An O(N2) Algorithm for Discovering Optimal Boolean Pattern Pairs Hideo Bannai, Heikki Hyyro, Ayumi Shinohara, Masayuki Takeda, Kenta Nakai and Satoru Miyano IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 1, NO. 4,

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An O(N2) Algorithm for DiscoveringOptimal
Boolean Pattern PairsHideo Bannai, Heikki
Hyyro, Ayumi Shinohara, Masayuki Takeda, Kenta
Nakai and Satoru MiyanoIEEE/ACM TRANSACTIONS ON
COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 1,
NO. 4, OCTOBER-DECEMBER 2004Presented
by,Sivaramakrishnan SubramanianGraduate
Student, CPSC, TAMU.siv_at_tamu.edu
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Motive

• Finding patterns conserved across a set of
  biologically related sequences to extract meaning
  is a common topic in Bioinformatics.
• More than one sequence element can affect the
  biological characteristics of the sequences.
• Past work on finding composite patterns-
  Structured Motifs, MITRA, Bioprospector

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Overview

• Given a set of sequences and numeric attribute
  values for each sequence, the problem is to find
  the optimal (w.r.t to a scoring function) pair of
  patterns combined with any Boolean function.
• Past work- finds combination of 2 patterns p and
  q where (pq) occur in each string
• this papers formulation allows all possible
  combinations such as (pq)conditions like
  presence of one element but absence of other
  can be specified.
• Thus this method can be used to find cooperative
  as well as competing sequence elements.
• O(N2) Algorithm and Implementation based on
  suffix arrays (this is the homework!!!) are the
  main contributions of this paper.

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Preliminaries

• Let ? be a finite alphabet e denote an empty
  string.
• Let ?(p,s) be a Boolean matching function? true
  only if p is a substring of s.
• Boolean pattern pair a triplet ltF,p,qgt where p
  and q are patterns and F is a 2-ary Boolean
  function.
• Matching function value for a pattern pair
  ?(ltF,p,qgt,s) is defined as F(?(p,s),?(q,s)).
• All possible F values are defined in the
  following table.

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All Candidate Boolean Operations on ltF,p,qgt
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Preliminaries

• A pattern or a Boolean pattern pair ? matches a
  string s if and only if ?(?,s) is true. Pattern e
  matches any string.
• For a given set of strings Ss1, . . ., sm let
  M(?,S) denote the set of indices of strings in S
  that ? matches, that is, M(?,S)i
  ?(?,si)true, and let its complement be denoted
  as M(?,S)i?(?,si) false.
• For each siS, we are given an associated numeric
  attribute value ri. Let R(?,S) ?iM(?,S)ri
  denote the sum of ri over all si that ? matches.
  Let M(?) and R(?) be a shorthand notation for
  M(?,S) and R(?,S), respectively. Note that
  M(e)m R(e)?i1 to mri.

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Scoring Function

• Objective is to find a pattern that maximizes a
  suitable scoring function score.
• The paper concentrates on scoring functions whose
  values for a pattern ? depend on values cumulated
  over the strings in S that match ?.
• Scoring function score takes parameters M(?)
  and R(?).
• Also assumed that the score value computation can
  be done in constant time if the parameter values
  are known.
• Specific choice for the scoring function highly
  depends on the particular application.

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Problem Definition

• Given a set Ss1, . . ., sm of strings, where
  each string si is assigned a numeric attribute
  value ri and a scoring function score
  RxRgtR, find the Boolean pattern pair
  ?ltF,p,qgt p,q?,FF0,,F15 that
  maximizes score(M(?),R(?)).

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Suffix tree GST

• Edges are labeled with substrings of s.
• For a node v, l(v) is the string obtained by
  concatenating edge labels from root to v.
• For each leaf node v, l(v) is a distinct suffix
  of s for each suffix there exists a leaf v.
• Each node has at least 2 children first
  character of the labels on the edges to its
  children are distinct.
• GST Given a set Ss1, . . ., sm GST is a
  suffix tree for the String s11. . .smm where
  each i is a distinct character that does not
  belong to ?.
• All paths are ended at the first appearance of i
  and each leaf is labeled with idi.
• O(N) space and time.

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Suffix tree
S caggaggaccat. The paths of the suffix tree
from the root to the leaves (suffixes) are sorted
in lexicographic order from left to right, each
leaf corresponding to a position in the suffix
array. The integer in the suffix array represents
the position in the string from which the
corresponding suffix starts. Asij indicates
sjn is the ith suffix in the lexicographic
ordering The lcp array represents the length of
the longest path that consecutive suffixes in the
suffix array share.
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GST (Generalized Suffix Tree)
A Generalized Suffix Tree and its corresponding
suffix array for the strings facct, gctt, ctctg.
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A Naïve O(N3) Algorithm

• Let N ?i1 to mlength(si)
• O(N) candidates for a single pattern? patterns of
  form l(v), where v is a node in the GST over the
  set S. (Why???)
• Hence O(N2) candidate pattern pairs
• For a given pair ltF,l(v1),l(v2)gt, the values
  M(?) and R(?) can be computed in O(N) time by
  any of the linear time string matching
  algorithms.
• Then scoring function value is calculated in
  constant time given M(?) and R(?).
• TimeO(N3). SpaceO(N) for Suffix tree.

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O(N2) Algorithm

• Two steps
• Find M(l(v)) and R(l(v)) for all nodes v of GST
  in O(N) time and space
• Solve optimal pair of substring patterns problem
  in O(N2) time and O(N) space for any scoring
  function score provided that it can be calculated
  in constant time given its inputs.

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Algorithm- First step

• If R(l(v)) for all v can be found in O(N) time so
  can be M(l(v). (when ri1 for all i,
  R(l(v)M(l(V))
• Let LF(v) be the set of all leaf nodes in the
  subtree rooted by node v.
• Let ci(v) denote the number of leaves in LF(v)
  that have the label idi.
• Let sum of leaf attributes be ?LF(v)ri.

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Algorithm- First step

• ?LF(v)ri ?iM(l(v))(ci(v).ri)
• R(l(v)) ?iM(l(v))ri ?LF(v)ri -
  ?iM(l(v))((ci(v)-1).ri) (1)
• Let correction factor be corr(l(v),S)?iM(l(v))((
  ci(v)-1).ri)
• In (1) ?LF(v)ri can be calculated for all v using
  a linear time post-order traversal as ?LF(v)ri
  ?v(?LF(v)ri v is a child node of v).

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Algorithm- First step

• How to remove the redundancies (correcting
  factors) in (1)?
• Let I(idi) be the list of all leaves with the
  label idi in the order they appear in the
  post-order traversal of the tree. Constructing
  the lists I can be done in linear time for all
  labels idi.
• The leaves in LF(v) with the label idi form a
  continuous interval of length ci(v) in the list
  I(idi).
• If ci(v) gt 0, a length-ci(v) interval in I(idi)
  contains (ci(v)-1) adjacent (overlapping) leaf
  pairs.
• If x,y LF(v), the node lca(x,y) belongs to the
  subtree rooted by v.
• For any si S, ?(l(v),si)true, that is, i
  M(l(v)) if and only if there is a leaf x LF(v)
  with the label idi.

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Algorithm- First step

• Initially correction value0 for all v.
• For each adjacent leaf pairs in I(idi) add ri to
  the correction value of the node lca(x,y).
• For each v, sum of correction values in the nodes
  of the sub-tree rooted by v is (ci(v)-1).ri.
• Repeat this for all lists I(idi)- the preceding
  total sum becomes ?iM(l(v))((ci(v)-1).ri)
  corr(l(v),S)
• Perform a linear time bottom-up (post-order)
  traversal to find R(l(v)).

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Algorithm- First step
Correction values at v1,v2,v3 set to r3,r2,r3
V3r3r2r3-r3 r2r3R(l(v3)) V2R(l(v3))r2-r2
r2r3R(l(v2)) V1r1R(l(v2))r3-r3
r1r2r3R(l(v1))
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Pseudo code for Step 1
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O(N2) Algorithm

• Two steps
• Find M(l(v)) and R(l(v)) for all nodes v of GST
  in O(N) time and space
• Solve optimal pair of substring patterns problem
  in O(N2) time and O(N) space for any scoring
  function score provided that it can be calculated
  in constant time given its inputs.

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Algorithm- Second step

• O(N) choices for the first pattern?l(v1)
• For each l(v1)? use a modified version of the
  previous algorithm for the O(N) choices for the
  second pattern,l(v2)
• given a fixed l(v1), we additionally label each
  string siS and the corresponding leaves in the
  GST with the Boolean value ?(l(v1),si)? O(N)
  time.
• Cumulate the sums and correction values
  separately for true and false values of the
  additional label.

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Algorithm- Second step

• ?iM(l(v2))(ri ?(l(v1),si) true)
  ?iM(l(v2))(ri ?(l(v1),si) true,
  ?(l(v2),si) true) R(ltF8,l(v1),l(v2)gt)
• ?iM(l(v2))(ri ?(l(v1),si) false)
  ?iM(l(v2))(ri ?(l(v1),si) false,
  ?(l(v2),si) true) R(ltF2,l(v1),l(v2)gt)
• ?iM(l(v2))(ri ?(l(v1),si) true)
  ?iM(l(v2))(ri ?(l(v1),si) true,
  ?(l(v2),si) false) R(ltF4,l(v1),l(v2)gt)
  R(l(v1)) - R(ltF8,l(v1),l(v2)gt)
• ?iM(l(v2))(ri ?(l(v1),si) false)
  ?iM(l(v2))(ri ?(l(v1),si) false,
  ?(l(v2),si) false) R(ltF1,l(v1),l(v2)gt)
  R(e) R(l(v1) - R(ltF2,l(v1),l(v2)gt)
• where R(e) R(l(v1) can be computed in linear
  time.

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Algorithm- Second step

• All cumulative values of the form ?i(ri
  ?(l(v1),si) b1, ?(l(v2),si)b2) where
  b1,b2true,false can be computed in linear
  time.
• Thus R(ltF,l(v1),l(v2)gt) and hence the score can
  be computed in linear time for all pairs of the
  form ltF,l(v1),l(v2)gt, given a fixed l(v1).
• Thus O(N2) for all pattern pairs.
• Since the O(N) calculations for each l(v1) is
  independent, the same GST can be reused. Hence
  the space complexity is O(N).

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Algorithm- Second step
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The rest of the paper in a nutshell

• Extension for k-ary Boolean function.
• Implementation using suffix arrays.
• Computational experiments and results.
• Algorithm Variations? Multiple String Attributes,
  Distance Restrictions.

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Homework

• Explain the implementation of the Optimal Boolean
  Pattern Pair problem using suffix arrays in your
  own words. Also explain why is it more efficient
  than the suffix tree approach.

Email siv_at_tamu.edu
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• THANK YOU