Sorting by Transpositions via Matrix Tightness

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Sorting by Transpositions via Matrix Tightness

• Tzvika Hartman and Elad Verbin

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Sorting by Reversals (SBR)

• INPUT A signed permutation
• OUTPUT A shortest sequence of reversals that
  transforms it to (1,2,...,n)

(3 4 -2 -5 1) (-4 -3 -2 -5 1) (-4 -3 -2 -1 5) (
1 2 3 4 5)
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SBR - History

• Introduced by Sankoff in 1990.
• First polynomial Algorithm by Hannenhalli-Pevzner
  in 1995. Time O(n4).
• Best algorithm so far Eric Tannier and
  Marie-France Sagot, 2004. O(n3/2log1/2n).

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SBR - motivation

• Reversals are the most common large-scale genome
  mutation
• The reversal distance is a measure for the
  genomic distance, e.g. between Cabbage and
  Turnip.

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Sorting by Transpositions (SBT)

• INPUT An unsigned permutation
• OUTPUT A shortest sequence of transpositions
  that transforms it to (1,2,...,n)

(3 4 2 5 1) (1 3 4 2 5) (1 2 3 4 5)
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SBT - History

• Introduced in 1995 by Bafna and Pevzner
• No polynomial algorithm yet.
• 1.5-approximations by Bafna and Pevzner, by
  Christie and by Hartman
• 1.375-approx by Elias and Hartman

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Talk Overview

• The paper gives a graphic model for SBT
• We reduce a hard problem (SBT) to a (maybe
  easier) problem the tightness problem.
• Part I The Graph Tightness Problem
• Part II How the reduction works

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Part I The Tightness Problem
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The Clicking Game

• You are given a graph.

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The Clicking Game

• A Click operation on a black vertex v is defined
  as
• 1. Flip the existence of all
• edges in v's neighborhood
• 2. Flip the colors of v's neighbors
• 3. Delete v

v
v
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The Clicking Game

• A Click operation on a black vertex v is defined
  as
• 1. Flip the existence of all
• edges in v's neighborhood
• 2. Flip the colors of v's neighbors
• 3. Delete v

v
v
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The Clicking Game

• A Click operation on a black vertex v is defined
  as
• 1. Flip the existence of all
• edges in v's neighborhood
• 2. Flip the colors of v's neighbors
• 3. Delete v

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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

1.
v1
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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

v2
2.
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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

v3
3.
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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

4.
v4
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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

5.
v5
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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

v6
6.
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The Clicking Game

• GOAL To sort the graph -- To turn it to a graph
  with only isolated white vertices. Is that
  possible?

7.
v7
8.
FINISHED
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The Graph Tightness Problem

• A graph is called tight iff it can be sorted in
  this way.
• The tightness problem
• INPUT A graph
• OUTPUT is it tight?

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The Graph Tightness Problem

• Hanenhalli and Pevzner reduced the SBR problem
  into the graph tightness problem, and then solved
  it in polynomial time.
• H-P Theorem A graph is tight iff every connected
  component either contains a black vertex or is an
  isolated white vertex.

NON-TIGHT
TIGHT
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Our results about SBT

• We reduce SBT into a tightness problem where
  the graph is much more complicated.
• I will present a simplification. We believe that
  if you solve it, you solve SBT. (and if it is
  NP-Hard then SBT also is)

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The Directed Clicking Game

• You are given a directed graph.

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The Directed Clicking Game

• A Click operation on a black vertex v is defined
  as
• 1. Flip the existence of all
• edges going from v's in-neighbors
• to v's out-neighbors
• 2. Flip colors of
• 2-directional
• neighbors of v
• 3. Delete v

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The Directed Clicking Game

• A Click operation on a black vertex v is defined
  as
• 1. Flip the existence of all
• edges going from v's in-neighbors
• to v's out-neighbors
• 2. Flip colors of
• 2-directional
• neighbors of v
• 3. Delete v

(and we get stuck in the next step)
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The Directed Graph Tightness Problem

• A graph is called tight iff it can be turned into
  a trivial graph using clicking operations
• The directed tightness problem
• INPUT A directed graph
• OUTPUT is it tight?

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The Directed Graph Tightness Problem

• Unfortunately, the characterization of the
  undirected case does not carry over
• A cycle of length n is tight iff it has n or n-1
  black vertices, and is not tight otherwise.
• Also, it's a non-monotone property adding edges
  might make the graph not tight

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Please solve it!

• Please solve the directed clicking game for us.
• We'll give you money.
• 128 divided by number
• of pages in the solution

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Part II How the reduction works(we'll show
how H-P got from SBR to the clicking problem. Our
paper is the first time someone works with SBT in
an analogous manner)
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Solving SBR

• (4 3 1 -5 -2)
• (frame the permutation by 0, n1)
• (0 4 3 1 -5 -2 6)
• a pair -- two consecutive numbers -- (1,-2),
  (4,3).
• good pair -- different signs (1,-2)
• bad pair -- same sign (4,3)
• A good pair can be used to create an adjacency
• Proposition A sequence that uses only good pairs
  is shortest

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Solving SBR

• How does using a good pair effect other pairs?
• (0 4 3 1 -5 -2 6)
• Using the good pair (1,-2) turns (4,-5) into bad
• The Overlap Graph models the effect of using
  pairs on the goodness of other pairs.
• We get the clicking game.

Permutation can be sorted using only good pairs
Overlap Graph is tight
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Analogous Model for SBT

• Instead of pairs we have triplets (forks)
• (0 1 3 5 4 6 2 7)
• Good forks, bad forks, need to model their
  effects on each other
• The model a matrix over a ring.
• The directed graph problem a simplification of
  this

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Thanks!

• To you!
• To Haim Kaplan and Ron Shamir.
• To whoever solves our problem (make mony
  fast!11!!)