On the average number of reversals needed to sort signed - PowerPoint PPT Presentation

Introduction Sorting signed permutations Conclusion On the average number of reversals needed to sort signed permutations 1 Thaynara Arielly de Lima & 1 , 2 Mauricio Ayala-Rinc´ on atica 1 & Ciˆ ao 2 Departamentos de Matem´ encia da Computa¸ c˜ Authors funded by CAPES and CNPq “XI Semin´ ario Informal, mas Formal!” November 2013 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Conclusion Introduction 1 Reversals Sorting signed permutations 2 Breakpoint Graph Cycles in Breakpoint Graphs and cycles in permutations Searching the average Conclusion 3 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Reversals Conclusion Genome Rearrangement Figure: A genome alignment of eight Yersinia (Figure in [DMR08]). Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Reversals Conclusion Reversals Figure: A most parsimonious rearrangement scenario for human and mouse X-chromosomes (Figure in [PT03]). Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Reversals Conclusion Genome Rearragement Problem Restricted to reversals... Finding the MINIMUM number of reversals needed to transform a permutation into identity permutation. Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Reversals Conclusion Complexities Operation Example Complexity +1 + 2 − 5 − 4 − 3 + 6 Reversals on signed permutations Polynomial +1 + 2+3 + 4 + 5 + 6 Reversals on unsigned permutations 125436 NP -hard 123456 This work is based in sorting signed permutations by reversals. Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Reversals Conclusion Average number of block interchange Block Interchange 1 45 3 2 6 Polynomial 123456 Consider an unsigned permutation π = π 1 π 2 · · · π n Mikl´ os B´ ona & Ryan Flynn [BF09] showed that: ⌊ ( n +2) / 2 ⌋ − � n 1 1 n − i =2 i a n = 2 where a n = average number of Block Interchange needed to sort permutations of length n . Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Sorting signed permutations Reversals Conclusion Our goal Search for the average number of reversals needed to sort signed permutations. Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Consider π = π 1 π 2 · · · π n . Extend π by adding π 0 = +0 e π n +1 = − 0. Associate to each π i the pair − π i + π i . +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; ii) there is a black edge between vertices with labels π i and − π i +1 , 0 ≤ i < n and π n and π n +1 . +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia

Introduction Breakpoint Graph Sorting signed permutations Cycles in Breakpoint Graphs and cycles in permutations Conclusion Breakpoint Graph Definition (Breakpoint Graph) The Breakpoint Graph G ( π ) of a permutation π is a bi-colored graph with 2 n + 2 vertices such that: i) there is a gray edge between vertices with labels + i and − ( i + 1) , 0 ≤ i < n and + n and − 0 ; ii) there is a black edge between vertices with labels π i and − π i +1 , 0 ≤ i < n and π n and π n +1 . +0 -2 +2 -3 +3 +1 -1 -4 +4 +5 -5 -0 Figure: Breakpoint Graph of permutation π = +2 + 3 − 1 + 4 − 5 Thaynara Arielly de Lima & Mauricio Ayala-Rinc´ on Average reversal distance Semin´ ario Informal Bras´ ılia