Medians of permutations and gene orders

Transcription

1 Medians of permutations and gene orders Sylvie Hamel Université de Montréal Work in collaboration with: Guillaume Blin Maxime Crochemore Stéphane Vialette from Université Marne-la-Vallée

2 Medians of permutations and gene orders Sylvie Hamel Université de Montréal Research trainees/programmers: Quentin Dejean Anthony Estebe Kelsey Lang from Université Monpellier II and University of Victoria

3 Biological context: : Permutations

4 Biological context: : Signed Permutations

5 The general problem: Given m permutations π, π,...,π m of {,,...,n} and a distance function d, the median problem is to find a permutation π that is the closest, under the distance d, to the m given permutations.

6 Biological context: phylogeny? A B C...

7 Blanchette and Sankoff*: In 997, they study the problem of finding a median of 3 genomes (permutations) under the breakpoint distance. The breakpoint distance between two permutations π and π is the number of pair of elements which are adjacent in π but not in π * D. Sankoff, M. Blanchette, The median problem for breakpoints in comparative genomics, LNCS 76, pp. 5-63, 997.

8 Blanchette and Sankoff*: In 997, they study the problem of finding a median of 3 genomes (permutations) under the breakpoint distance. π = [,, 3, 4, 5, 6, 7, 8] distance = 4 π = [, 5, 6, 8, 4, 3,, 7] distance = 4 * D. Sankoff, M. Blanchette, The median problem for breakpoints in comparative genomics, LNCS 76, pp. 5-63, 997.

9 Blanchette and Sankoff*: They described efficient heuristics to find a median of three circular genomes that have, or do not have, the same gene content Pe er and Shamir**: They show that the median problem of three permutations or signed permutations under the breakpoint distance is NP-complete. * D. Sankoff, M. Blanchette, The median problem for breakpoints in comparative genomics, LNCS 76, pp. 5-63, 997. ** I. Pe er, R. Shamir, The median problems for breakpoints are NP-complete, Elec. Colloq. on Comput. Complexity, vol. 7, 998.

10 The Kendall- τ distance: Counts the number of pairwise disagreements between two permutations i.e Maurice Kendall d KT (π, σ) = (i, j) :i < j and [(π[i] < π[j] and σ[i] > σ[j]) or (π[i] > π[j] and σ[i] < σ[j]) τ The Kendall- distance is equivalent to the bubble-sort distance i.e. the number of transpositions needed to transform one permutation into the other one. We have d KT (π,ı)=inv(π)

11 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) =

12 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = +

13 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = + +

14 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = + +

15 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = + + +

16 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = + + +

17 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = + + +

18 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) =

19 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) =

20 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) =

21 Example: π = [, 4,, 5, 3] σ = [3, 4,,, 5] d KT (π, σ) = = 5

22 τ Kendall- distance between a permutation and a set of permutations A = {π,...,π m } : π m d KT (π, A) = d KT (π, π i ) i=

23 The problem of finding the median of a set of m permutations of {,,...,n} under the Kendall- τ distance is best known in the literature as the Kemedy Score Problem In this problem, we have m voters that have to order n candidates from their best-liked candidate to their least-liked one. The problem then consists in finding a Kemedy consensus i.e. and order of the candidates that agree the most with the orders of the voters.

24 =,3] Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π [4,, π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : d KT (π,π ) d KT (π,π 3 ) G(π) : 4 3 π

25 =,3] Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π [4,, π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : G(π) : d KT (π,π ) d KT (π,π 3 ) π 4 3

26 =,,3] Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π [4, π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : G(π) : d KT (π,π ) d KT (π,π 3 ) π 4 3

27 =, Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π [4,,3] π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : G(π) : d KT (π,π ) d KT (π,π 3 ) π 4 3

28 = [4,,3] Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π, π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : G(π) : d KT (π,π ) d KT (π,π 3 ) π 4 3

29 = [4, Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π,,3] π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : G(π) : d KT (π,π ) d KT (π,π 3 ) π 4 3 0

30 = [4,, Example: π = [,, 3, 4] π = [4,,, 3] d KT (π,π ) π,3] π 3 = [4,, 3, ] Disagreement graph G(π) of a permutation with respect to a set of permutations A : G(π) d KT (π,π ) d KT (π,π 3 ) : d KT =5 π 4 3 0

31 Our problem: Given a set of m permutations A S n to find a permutation π such that, we want d KT (π,a) d KT (π, A), for all π S n. If m is the cardinality of the set A S n, the problem is NP-complet for m 4, m even* We will then considered the case m 3, m odd * C. Dwork et al., Rank Aggregation Methods for the Web, Tenth International World Wide Web Conference, Hong Kong, 00.

32 Example where there is more than one median: A = { π = [,, 3], π = [3,, ], π 3 = [, 3, ] } d KT =4 d KT =5 d KT = d KT =4 d KT =4 d KT =5

33 Finding a median with a brute force algorithm: n! permutations de {,, 3,... n} n=5, 0 permutations n=6, 70 permutations n=7, 5040 permutations n=8, permutations n=9, permutation n=0, permutations n=, permutations...

34 Some properties of the medians: Lemma : If a pair of elements appear in the same order in all permutations of the set A, then they also appear in that order in all medians π Lemma follows directly from the Extended Condorcet criterion (Truchon, 998): If there is a partition (C, C ) of {,,, n} such that for any x in C and y in C the majority prefers x to y, then x must be ranked above y. The original Cordorcet criterion, proposed by Marie Jean Antoine Nicolas de Caritat in 785, marquis de Condorcet, stated that if there is some element of {,,..., n} that defeats every other in pairwise simple majority voting, then this element should be ranked first. M. Truchon, An extension of the Condorcet criterion and Kemeny orders, report 98-5, Centre de Recherche en Économie et Finance Appliquées, 998 M.-J. Condorcet, Essai sur l application de l analyse à la probabilité des décisions rendues à la pluralité des voix, 785.

35 Some properties of the medians: Lemma : If a pair of elements appear in the same order in all permutations of the set A, then they also appear in that order in all medians π Lemma gives us a set of constraints which satisfy π must Lemma imply that in the disagreement graph of π, there are no arcs of weight m

36 Some properties of the medians: Lemma : The order of any adjacent pair of elements in a median π agree with the order of this pair of element in the majority of the permutations in A So in the disagreements graph of a median, all adjacent nodes are linked by an edge of weight m, where m is the cardinality of A

37 04 ~ 7 minutes Comparaison des temps de l algorithme force brute et algorithme force brute + contraintes

38 Some definitions for our heuristic: Definition : Given π = π π...π n, we call cyclic movement of a segment π[i..j], denoted c[i..j](π), the cycling shifting of the elements of the segment to the right or to the left : c r [i, j](π) =π...π i π j π i...π j π j+...π n c [i, j](π) =π...π i π i+...π j π i π j+...π n

39 Some definitions for our heuristic: Definition : Given our set of permutations A, we say that a cyclic movement is a k-move if d KT (c[i, j](π),a)=d KT (π, A)+k Definition 3: A good cyclic movement is a k-move, where k < 0

40 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π i j

41 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π i j

42 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π i j

43 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, 4 ] π i j

44 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π i j

45 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π i j

46 Example of k-move: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π 3 Total = k = - Total = i j

47 Our to find k-moves: π = [, 3, 8, 4, 6,, 7, 5] π = [7,,, 3, 4, 5, 6, 8] = [5, 3,, 8, 7, 6, 4, ] π 3 Total = i=3 j=7 Theorem : We have that k = 3(j i) c r [i, j](π) j t=i is a k-move iff w G(π) (π t, π j )

48 Our heuristic: Given a set of permutations A = {π,...,π m } : ) Take π as a starting point for the heuristic ) Find and execute a good move, if any 3) Repeat ) till there is no more good moves and keep the result as a possible median for A 4) Repeat ) with π i, i m 5) Take the best of the m result as one median of A

49 Why do we execute our heuristic on each permutations of A? π π = [3, 6, 4,,, 7, 5] = [4, 6, 5,,, 7, 3] π = [6, 4,, 7, 5, 3, ] π 3 = [, 7, 5, 3, 6,, 4] π [3, 6, 4,,, 7, 5] d KT = 3 π [6, 4,, 5,, 7, 3] d KT = 3 d KT = π 3 π

50 Some results of our heuristic for set of permutations A of cardinality 3: n number of considered sets % of errors 0% 0% 0% 0% 0.05% 0.5% distance difference

51 Some results of our heuristic for set of permutations A of cardinality 3: % 0% 0.05% 0.5% 0.35% 0.6%.%.6% 0 0

52 Adding 0-moves: π π π 3 = [3, 6, 4,,, 7, 5] = [4, 6, 5,,, 7, 3] = [, 7, 5, 3, 6,, 4] π = [6, 4,, 7, 5, 3, ] π [6, 4,, 5,, 7, 3]

53 Good 0-moves: A good 0-move is a 0-move that can be immediately follow by any good -k move π π π 3 = [3, 6, 4,,, 7, 5] = [4, 6, 5,,, 7, 3] = [, 7, 5, 3, 6,, 4] π = [6, 4,, 7, 5, 3, ]

54 Good 0-moves: A good 0-move is a 0-move that can be immediately follow by any good -k move π π π 3 = [3, 6, 4,,, 7, 5] = [4, 6, 5,,, 7, 3] = [, 7, 5, 3, 6,, 4] π = [6, 4,, 7, 5, 3, ]

55 Good 0-moves: c r [i, j](π) is good 0-move if it is a 0-move and w G(π) (π j, π j+ )=or w G(π) (π i, π j )= π π π 3 = [3, 6, 4,,, 7, 5] = [4, 6, 5,,, 7, 3] = [, 7, 5, 3, 6,, 4] π = [6, 4,, 7, 5, 3, ]

56 Results of our heuristic with 0-moves: We applied our heuristic on triplets of permutations of {,,..., n} for n between 6 and 0 and for each of these triplet, we computed the number of required 0-moves to get to the median The maximal number of 0-move permitted was 3

57 What s left to do: Understand why for some given set of permutations there is only one median and for others more than one (we even found 33 medians for a triplet of permutations of {,,..., 0}) Study the problem under important biological distances on permutations and signed permutations Is the median problem of a set of m permutations, m odd, under the Kendall- τ distance polynomial? If we know the medians of a set of permutations A and the medians of set of permutations B, does it tell us something on the set of medians of A B, A B, A \ B, A,...

58 ??????? Questions????????