An Optimal Algorithm for MAX-SUM SEGMENT

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An Optimal Algorithm for MAX-SUM SEGMENT Its
Applications in Bioinformatics

• Hsueh-I Lu
• Academia Sinica, Taiwan

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This is joint work with

• Institute of Statistics, National Central
  University
• Tsai-Hung Fan
• Tsung-Shan Tsou
• Institute of Biomedical Sciences, Academia Sinica
• Adam Yao
• Shufen Lee
• Tsai-Cheng Wang

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Outline

• The MAX-SUM SEGMENT problem
• Previous work and our algorithm
• A feature of our algorithm
• Processing the input sequence in an on-line
  manner
• An application in bioinformatics
• Finding new repeats in genomic sequences

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The MAX-SUM SEGMENT Problem

• Input
• a sequence S of n numbers
• two length bounds L and U.
• Output
• a segment Si, j with maximum sum over all
  segments of S with length at least L and at most
  U.

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Example
sum 2
sum 4

• S -3 2 -2 5 -4 1 -2 3 1
• L 2 U 2
• L 2 U 8

sum 5
sum 3
A feasible segment
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Simple Observations

1. There are only O(n2) segments Si, j in S.
2. The sum of each segment Si, j can be computed
   in O(1) time.

O(n) values for i and O(n) values for j.
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Simple Observations

1. There are only O(n2) segments Si, j in S.
2. The sum of each segment Si, j can be computed
   in O(1) time.

With appropriate linear-time preprocessing
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Prefix sums of S

• Define prefix-sum(i) sum of S1, i
• O(n) time to pre-compute prefix-sum(i) for all
  indices i.
• Sum of Si, j
• prefix-sum(j) prefix-sum(i 1)

j
i
1
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As a result

• The problem has a naïve O(n2)-time algorithm
• Go through all O(n2) feasible segments and output
  one segment with maximum sum.

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Previous work

• Lin, Jiang, and Chao JCSS 2002 gave the first
  known O(n)-time algorithm for the MAX-SUM SEGMENT
  problem.

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Lin-Jiang-Chao

• Based upon a clever but somewhat complicated
  technique called left-negative decomposition of
  the input sequence.

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Our result

• An alternative linear-time algorithm
• Bypassing the pre-processing step of
  left-negative segment decomposition.
• Has the capability to process the input sequence
  in an on-line manner

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On-line processing

• Suppose the input sequence S is given in
  iterations.
• For each i 1, 2, , n, the number Si is given
  in the i-th iteration.
• When Si is received, our algorithm is capable
  of interatively outputing a max-sum segment of
  S1,i.
• The required working space is O(U L 1).

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Our algorithm
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Geometric representation
height of j-th point prefix-sum(j)
S 2 -1 1 1 -2 3 -2
1
2
j
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Sum of Si, j prefix-sum(j) prefix-sum(i
1)

• For each ending index j,
• Maximizing the sum of Si, j is equivalent to
  minimizing prefix-sum(i 1).

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Lowest point in j-U, j-L
valley(j)
Feasible index set F(j) for index j.
Feasible index set F(j) for index j.
j
j U
j L
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valley(j)

• Clearly, for each ending index j, 1valley(j) has
  to be the best starting index.
• In other words, 1valley(j) maximizes the sum of
  Si, j over all feasible starting indices i.

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As a result

• The MAX-SUM SEGMENT problem is linear-time
  reducible to computing valley(j) for all indices
  j.
• More specifically, the solution to MAX-SUM
  SEGMENT is simply the segment S1valley(j), j
  with maximum sum over all indices j.

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Computing valley(j) for all indices j in O(n) time

• Case 1 U is ineffective (e.g., U n)
• Case 2 U is effective

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Case 1 U is ineffective
j
F(j)
j 1
F(j 1)
j 2
F(j 2)
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F(j 1) F(j) ? j-L1

• if prefix-sum( jL1) lt prefix-sum(valley(j))
• let valley(j 1) j L 1
• otherwise,
• let valley(j 1) valley(j)

O(n) time to compute valley(j) for all j
O(n) time to compute a max-sum segment.
O(1) time to compute valley(j 1) from valley(j)
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Case 2 U is effective
F(j)
j
j 1
j 1
F(j 1)
F(j 2)
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We need a data structure D(j) for the indices in
F(j)

• The specification
• valley(j) can be obtained from D(j) in O(1)
  time.
• D(j 1) can be updated from D(j) in amortized
  O(1) time.

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A solution

• Let D(j) be a list of indices X(1), X(2) such
  that
• X(1) is the lowest point in F(j) j U, jL
• That is, X(1) valley(j).
• X(2) is the lowest point in X(1)1, jL
• X(3) is the lowest point in X(2)1, jL

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j - U
j - L
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Spec. 1

• valley(j) can be obtained from D(j) in O(1)
  time?
• Yes! Just read the first number in D(j).

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Spec. 2

• D(j1) can be updated from D(j) in amortized O(1)
  time?
• Yes! Two steps
• (1) If valley(j) is not in F(j1), we just delete
  the first number from the list.
• (2) Scan the list of indices in D(j) from right
  to left to see where j-U1 fits in.

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time complexity

• The overall time complexity is O(n), although
  some iterations may take more than constant time.

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On-line processing

• In the j-th iteration, when Sj is received,
• prefix-sum(j) can be computed on the fly
• valley(j) can be computed interatively,
• So, our algorithm can process the input sequence
  in an online manner.
• the required working space is O(U L 1).

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An application in bioinformatics

• Identifying repeats in genomic sequences.

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Repeats

• DNA repeats usually contain important biological
  information.
• TIGR (The Institute of Genomic Research)
  maintains databases for repeats in various
  genomes.

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Finding repeats via MAX-SUM SEGMENT

• Input a DNA sequence R
• Output a segment Ri,j of R that is likely to
  be a previously unknown repeat in R.
• Step 1. Filter out known repeats
• Step 2. Self-alignment and reducing the 2D
  alignment scores into a sequence S of numbers.
• Step 3. Run MAX-SUM SEGMENT algorithm on S

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1.Masking Repeats of Chromosome 1 listed in the
Tigr Rice Repeat Database
Unmasked Chromosome 1
Masked Chromosome 1
Masked region
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2.Filtering Simple Regions of Chromosome 1
Masked Chromosome 1
Filtered and Masked Chromosome 1
Masked region
Filtered region
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Aligning Filtered and Rice Chromosome 1 against
itself
Negative scores to penalize unaligned area.
E lt 1e -10
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Max-Sum
Adding the scores in the same column into a
score.
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left36345576 right36345588 Score253
Length44744612 4 1 30 144 56 2 9 2 1 1 9
-3 36345580
ataaaaaatattagaagaaaagtatagagtgcatatagaaatataattaa
gaaataatagaaattcggaattagaaaacaacagatattagaagaagagt
atagagtccatataagaatttagaatgaactaaaattcggaataaaaatt
aaaattaaagatagaatttagagtctata
over 80 homologues (e-value lt e-10 ) in rice
chromosome 1 but none in the TIGR Rice Repeat
Database released on July 29, 2002
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Conclusion

• We give a new O(n)-time algorithm for the MAXSUM
  SEGMENT problem.
• Our algorithm can handle the input sequence in an
  online manner, which is an important feature for
  handling genome-sized input. The working space
  required is only O(U-L1).
• Our algorithm can be an important subroutine in
  finding new repeats in genomic sequences.