Suffix Trees

1
Suffix Trees

• Suffix trees
• Linearized suffix trees
• Virtual suffix trees
• Suffix arrays
• Enhanced suffix arrays
• Suffix cactus, suffix vectors,

2
Suffix Trees

• String any sequence of characters.
• Substring of string S string composed of
  characters i through j, i lt j of S.
• S cater gt ate is a substring.
• car is not a substring.
• Empty string is a substring of S.

3
Subsequence

• Subsequence of string S string composed of
  characters i1 lt i2 lt lt ik of S.
• S cater gt ate is a subsequence.
• car is a subsequence.
• The empty string is a subsequence.

4
String/Pattern Matching

• You are given a source string S.
• Answer queries of the form is the string pi a
  substring of S?
• Knuth-Morris-Pratt (KMP) string matching.
• O(S pi ) time per query.
• O(nS Si pi ) time for n queries.
• Suffix tree solution.
• O(S Si pi ) time for n queries.

5
String/Pattern Matching

• KMP preprocesses the query string pi, whereas the
  suffix tree method preprocesses the source string
  S.
• An application of string matching.
• Genome project.
• Databank of strings (gene sequences).
• Character set is ATGC.
• Determine if a new sequence is a substring of a
  databank sequence.

6
Definition Of Suffix Tree

• Compressed trie with edge information.
• Keys are the nonempty suffixes of a given string
  S.
• Nonempty suffixes of S sleeper are
• sleeper
• leeper
• eeper
• eper
• per, er, and r.

7
String Matching Suffixes

• pi is a substring of S iff pi is a prefix of some
  suffix of S.
• Nonempty suffixes of S sleeper are
• sleeper
• leeper
• eeper
• eper
• per, er, and r.
• Which of these are substrings of S?
• leep, eepe, pe, leap, peel

8
Last Character Of S Repeats

• When the last character of S appears more than
  once in S, S has at least one suffix that is a
  proper prefix of another suffix.
• S creeper
• creeper, reeper, eeper, eper, per, er, r

• When the last character of S appears more than
  once in S, use an end of string character to
  overcome this problem.
• S creeper
• creeper, reeper, eeper, eper, per, er, r,
  

9
Suffix Tree For S abbbabbbb
10
Suffix Tree For S abbbabbbb
1
2
5
10
3
1
5
9
4
4
8
3
abbbabbbb
7
2
6
12345678910
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Suffix Tree For S abbbabbbb
1
1
4
5
2
10
1
3
8
1
5
9
4
4
2
8
3
abbbabbbb
7
2
6
12345678910
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Suffix Tree Construction

• See Web write up for algorithm.
• Time complexity
• S n, alphabet size r.
• O(nr) using array nodes.
• This is O(n) for r a constant (or r lt c).
• O(n) expected time using a hash table.
• O(n) time algorithm for large r in reference
  cited in Web write up.

13
Suffix Array

• Array that contains the start position of
  suffixes in lexicographic order.
• abbbabbbb
• Assume lt a lt b
• lt abbbabbbb lt abbbb lt b lt babbbb lt bb lt
  bbabbbb lt bbb lt bbbabbbb lt bbbb
• SA 10, 1, 5, 9, 4, 8, 3, 7, 2, 6
• LCP length of longest common prefix between
  adjacent entries of SA.
• LCP 0, 4, 0, 1, 1, 2, 2, 3, 3, -

14
Suffix Array

• Less space than suffix tree
• Linear time construction
• Can be used to solve several of the problems
  solved by a suffix tree with same asymptotic
  complexity.
• Substring matching ? binary search for p using
  SA.
• O(p log S).

15
O(pi) Time Substring Matching
babb
abbba
baba
16
Find All Occurrences Of pi

• Search suffix tree for pi.
• Suppose the search for pi is successful.
• When search terminates at an element node, pi
  appears exactly once in the source string S.

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Search Terminates At Element Node
abbbb
18
Search Terminates At Branch Node

• When the search for pi terminates at a branch
  node, each element node in the subtree rooted at
  this branch node gives a different occurrence of
  pi.

19
Search Terminates At Branch Node
ab
20
Find All Occurrences Of pi

• To find all occurrences of pi in time linear in
  the length of pi and linear in the number of
  occurrences of pi, augment suffix tree
• Link all element nodes into a chain in inorder.
• Each branch node keeps a pointer to the left most
  and right most element node in its subtree.

21
Augmented Suffix Tree
b
22
Longest Repeating Substring

• Find longest substring of S that occurs more than
  m gt 1 times in S.
• Label branch nodes with number of element nodes
  in subtree.
• Find branch node with label gt m and max char
  field.

23
Longest Repeating Substring
m 2
m 5
24
Longest Common Substring

• Given two strings S and T.
• Find the longest common substring.
• S carport, T airports
• Longest common substring rport
• Longest common subsequence arport
• Longest common subsequence may be found in
  O(ST) time using dynamic programming.
• Longest common substring may be found in
  O(ST) time using a suffix tree.

25
Longest Common Substring

• Let be a new symbol.
• Construct the suffix tree for the string U
  ST.
• U carportairports
• No repeating substring includes .
• Find longest repeating substring that is both to
  left and right of .
• Find branch node that has max char and has at
  least one element node in its subtree that
  represents a suffix that begins in S as well as
  at least one that begins in T.