Edit Distances and Factorisations of Even Permutations
32
0
0
Texte intégral
(2) Edit Distances and Factorisations of Even Permutations Introduction. Edit distances Edit operations: given fixed set of allowed operations; Edit distance: minimum number of edit operations needed to transform X into Y ;. . ←→. ···. . ←→. Many applications: spelling correction (example: type “dsitnace” in Google); genome rearrangements; interconnection networks; .. ..
(3) Edit Distances and Factorisations of Even Permutations Introduction Permutations. Permutations Permutations can model: genomes and mutations [Hannenhalli and Pevzner, 1999]. devices in interconnection networks h2 3 1 4i. h4 2 1 3i. h1 2 4 3i. h3 2 4 1i. h2 3 4 1i. h4 3 2 1i. h3 2 1 4i. h1 2 3 4i. h4 2 3 1i. h2 4 3 1i. h3 4 2 1i. h1 4 2 3i. h1 3 2 4i. h3 1 2 4i. h2 1 3 4i. h4 1 3 2i. h1 4 3 2i. h3 4 1 2i. h2 4 1 3i. h4 1 2 3i. h2 1 4 3i. h3 1 4 2i. h1 3 4 2i. h4 3 1 2i.
(4) Edit Distances and Factorisations of Even Permutations Introduction Permutations. Permutations: basic definitions Permutation: linear ordering of {1, 2, . . . , n}; Disjoint cycle decomposition: 1 2 3 4 5 6 7 = (1, 4, 2)(3, 6, 7)(5). 4 1 6 2 5 7 3 The graph of permutation π, denoted by Γ(π):. 4. 1. 6. 2. 5. 7. 3. π is even if Γ(π) has an even number of even cycles; Conjugacy class: permutations with the same decomposition; 1-cycles (or fixed points) are often omitted;.
(5) Edit Distances and Factorisations of Even Permutations Introduction The problem(s). The problem(s). Let: π be a permutation of {1, 2, . . . , n}; S = {s1 , s2 , . . .} be a set of permutations of {1, 2, . . . , n} (the edit operations); ι be the identity permutation h1 2 · · · ni;. We want to: 1. “sort π by S”: find a sequence of elements of S that sorts π and is as short as possible: π ◦ x1 ◦ x2 ◦ · · · ◦ xt = ι where x1 , . . . , xt ∈ S and t is minimal. 2. “compute the S-distance dS (π, ι)”: find the length of such a sequence;.
(6) Edit Distances and Factorisations of Even Permutations Introduction The problem(s). Some edit operations. From genome rearrangements: reversals: transpositions: block-interchanges:. h3 2 5 4 1i → h3 2 1 4 5i h3 2 5 4 1 i → h 3 4 1 2 5i h 5 4 3 2 1i → h 3 4 5 2 1 i. From interconnection networks: prefix reversals: prefix transpositions:. h2 3 5 4 1i → h4 5 3 2 1i h 3 2 5 4 1 i → h4 1 3 2 5i.
(7) Edit Distances and Factorisations of Even Permutations Introduction The problem(s). classical. Operation reversals signed reversals block-interchanges transpositions2. Sorting NP-hard O(n3/2 ) O(n log n) ?. Distance NP-hard O(n) O(n) ?. Diameter n−1 n+1 n/2 n 2 ≤ ? ≤. prefix. Background. reversals signed reversals transpositions. ? ? ?. ? ? ?. 15n 14 ≤ ? 3n 2 ≤ ? ≤ 2n 3 ≤ ? ≤. All three prefix variants are 2-approximable; 2. 11/8-approximable [Elias and Hartman, 2006]. 2n 3 ≤ 18n 11. 2(n − 1) n − log8 n.
(8) Edit Distances and Factorisations of Even Permutations Introduction The problem(s). Results. Expression of the “cycle graph” [Bafna and Pevzner, 1998] of π as an even permutation π; Reformulation of every edit distance problem on π in terms of particular factorisations of π; Simple recovery of previous results; New lower bound on the prefix transposition distance, which outperforms previous results; Improved lower 3n+1 bound on the maximal value of that distance ( 2n → ); 3 4.
(9) Edit Distances and Factorisations of Even Permutations Results The “cycle graph”. The “cycle graph” [Bafna and Pevzner, 1998] The “cycle graph” of π, denoted by G (π): π5. π6 7 5. π7 3 (here π = h4 1 6 2 5 7 3i). π4 2. π3. 0 π0 6 1 π2. 4 π1. 1. V (G ) = (π0 = 0, π1 , π2 , . . . , πn );. 2. E (G ) =.
(10) Edit Distances and Factorisations of Even Permutations Results The “cycle graph”. The “cycle graph” [Bafna and Pevzner, 1998] The “cycle graph” of π, denoted by G (π): π5. π6 7 5. π7 3 (here π = h4 1 6 2 5 7 3i). π4 2. π3. 0 π0 6 1 π2. 4 π1. 1. V (G ) = (π0 = 0, π1 , π2 , . . . , πn );. 2. E (G ) ={black arcs}.
(11) Edit Distances and Factorisations of Even Permutations Results The “cycle graph”. The “cycle graph” [Bafna and Pevzner, 1998] The “cycle graph” of π, denoted by G (π): π5. π6 7 5. π7 3 (here π = h4 1 6 2 5 7 3i). π4 2. π3. 0 π0 6 1 π2. 4 π1. 1. V (G ) = (π0 = 0, π1 , π2 , . . . , πn );. 2. E (G ) ={black arcs} ∪ {grey arcs};.
(12) Edit Distances and Factorisations of Even Permutations Results The “cycle graph”. The “cycle graph” [Bafna and Pevzner, 1998] The “cycle graph” of π, denoted by G (π): π5. π6 7 5. π7 3 (here π = h4 1 6 2 5 7 3i). π4 2. π3. 0 π0 6 1 π2. 1. V (G ) = (π0 = 0, π1 , π2 , . . . , πn );. 2. E (G ) ={black arcs} ∪ {grey arcs};. 4 π1. Unique decomposition into “alternating cycles”;.
(13) Edit Distances and Factorisations of Even Permutations Results G (π) as an even permutation. G (π) as an even permutation “Monochrome” decomposition: 7 5. 7 3. 2. 5 0. 6. 4 1. 3. 2. 0 6. 4 1.
(14) Edit Distances and Factorisations of Even Permutations Results G (π) as an even permutation. G (π) as an even permutation “Monochrome” decomposition: 7 5. 7 3. 2. 5 0. 6. 4 1 π̂ = (0, 3, 7, 5, 2, 6, 1, 4). 3. 2. 0 6. 4 1 π̇ = (0, 1, 2, 3, 4, 5, 6, 7).
(15) Edit Distances and Factorisations of Even Permutations Results G (π) as an even permutation. G (π) as an even permutation “Monochrome” decomposition: 7 5. 7 3. 2. 5 0. 2. 6. π=. 4 1 π̂ = (0, 3, 7, 5, 2, 6, 1, 4). 3 0. 6. ◦. 4 1 π̇ = (0, 1, 2, 3, 4, 5, 6, 7).
(16) Edit Distances and Factorisations of Even Permutations Results G (π) as an even permutation. G (π) as an even permutation “Monochrome” decomposition: 7 5 2. 3. 5 0. 6. π= =. 7 3. 2. 4 6 4 1 1 π̂ π̇ = = (0, 3, 7, 5, 2, 6, 1, 4) ◦ (0, 1, 2, 3, 4, 5, 6, 7) (0, 4, 2, 7, 3)(1, 6, 5). 0.
(17) Edit Distances and Factorisations of Even Permutations Results G (π) as an even permutation. G (π) as an even permutation We have π = π̂ ◦ π̇, with Γ(π) ' G (π); indeed: 7 5. 3. 2. 0 6. π= =. 4. 1 π̂ π̇ = = (0, 3, 7, 5, 2, 6, 1, 4) ◦ (0, 1, 2, 3, 4, 5, 6, 7) (0, 4, 2, 7, 3)(1, 6, 5).
(18) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. A general lower bounding technique Note that “sorting by S” is equivalent to “factorising by S”: −1 π ◦ x1 ◦ x2 ◦ · · · ◦ xt = ι ⇔ π = xt−1 ◦ xt−1 ◦ · · · ◦ x1−1 | {z } | {z } x1 ,x2 ,...,xt ∈S. x1−1 ,x2−1 ,...,xt−1 ∈S. Theorem 1 Let: 1. S ⊂ Sn , with S = {s1 , s2 , . . .},. 2. S 0 = {s1 , s2 , . . .},. 3. C the set of conjugacy classes that intersect S 0 .. Then for all π in Sn , every factorisation of π into t elements of S yields a factorisation of π into t elements of C..
(19) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. Theorem 1 in action. Our problem: computing dS (π):. π. π ◦ x1. π ◦ x1 ◦ x2. ···. ι.
(20) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. Theorem 1 in action. Our problem: computing dS (π):. New problem: computing dC (π):. π. π. π ◦ x1. π ◦ x1 ◦ x2. ···. ι. ι.
(21) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. Theorem 1 in action. Our problem: computing dS (π):. π. π ◦ x1. π ◦ x1 ◦ x2. π. π ◦ y1. π ◦ y1 ◦ y2. ···. ι. dC (π) ≤ dS (π) New problem: computing dC (π):. ··· ι.
(22) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. Theorem 1 in action. Our problem: computing dS (π):. π. π ◦ x1. π ◦ x1 ◦ x2. π. π ◦ y1. π ◦ y1 ◦ y2. ···. ι. dC (π) ≤ dS (π) New problem: computing dC (π):. ···. Lemma 2 For all π, σ in Sn : π ◦ σ = π ◦ σ ◦ π −1 ◦ π = σ π ◦ π.. ι.
(23) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. A lower bound on the block-interchange distance Example 3 (lower bound on bid(π)) S ={block-interchanges}, denoted by β(i, j, k, l); . 1 ··· i − 1 i ··· j − 1 j j + 1 ··· k − 1 k ··· l − 1 l l + 1 ··· n 1 ··· i − 1 k ··· l − 1 j j + 1 ··· k − 1 i ··· j − 1 l l + 1 ··· n. ..
(24) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. A lower bound on the block-interchange distance Example 3 (lower bound on bid(π)) S ={block-interchanges}, denoted by β(i, j, k, l); we have: β(i, j, k, l) = (i −1, k −1)(j −1, l −1). (1 ≤ i < j ≤ k < l ≤ n+1).. We have S 0 ⊆ C, where C contains all pairs of 2-cycles; We have dC (π) =. |π|−c(Γ(π)) ; 2. Therefore, we recover the result of [Christie, 1996]: ∀ π ∈ Sn : bid(π) ≥. n + 1 − c(Γ(π)) . 2.
(25) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. A lower bound on the transposition distance Example 3 (lower bound on td(π)) S ={transpositions}, denoted by τ (i, j, k); 1 ··· i − 1 i i + 1 ··· j − 2 j − 1. 1 ··· i − 1 j j + 1 ··· k − 1. j j + 1 ··· k − 1 k ··· n. i i + 1 ··· j − 2 j − 1 k ··· n. ..
(26) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. A lower bound on the transposition distance Example 3 (lower bound on td(π)) S ={transpositions}, denoted by τ (i, j, k); we have: τ (i, j, k) = (i − 1, k − 1, j − 1). (1 ≤ i < j < k ≤ n + 1).. We have S 0 ⊆ C, the set of all 3-cycles; We have dC (π) =. |π|−codd (Γ(π)) ; 2. Therefore, we recover the result of [Bafna and Pevzner, 1998]: ∀ π ∈ Sn : td(π) ≥. n + 1 − codd (Γ(π)) . 2.
(27) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. A new lower bound on the prefix transposition distance Example 3 (lower bound on ptd(π)) S ={prefix transpositions}; we get: τ (1, j, k) = (0, k − 1, j − 1). (1 < j < k ≤ n + 1).. We have S 0 ⊆ C, the set of all 3-cycles that contain 0; We can compute dC (π), and this yields the following new lower bound : n + 1 + c(Γ(π)) 0 if π1 = 1, ∀ π ∈ Sn : ptd(π) ≥ −c1 (Γ(π))− 1 otherwise. 2.
(28) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. Quality of the results Block-interchanges: the lower bound is the exact distance; Prefix transpositions: the new result:. Percentage of permutations tight with respect to the given lower bound. always outperforms [Dias and Meidanis, 2002]; “often” outperforms [Chitturi and Sudborough, 2008]: Dias and Meidanis Chitturi and Sudborough Labarre. 100. 80. 60. 40. 20. 0 1. 2. 3. 4. 5. 6. Values of n. 7. 8. 9. 10.
(29) Edit Distances and Factorisations of Even Permutations Results A general lower bounding technique. The prefix transposition diameter of Sn , thereby These permutations satisfy ptd(π) ≥ 3n+1 4 improving on the lower bound of 2n/3 by [Chitturi and Sudborough, 2008]:. (a) 0. 3. 2. 1. 4. 7. 6. 5. 8. 0. 1. 4. 3. 2. 5. 8. 7. 6. 9. 0. 1. 4. 3. 2. 5. 8. 7. 6. 9. 10. 0. 2. 1. 3. 6. 5. 4. 7. 10. 9. 8. (b). (c). (d) 11.
(30) Edit Distances and Factorisations of Even Permutations Future work. Future work. Complexity/approximation issues (transpositions, prefix operations); Can the π model provide upper bounds? Extending the π model to signed permutations and/or other structures;.
(31) Edit Distances and Factorisations of Even Permutations Future work. Thank you!.
(32) Edit Distances and Factorisations of Even Permutations References. Bafna, V. and Pevzner, P. A. (1998). Sorting by transpositions. SIAM Journal on Discrete Mathematics, 11(2):224–240 (electronic). Chitturi, B. and Sudborough, I. H. (2008). Bounding prefix transposition distance for strings and permutations. In HICSS, page 468, Los Alamitos, CA, USA. Christie, D. A. (1996). Sorting permutations by block-interchanges. Information Processing Letters, 60(4):165–169. Dias, Z. and Meidanis, J. (2002). Sorting by prefix transpositions. In SPIRE, volume 2476 of LNCS, pages 65–76. Springer-Verlag. Elias, I. and Hartman, T. (2006). A 1.375-approximation algorithm for sorting by transpositions. IEEE/ACM Trans. Comput. Biol. Bioinform., 3(4):369–379. Hannenhalli, S. and Pevzner, P. A. (1999). Transforming cabbage into turnip: Polynomial algorithm for sorting signed permutations by reversals. Journal of the ACM, 46(1):1–27..
(33)
Documents relatifs
In summary, we have investigated the possibility of assessing the - electronic couplings from charge transport measurements in a connected device, using a statistical
o exchange. Tn general unique solutions are found or solutions with overlapping errors as it can be seen in fig. In the D t u wave there is weaker but very significant
Puis, complète les calculs pour déterminer le
with electrode-molecule coupling constants ' & and ' - (assumed equal for all molecules). ] where is each molecule energy level and t is a nearest neighbor
[r]
Shanthi Ameratunga, Eric Bernes, Chris Baguley, Christine Branche, Frances Bunn, Jose Capel Ferrer, Witaya Chadbunchachai, Ann Dellinger, Kathleen Elsig, Veronique Feypell,
[r]
Un cône de révolution de hauteur 6 cm et dont la base a pour diamètre
