Computer Science Department Technion

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Computer Science DepartmentTechnion Israel
Institute of TechnologyGenomic Sorting with
Length-Weighted ReversalsRon Y.
PinterTechnionSteve SkienaSUNY Stony Brook
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Genome Rearrangement

• events
• duplication
• translocation
• reversal (inversion)
• occur primarily during reproduction
• allow large-scale genomic comparisons

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Sorting by Reversals

• genome represented as a permutation on
• 1, 2, , n
• n homologous genes among species
• assumptions
• can identify genes
• genes are distinct
• operation reversal of a subsequence (of genes)
• models inversion (occurs during crossover)
• one of the permutations can be 1, 2, , n
• appropriately relabel others

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Example
4 3 2 8 7 1 5 6 11 10 9

4 3 2 1 7 8 5 6 9 10 11

1 2 3 4 8 7 6 5 9 10 11

1 2 3 4 5 6 7 8 9 10 11
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Our Model

• unsigned
• cost of reversal of subsequence of length l is
  f(l)
• total sorting cost (or distance) is
• f (length(sj))

S Sj are reversed subsequences
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Cost Functions

• additive
• f(xy) f(x) f(y)
• subadditive
• f(xy) lt f(x) f(y)
• superadditive
• f(xy) gt f(x) f(y)
• other
• e.g. bitonic

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Problems

• algorithm to sort any permutation
• worst-case min cost
• approximate min cost for a given permutation

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Extremal Costs

• highly subadditive e.g. unit cost, f(l) 1
• NP complete Caprara, 97
• series of approximation ratios 2, 1.75, 1.375
• highly superadditive f(l) gt l2
• essentially bubblesort

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Our Results

• additive cost function
• specifically f(l) l
• QuickSort-like algorithm for worst-case
• complexity O(n lg2n)
• min cost approximation ratio of O(lg2n)

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MedianEject(a,b)

• find r maximal blocks of wrong-sided elements
  with respect to median
• for lg r do flip every other pair of blocks
  of wrong-sided and adjacent blocks
• move wrong-sided blocks to median boundary
• reverse left and right blocks

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Sample Run

• complexity O((b-a) lg r)

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ReversalSort(a,b)

• MedianEject (a,b)
• ReversalSort (a, )
• ReversalSort ( ,b)
• Complexity
• T(n) 2 ? T ( ) O(f(n) lg n) O(f(n)lg2n)
• O(n lg2n) for f(n)n

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Algorithmic Improvements

• I simplify short phases
• II merge 2 last steps of MedianEject
• when possible (2pq vs. 3pq)
• III apply II recursively

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Approximation Ratio

• M(p) is the maximal total distance between pairs
  of out-of order elements
• Lemma 4 min cost is ?(M(p))
• but
• Lemma 6 of out-of order elts lt 3 ? M(p)
• Lemma 7 MedianEject touches only elements within
  linear range from
• out-of-order elements
• yields
• each round of MedianEject takes O(M(p) ? lg2 n)
• ReversalSort costs O(M(p) ? lg2 n)
• ReversalSort is at most O((lg2 n) times optimal

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Bioinformatic Validation

• use our cost ( distance) to build phylogenetic
  trees
• 4 plants (chloroplastic genes)
• consistent with Martin et al., PNAS Sept 02
• work in progress M. Shoham

Cyanophora
Cyanidium
Guilardia
Porphyra
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Open Problems Algorithmic

• weighted genes
• tighter approximation ratio
• close to O(lg n)
• can get to constant?
• other cost functions (incl. bitonic)
• the signed case

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Open Problems Modeling

• chromosomal ordering
• what is the right cost function?
• consider cost(l) ld
• combine with constant-based models
• restricted regions
• undesired reversal sequences
• deal with duplication and translocation events