214: Algorithms and Complexity Shanghai Jiao Tong University 2016.12.22
Q: How to find a match of S in a target DNA sequence?
S: DNA:
Suffix tree and Suffix array Karatsuba CS214: Algorithms and - - PowerPoint PPT Presentation
Suffix tree and Suffix array Karatsuba CS214: Algorithms and - - PowerPoint PPT Presentation
Suffix tree and Suffix array Karatsuba CS214: Algorithms and Complexity Shanghai Jiao Tong University 2016.12.22 Q: How to find a match of S in a target DNA sequence? S: DNA: Q: How to find a match of S in a target DNA sequence? S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
Average time complexity: O(|S| ∗ |DNA|)
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Q: How to find a match of S in a target DNA sequence?
S: DNA:
Too Slow!
Average time complexity: O(|S| ∗ |DNA|)
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We need a much more efficient algorithm!
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Structure
- Suffix Tries
– Suffix Trees
- Suffix Array
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What’s suffix?
Suffix Tries
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What’s suffix?
Example:
abcdefg S: Suffix Tries
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What’s suffix?
Example:
abcdefg S: Suffix Tries
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What’s suffix?
Example:
abcdefg S: Suffix Tries
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What’s suffix?
Example:
abcdefg S: Suffix Tries
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What’s suffix?
Example:
abcdefg S: Suffix Tries
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What’s suffix?
Example:
abcdefg S: Suffix Tries And so on.
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What’s suffix?
Example:
abcdefg S: Suffix Tries And so on.
Attention: We use $ to represent empty suffix!
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Suffix Tries
Suffix trie are a space-efficient data structure to store a string that allows many kinds of queries to be answered quickly.
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 5
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 5
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 5
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a 5
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4 $ 5
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4 $ 5 b
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4 $ 5 b 2 3 a
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4 $ 5 b 2 3 4 a a
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4 $ 5 b 2 3 4 a a 5 $
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 01234 $ 5 134 a a 5 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 $ a a 5 $ b a a $ $ b a a $
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 $ a a 5 $ b a a $ $ b a a $
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Suffix Tries
S= a b a a $ Suffix: $ a$ aa$ baa$ abaa$
0 1 2 3 4 $ a a 5 $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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Suffix Tries
S=abaa$
Every suffix of s is represented by some path from the root to a leaf node.
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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Application of Suffix Tries (1)
S=abaa$ What can we do with this?
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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S=abaa$ What can we do with this?
- 1. Is ”ba” a substring of S?
Application of Suffix Tries (1)
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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S=abaa$ What can we do with this?
- 1. Is ”ba” a substring of S?
Application of Suffix Tries (1)
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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S=abaa$ What can we do with this?
- 1. Is ”ba” a substring of S?
- 2. Is ”ba” a suffix of S?
Application of Suffix Tries (1)
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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S=abaa$ What can we do with this?
- 1. Is ”ba” a substring of S?
- 2. Is ”ba” a suffix of S?
- 3. How many times does ”ba” appear?
Application of Suffix Tries (1)
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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S=abaa$ What can we do with this?
- 1. Is ”ba” a substring of S?
- 2. Is ”ba” a suffix of S?
- 3. How many times does ”ba” appear?
... ...
Application of Suffix Tries (1)
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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Suffix Tries
S=abaa$ What we can not do
Find the longest common substring of S and t S= abaa$ t= bba
$ a a $ b a a $ $ b a a $ abaa$ aa$ a$ $ baa$
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Links
S=abaa$
Suffix links connect node representing xα to a node representing α.
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Tires with Links
S=abaa$
01234 $ 5 134 a a 4 $ 5 b 2 3 4 a a 5 $ 5 $ b 2 3 4 a a 5 $
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Suffix Tires with Links
S=abaa$
$ a a $ b a a $ $ b a a $
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=$
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=$
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=$
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=$
Ooops!
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=ba
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=ba What’s the semantic of jumping?
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=ba LCS(t1,...,tk,S)= either (t1,...ti)
- r LCS(t2,...,tk,S)
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=ba
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=ba
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=ba
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S=abaa$
$ a a $ b a a $ $ b a a $
Find the longest common substring of S and t
S= abaa$ t= babaa
Application with Suffix Tries (2)
max commom=abaa
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Constructing Suffix Tries
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First build a suffix tries of S[0] Add char one by one into the suffix tries To build a suffix tries of S
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r
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r
a
a
S=abaa$
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r
a
a
S=abaa$ Suffix: a
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r
ab
a b
S=abaa$ Suffix: ab
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r
ab
a b b
S=abaa$ Suffix: ab b
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r
ab
a b b
S=abaa$ Suffix: ab b
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r
ab
a b b
S=abaa$ Suffix: ab b
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r
aba
a b b a
S=abaa$ Suffix: aba b
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r
aba
a b b a a
S=abaa$ Suffix: aba ba
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r
aba
a b b a a
S=abaa$ Suffix: aba ba
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r
aba
a b b a a
S=abaa$ Suffix: aba ba a
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r
aba
a b b a a
S=abaa$ Suffix: aba ba a
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r
abaa$
a b b a a a a a
$ $ $ $ $
S=abaa$ Suffix: abaa$ baa$ aa$ a$ $
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How many nodes can a suffix trie have?
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Space-Efficient Suffix Trees
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A More Compact Represntation
r a b b a a a a a
$ $ $ $ $
S=abaa$
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A More Compact Represntation
r a baa$ a$
$ $
baa$
S=abaa$ 12345
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A More Compact Represntation
r 3:3 2:4 4:5
5:5
2:4
S=abaa$ 12345
5:5
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How to construct suffix tree in Linear time
Further reading: Ukkonens Algorithm
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Suffix arrays
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Suffix Array Example str = catttcat $
1 catttcat$ 2 attcat$ 3 ttcat$ 4 tcat$ 5 cat$ 6 at$ 7 t$ 8 $
sort the suffixes alphabetically
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
What can we do with this?
- 1. Counting:
how many times does ’at’ occur? All the suffixes that start with ’at’ will be next to each other in the array. Binary search to find ’at’
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
What can we do with this?
- 2. K-mer counting:
k-length substring that occurs exactly i times.
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
K = 2 CurrentCount
1 8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
K = 2 CurrentCount
1 1 8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
K = 2 CurrentCount
1 1 2 8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
K = 2 CurrentCount
1 1 2 1 (at,2) 8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
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Suffix Arrays
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
K = 2 CurrentCount
1 1 2 1 2 (at,2)
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Suffix Arrays
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
K = 2 CurrentCount
1 1 2 1 2 1 (at,2) (ca,2)
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Suffix Arrays
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
K = 2 CurrentCount
1 1 2 1 2 1 1 (at,2) (ca,2) (t$,1)
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Suffix Arrays
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
K = 2 CurrentCount
1 1 2 1 2 1 1 1 (at,2) (ca,2) (t$,1) (tc,1)
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Suffix Arrays
8 $ 6 at$ 2 attcat$ 5 cat$ 1 cattcat$ 7 t$ 4 tcat$ 3 ttcat$
K = 2 CurrentCount
1 1 2 1 2 1 1 1 1 (at,2) (ca,2) (t$,1) (tc,1) (tt,1)
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Constructing Suffix Arrays
- Easy O(n2 log n) algorithm:
Sort the n suffixes, which takes O(n log n) comparisons, where each comparison takes O(n).
- We can do better: