Counting Suffix Arrays and Strings

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Counting Suffix Arrays and Strings
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Suffix Array Data Structure
Suffix Array lexicographically sorted list of
all suffixes
13 - 12 - C 10 - CTC 5 - CTCTTCTC 7 - CTTCTC
2 - CTTCTCTTCTC 11 - TC 9 - TCTC
4 - TCTCTTCTC 6 - TCTTCTC 1 - TCTTCTCTTCTC 8
- TTCTC 3 - TTCTCTTCTC
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Overview

• Classify strings sharing same suffix array
• Counting strings sharing same suffix array
• Counting suffix arrays? Lower bound suffix array
  compression
• Summation identities

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1. Classify Strings for Suffix Array

• t - string of length n,
• P - permutation of 1,..., n,
• R - inverse of P.
• Theorem
• P is the suffix array of t if and only if
• for all i ?1,...,n
• tPi ? tPi1 and
• tPi tPi1 ? RPi1 ? RPi11
• same as
• RPi1 gt RPi11 ? tPi lt tPi1

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1. Classify Strings for Suffix Array
a) tPi ? tPi1 and b) RPi1 gt
RPi11 ? tPi lt tPi1
Text to be indexed
R-descent
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1. Classify Strings for Suffix Array
Equivalences between strings
Text to be indexed
(order-equivalent)
(order-distinct)
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2. Counting Strings for Suffix Array
Text to be indexed
Base string
Non-decreasing sequences
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2. Counting Strings for Suffix Array

• Suffix array P of length n with d R-descents.
• Number of strings over alphabet of size a for P
• Number of non-decreasing sequences overa-d
  elements

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2. Counting Strings for Suffix Array

• Suffix array P of length n with d R-descents.
• Number of strings composed of exactly k distinct
  characters for P is

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2. Counting Strings for Suffix Array

• Number of strings over alphabet size 20 for
  suffix arrays of length n with 10 R-descents

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2. Counting Strings for Suffix Array

• Suffix array P of length n with d R-descents
• Number of order-distinct strings over alphabet of
  size a is
• Number of order-distinct strings where all k
  distinct characters must appear is

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3. Counting Suffix Arrays

• Definition
• Let P permutation of 1,..., n.
• Position i?1,...,n-1 is a permutation descent
• if Pi gt Pi1.
• Definition
• The Eulerian number gives the number of
• permutations of 1,...,n with exactly d
• permutation descents.

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3. Counting Suffix Arrays

• Well-known fact
• Recursive enumeration of Eulerian numbers
• ,
• for n ? d, and

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3. Counting Suffix Arrays

• Definition
• Let A(n,d) be the number of permutations of
  length n with d R-descents.
• Observation
• A(n,0) 1
• A(n,d) 0 for n ? d
• see next

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3. Counting Suffix Arrays
Text to be indexed
(d1) possible positions without additional
R-descent
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3. Counting Suffix Arrays
Text to be indexed
(d1) possible positions without additional
R-descent
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3. Counting Suffix Arrays

• Together
• A(n,0) 1,
• A(n,d) 0 for n ? d, and
• A(n,d) (d1) A(n-1,d) (n-d) A(n-1,d-1)
• Theorem
• The number A(n,d) of permutations of length n
• with d R-descents is the Eulerian number
  .

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3. Counting Suffix Arrays

• The number of distinct suffix arrays of length n
  for strings over alphabet of size a
• Lower bound for compressibility of suffix arrays
  in the Kolmogorov sense

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3. Counting Suffix Arrays

• Number of distinct suffix arrays of length n for
  strings over alphabet of size 20

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3. Counting Suffix Arrays

• Number of distinct suffix arrays of length n for
  strings over alphabet of size 4

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4. Summation Identities

• Worpitzkis identity by summing up the number of
  strings of length n for each suffix array
• Summation rule for Eulerian numbers to generate
  the Stirling numbers of second kind

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Summary

• Constructive proofs to count strings sharing the
  same suffix array
• Constructive proof to count distinct suffix
  arrays yielding lower bound for suffix array
  compression
• Constructive proofs for Worpitzkis identity and
  the summation rule of Eulerian numbers to count
  Stirling numbers of second kind

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Outlook

• Efficient enumeration algorithm for suffix arrays
• Compressed suffix arrays for fast querying in
  bioinformatics applications
• Average case analysis under non-uniform model

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• Thank you for
• your attention!