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Suffix Trees and String Matching, Slides of Computer Science

Alagappa UniversityComputer Science

Suffix trees and their application in string matching. Suffix trees are a data structure used to efficiently determine if a given pattern is a substring of a larger string. Various types of suffix trees, including suffix arrays and enhanced suffix arrays, and their use in substring matching. It also mentions the knuth-morris-pratt algorithm as an alternative string matching method and its comparison to suffix trees. Examples of suffix trees for different strings and explains how to construct them.

Typology: Slides

2012/2013

Uploaded on 03/19/2013

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Suffix Trees
Suffix trees
•Linearized suffix trees
•Virtual suffix trees
Suffix arrays
•Enhanced suffix arrays
•Suffix cactus, suffix vectors, …
Suffix Trees
•String … any sequence of characters.
•Substring of string S… string composed of
characters ithrough j, i <= j of S.
S = cater => ate is a substring.
car is not a substring.
Empty string is a substring of S.
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Suffix Trees

Suffix trees

  • Linearized suffix trees
  • Virtual suffix trees

Suffix arrays

  • Enhanced suffix arrays
  • Suffix cactus, suffix vectors, …

Suffix Trees

  • String … any sequence of characters.
  • Substring of string S … string composed of

characters i through j, i <= j of S.

 S = cater => ate is a substring.

 car is not a substring.

 Empty string is a substring of S.

Subsequence

  • Subsequence of string S … string composed

of characters i

1

< i

2

< … < i

k

of S.

 S = cater => ate is a subsequence.

 car is a subsequence.

 The empty string is a subsequence.

String/Pattern Matching

  • You are given a source string S.
  • Answer queries of the form: is the string p i

a

substring of S?

  • Knuth-Morris-Pratt (KMP) string matching.

 O(|S| + | p i

|) time per query.

 O(n|S| + Σ i

| p i

|) time for n queries.

  • Suffix tree solution.

 O(|S| + Σ

i

| p i

|) time for n queries.

String Matching & Suffixes

  • p i

is a substring of S iff p

i

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

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#, #

Suffix Tree For S = abbbabbbb#

abbb

b

abbbb# b#

abbbb# #

b

abbbb#

abbbb#

b

b#

Suffix Tree For S = abbbabbbb#

abbb

b

abbbb# b#

abbbb# #

b

abbbb#

abbbb#

b

b#

abbbabbbb#

Suffix Array

  • Array that contains the start position of suffixes in

lexicographic order.

  • abbbabbbb#

 Assume # < a < b

 # < abbbabbbb# < abbbb# < b# < babbbb# <

bb# < bbabbbb# < bbb# < bbbabbbb# < 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, -]

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|).

O(|p

i

|) Time Substring Matching

babb abbba baba

abbb

b

abbbb# b#

abbbb# #

b

abbbb#

abbbb#

b

b#

abbbabbbb#

Find All Occurrences Of p

i

  • Search suffix tree for p i
  • Suppose the search for p i

is successful.

  • When search terminates at an element node, p i

appears exactly once in the source string S.

Search Terminates At Branch Node

abbb

b

abbbb# b#

abbbb# #

b

abbbb#

abbbb#

b

b#

abbbabbbb#

ab

Find All Occurrences Of p

i

  • To find all occurrences of p i

in time linear in

the length of p

i

and linear in the number of

occurrences of p

i

, 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.

Augmented Suffix Tree

abbb

b

abbbb# b#

abbbb# #

b

abbbb#

abbbb#

b

b#

abbbabbbb#

b

Longest Repeating Substring

  • Find longest substring of S that occurs more

than m > 1 times in S.

  • Label branch nodes with number of element

nodes in subtree.

  • Find branch node with label >= m and max

char# field.

Longest Common Substring

  • Let $ be a new symbol.
  • Construct the suffix tree for the string U = S$T#.

 U = carport$airports#

 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.