1. String Data Structures and Algorithms: Suffix Trees and Suffix Arrays
David Fernández-Baca
UNAM (Mexico)
(based on notes by Srinivas Aluru)
slightly modified by Benny Chor

2. BBSI Summer School - Iowa State University
Why Strings?
Biological sequences can be viewed as strings, or finite series of characters, over an alphabet Σ.
There is a wealth of algorithmic theory developed for general strings that we can apply to specific biological problems.
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Look-up Tables
Strings of length k over Σ can be represented by an integer index i, 0 ≤ i ≤ Σk – 1.
DNA is composed of four characters.
Σ = {A, G, C, T} |Σ| = 4
We can preprocess a database into a lookup table to locate all occurrences of a query index.
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Example
Let:
A = 0 (00) C = 1 (01) G = 2 (10) T = 3 (11)
Strings are converted based on the binary string they represent
String Binary Integer
AAA = = 0
ATA = = 12
AAC = = 1
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Search using an Index
Size:
|Σ|k
Linked list of occurrences
Query
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6. Applications of Indexing
Seeds for searching sequence databases
BLAST
Pair generation for fragment assembly in sequencing projects
CAP3 sequence assembly program
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Indexing
Using a sparse representation, a database can be preprocessed in linear time to allow locating all instances of a short string.
Major limitation: search is restricted to fixed length strings.
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Suffix Trees
Paths from root to leaves
represent all suffixes of S
S = M A L A Y A L A M $ A $ M LA
YALAM$ AL 5 10
YALAM$ $M $M
YALAM$ $
ALAYALAM$ $M 8 4 7 3
YALAM$ 1 9 6 2
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9. Suffix tree properties
For a string S of length n, there are n+1 leaves and at most n internal nodes.
therefore requires only linear space,
provided edge labels are O(1) space
Each leaf represents a unique suffix.
Concatenation of edge labels from root to a leaf spells out the suffix.
Each internal node represents a distinct common prefix to at least two suffixes.
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Edge Encoding
S = M A L A Y A L A M $
(2, 2)
(10, 10)
(3, 4)
(5, 10)
(1, 1) 10
(3, 4)
(5, 10) 5
(5, 10)
(9, 10)
(2, 10)
(10, 10)
(9, 10) 8 4 7 3 1 9
(9, 10)
(5, 10) 6 2
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11. Näive Suffix Tree Construction1
MALAYALAM$ 2
ALAYALAM$ 3
LAYALAM$ 4
AYALAM$ 5
YALAM$ 6
ALAM$ 7
LAM$ 8
AM$ 9 M$ 10 $
Before starting: Why exactly
do we need this $, which is
not part of the alphabet?
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12. Näive Suffix Tree Construction2 3 4 1
MALAYALAM$ 2
ALAYALAM$ 3
LAYALAM$ 4
AYALAM$ 5
YALAM$ 6
ALAM$ 7
LAM$ 8
AM$ 9 M$ 10 $ A
$MALAYALAM
LAYALAM$
LAYALAM$
YALAM$ 2 1 3 4
etc.
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13. Application: Finding a short Pattern in a long String
Build a suffix tree of the string.
Starting from the root, traverse a path matching characters of the pattern.
If stuck, pattern not present in string.
Otherwise, each leaf below gives a position of the pattern in the string.
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14. Finding a Pattern in a String
Find “ALA” A $ M LA
YALAM$ AL 5 10
YALAM$ M$ M$
YALAM$ $
ALAYALAM$ 3 M$ 8 4 7
YALAM$ 1 9
Two matches - at 6 and 2 6 2
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15. Finding Common SubStrings
Construct a generalized suffix tree for two strings (each suffix of each string is represented).
Label each leaf with the suffix number and string label.
Each internal node with a leaf from both strings in its subtree gives a common substring.
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16. Generalized Suffix Tree
WINDOW$ INDIGO$ $ D
$OG I ND O W
(2, 5)
(1, 7)
(2, 7) ND
OW$
$OGI
$OG
$OGI
OW$ $ $W $
INDOW$
(2, 4)
(2, 2)
(1, 3)
(1, 5)
(2, 6)
(2, 3)
(1, 4)
$OGI
OW$
(1, 6)
(1, 1)
(2, 1)
(1, 2)
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17. Lowest Common Ancestors
The lowest common ancestor (lca) of two nodes x and y in a rooted tree is the deepest node (farthest away from root) that is an ancestor of both x and y
Concatenation of edge labels from root to the lca of two leaves spells out the longest common prefix (lcp) of two strings
lca(x,y) an be found in constant time after linear preprocessing [Bender00]
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A Useful Property
String depth (lca (i , j)) = lcp (suffixi, suffixj) A A $
String depth = 3 M LA
YALAM$ AL AL
lca 5 10
YALAM$ M$ M$
YALAM$ $
ALAYALAM$
lcp = longest
common prefix M$ 8 4 7 3
YALAM$ 1 9 6 2
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19. Longest Common Extension
RAI
RAILWAY$
GRAINY$
RAI
lce(1,1) = 0
lce(2,1) = 3
We’ll soon find lce’s useful in reconstructing phylogenetic
trees based on whole genome/proteome sequences
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lce’s and lca’s
To compute lce’s for two strings S1 and S2
Build generalized suffix tree, T, of S1 and S2
Compute string depth for each node in T
Preprocess T for lca queries
lce(i,j) = string depth of lca of suffix i ofS1 and suffix j ofS2
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Example
WINDOW$ INDIGO$ $ D
$OG I ND O W
(2, 5)
(1, 7)
(2, 7) ND
OW$
$OGI
$OG
$OGI
OW$ $ $W $
INDOW$
(2, 4)
(2, 2)
(1, 3)
(1, 5)
(2, 6)
(2, 3)
(1, 4)
$OGI
OW$
(1, 6)
(1, 1)
(2, 1)
(1, 2)
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lce’s, revisited
Given two strings S1 and S2 , we are now
interested in finding, for each i, the index j
such that lce (i, j) is maximal.
What is the meaning of this task?
How do we accomplish it efficiently?
Notice that computing the values
lce (i, j) for all j is very inefficient!
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Palindromes
A palindrome is a string that reads the same in both directions
E.g., CATGTAC
red rum, sir, is murder
Palindrome problem: Find all maximal palindromes in a string S
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24. Finding Palindromes in S
Construct the reverse S’ of S
Build generalized suffix tree of S and S’
Preprocess T for lce queries
Now what?
Left as homework
Requirement: Linear time (const. per query) S
q + 1
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25. Palindromes in DNA sequences
We sometimes need to deal with
Crick-Watson complemented palindromes A  T C  G
E.g., ATCATGAT is a complemented palindrome
All complemented palindromes in S can be found using a GST of S and the complement of S’
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26. Suffix Array – Reducing Space6
ALAM$ 2
ALAYALAM$ 8
AM$ 4
AYALAM$ 7
LAM$ 3
LAYALAM$ 1
MALAYALAM$ 9 M$ 5
YALAM$ 10 $
M A L A Y A L A M $ 6 2 8 4 7 3 1 9 5 10
Suffix Array:
Lexicographic ordering of suffixes 3 1 2 -
Derive Longest Common Prefix array
Suffix 6 and 2 share “ALA”
Suffix 2,8 share just “A”. lcp achieved for successive pairs.
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27. Pattern Search in Suffix Array
All suffixes that share a common prefix appear in consecutive positions in the array.
Pattern P can be located in the string using a binary search on the suffix array.
Naïve Run-time = O(|P|  log n).
Improved to O(|P| + log n) [Manber&Myers93], and to O(|P|) [Abouelhoda et al. 02].
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Example
Text M A L Y $
Position 1 2 3 4 5 6 7 8 9 10
Suffix Array 3 7 4 10 5 8 9 1 2 6
lcp Array 3 1 1 2 1 6
ALAM$ 2
ALAYALAM$ 8
AM$ 4
AYALAM$ 7
LAM$ 3
LAYALAM$ 1
MALAYALAM$ 9 M$ 5
YALAM$ 10 $
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29. Suffix Trees vs. Suffix Arrays
Suffix Array = Lexicographic order of the leaves of the Suffix Tree
Suffix Tree  Suffix Array + lcp Array
(why? Wait for next slide)
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30. Building a ST from a SA and a lcp6
ALAM$ 2
ALAYALAM$ 8
AM$ 4
AYALAM$ 7
LAM$ 3
LAYALAM$ 1
MALAYALAM$ 9 M$ 5
YALAM$ 10 $ A LA
D = 1
D = 2 AL
YALAM$ $M $M
YALAM$
D = 3 $M 8 4 7 3
YALAM$ 6 2 SA 6 2 8 4 7 3 1 9 5 10
lcp 3 1 2 -
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31. Known (amazing) Results
Suffix tree can be constructed in O(n) time and O(n  |∑|) space [Weiner73, McCreight76, Ukkonen92].
Suffix arrays can be constructed without using suffix trees in O(n) time [Pang&Aluru03].
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More Applications
Suffix-prefix overlaps in fragment assembly
Maximal and tandem repeats
Shortest unique substrings
Maximal unique matches [MUMmer]
Approximate matching
Phylogenies based on complete genomes
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Dealing with errors
The basic string data structures can only extract information in the absence of errors.
To deal with errors, decompose into parts that do not involve errors.
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34. The k-mismatch problem
Given a pattern P, a text T, and a number k, find all occurrences of P in T with at most k mismatches
Example
P = bend, T = abentbananaend, k = 2
Match 1: bent
Match 2: bana
Match 3: aend
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Solution
Build GST of P and T and preprocess it for lce queries
For each starting index i in T, do at most k lce queries to determine if there is a k-mismatch beginning at i T P
Time = O(k |T |)
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References
M. I. Abouelhoda, S. Kurtz and E. Ohlebusch, The enhanced suffix array and its applications to genome analysis, 2nd Workshop on Algorithms in Bioinformatics, pp , 2002.
M. A. Bender and M. Farach-Colton, The LCA Problem Revisited, LATIN, pages 88-94, 2000.
P. Ko and S. Aluru, Linear time suffix sorting, CPM, pages ,
U. Manber and G. Myers. Suffix arrays: a new method for on-line search, SIAM J. Comput., 22: , 1993.
E. M. McCreight, A space-economical suffix tree construction algorithm, J. ACM, 23(2): , 1976.
E. Ukkonen, Constructing suffix trees on-line in linear time. Intern. Federation of Information Processing, pp ,1992. Also in Algorithmica, 14(3): , 1995.
P. Weiner, Linear pattern matching algorithms, Proc. of the 14th IEEE Annual Symp. on Switching and Automata Theory, pp. 1-11, 1973.
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