Matrix Tightness – A Linear Algebraic Framework for Sorting

by Transpositions

Tzvika Hartman Elad VerbinBar Ilan University Tel Aviv University

Sorting by Transpositions (SBT)• A transposition exchanges between 2

consecutive segments of a perm.• Example : 1 2 3 4 5 6 7 8 9

1 2 6 7 3 4 5 8 9 Transposition distance dt(π): The length of the

shortest sequence of transpositions that transform the given n-permutation to 1,2,…n

SBT – Previous Results

• Complexity of the problem is still unknown.• Best approximation algorithm has ratio

1.375 [EliasHartman05].

Reversal distance dr(π): The length of the shortest sequence of reversals that transform the given n-permutation to 1,2,…n

A Signed Permutation:4 -3 1 -5 -2 7 6Reversal r(i,j): Flip order, signs of numbers in positions i,i+1,..jAfter r(4,6):4 -3 1 -7 2 5 6

Sorting by Reversals (SBR)

Hannenhali & Pevzner (95) gave the first polynomialHannenhali & Pevzner (95) gave the first polynomial solutionsolution..

Basic Components of HP Theory

PermutationPermutation ((ππ11 , … , , … , ππnn))

Breakpoint GraphBreakpoint Graph

Overlap GraphOverlap Graph

Bafna & Pevzner Theory for SBT

PermutationPermutation ((ππ11 , … , , … , ππnn))

Breakpoint GraphBreakpoint Graph

??

Our Results

• Formulation of SBR as a Graph (Matrix) Tightness problem; link to linear algebra.

• A novel combinatorial model (overlap graph) for Sorting by Transpositions.

• Formulation of SBT as Matrix Tightness Problem.

• More about matrix tightness.

Overview of theTalk

• Graph clicking.• Tightness of matrices (link to linear

algebra).• Formulation of SBT as tightness problem.

Graph Clicking

Given a bi-colored graph, define a clicking operation on a black vertex:

Graph Clicking

Define a click operation on a black vertex v:1. Flip the colors in v’s neighborhood

v

Graph Clicking

Define a click operation on a black vertex v:1. Flip the colors in v’s neighborhood

v

Graph Clicking

Define a click operation on a black vertex v:1. Flip the colors in v’s neighborhood2. Flip the existence of edges in v’s

neighborhood

v

Graph Clicking

Define a click operation on a black vertex v:1. Flip the colors in v’s neighborhood2. Flip the existence of edges in v’s

neighborhood

v

Graph Clicking

Define a click operation on a black vertex v:1. Flip the colors in v’s neighborhood2. Flip the existence of edges in v’s

neighborhood3. Delete v

v

Graph Clicking

Define a click operation on a black vertex v:1. Flip the colors in v’s neighborhood2. Flip the existence of edges in v’s

neighborhood3. Delete v

Graph Clicking

v11.

Graph Clicking

2.

v2

Graph Clicking

3.

v3

Graph Clicking

4.v4

Graph Clicking

5.

v5

Graph Clicking

6.v6

Graph Clicking

7.v7

FINISHED8.

Graph Tightness by Clicking

• A graph is called tight if it can be turned into the empty graph by a sequence of n clicking operations.

• The HP-Theorem: An SBR-realizable graph is tight iff every connected component has a black vertex.

Tightness of Matrices

• Let A be the nxn binary adjacency matrix of G, with 1 in the diagonal iff the corresponding vertex is black.

0100000

1110000

0111011

0010100

0001100

0010001

0010011

1

2

3

4

57

6

Clicking vi equals ti iA A v v

0100000

1110000

0111011

0010100

0001100

0010001

0010011

1

2

3

4

57

6

Clicking vi equals ti iA A v v

0100000

1110000

0111011

0010100

0001100

0010001

0010011

1

2

3

4

57

6

0100000

1011011

0111011

0111111

0001100

0111010

0111000

1

2 4

57

6

Tightness of Matrices

• A matrix is tight if it can be turned into the zero matrix by clicking operations.

• Connected to Gaussian elimination and matrix decompositions.

• Gives a novel framework for SBR.

SBT as Matrix Tightness

• Good news: SBT is a matrix tightness problem over the ring M2,2(Z2).

• Bad news: Can’t solve tightness over rings (yet…).

• Possible direction: Solve tightness on non-symmetric matrices over Z2 (equivalent formulation on directed graphs).

Summary

• A full combinatorial model for SBT.• A unified algebraic model for SBR & SBT.• Can be extended to other rearrangement

operations.• We presented some results on the

tightness problem.• This is only the beginning…

Future Directions

• Find more properties of the tightness problem, and solve on more rings/fields.

• Extend the model for the general sorting problem (here we considered only tightness).

• Solve other gr problems by this model.• Solve SBT !!!

Acknowledgements• Ron Shamir.• Haim Kaplan.• Isaac Elias.

Thank you!