Tzvika Hartman Elad Verbin Bar Ilan University Tel Aviv University Matrix Tightness – A Linear Algebraic Framework for Sorting by Transpositions Tzvika Hartman Elad Verbin Bar 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].

Sorting by Reversals (SBR) A Signed Permutation: 4 -3 1 -5 -2 7 6 Reversal r(i,j): Flip order, signs of numbers in positions i,i+1,..j After r(4,6): 4 -3 1 -7 2 5 6 Reversal distance dr(π): The length of the shortest sequence of reversals that transform the given n-permutation to 1,2,…n Hannenhali & Pevzner (95) gave the first polynomial solution.

Basic Components of HP Theory Permutation (π1 , … , πn) Breakpoint Graph Overlap Graph

Bafna & Pevzner Theory for SBT Permutation (π1 , … , πn) Breakpoint 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: Flip the colors in v’s neighborhood v

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

Graph Clicking Define a click operation on a black vertex v: Flip the colors in v’s neighborhood Flip the existence of edges in v’s neighborhood v

Graph Clicking Define a click operation on a black vertex v: Flip the colors in v’s neighborhood Flip the existence of edges in v’s neighborhood v

Graph Clicking Define a click operation on a black vertex v: Flip the colors in v’s neighborhood Flip the existence of edges in v’s neighborhood Delete v v

Graph Clicking Define a click operation on a black vertex v: Flip the colors in v’s neighborhood Flip the existence of edges in v’s neighborhood Delete v

Graph Clicking 1. v1

Graph Clicking v2 2.

Graph Clicking v3 3.

Graph Clicking 4. v4

Graph Clicking 5. v5

Graph Clicking v6 6.

Graph Clicking 7. v7 8. FINISHED

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. 4 2 1 1 3 5 7 6

Clicking vi equals 1 4 2 1 3 5 7 6

Clicking vi equals 1 4 2 1 3 5 7 6 2 4 1 1 5 7 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 !