1. Introduction Sorting permutations with reversals in order to reconstruct evolutionary history of genome Reversal mutations occur often in chromosomes where each reverses the order of an interval of genes A shortest reversal sequence sorting one genome to another corresponds to the most likely evolutionary path between them

2. Introduction Sorting permutations and circular permutations using as few fixed-length reversals as possible Limiting the the transformations to reversals of length exactly k can be very restrictive

3. Can k-reversal sort ? Can the permutation {1,3,2,4,5} be sorted using k-reversals ? k=1 ? well …… k=2 ?  {1,3,2,4,5}  Bubble sort k=3 ? k=4 ? later on

4. Sorting {1,3,2,4,5} with k=3 Since 1 and 2 are separated by an odd number of items and any 3-reversal change this distance by either 0 or 2 it cannot be done !!! {2,3,1,4,5} – distance change 0 {1,4,2,3,5} – distance change 0 {1,3,5,4,2} – distance change 2

5. Sorting {1,3,2,4,5} with k=3 {2,3,1,4,5} – distance change 0 {1,4,2,3,5} – distance change 0 {1,3,5,4,2} – distance change 2 3-rev can change position of odd elements only and even elements only 3-rev is actually bubble sort for odd/even elements inside the permutation

6. Notation PG(k,n) – permutation group of size n using k- reversals The k-reversal operation on a permutation starting the n element Rev(i): {1, …,i-1,i+k-1, i+k-2, …, i+1,i,i+k, …,n} d – the diameter : max{ shortest path in cayley graph } or minimum reversals to get from p to q

7. Notation The Cayley graph is the graph whose vertices are the elements of G, with an edge between vertices p and q iff Cayley graph of PG(3,4): 13242314 24131423 13424312 42131243 12343214 34121432 21343124 34212431 34212431 21343124 32414231 41323142

8. Equivalent Transformations in PG(k,n) 4l – reversal ↔ 4 – reversal ↔ ζ 1,2, ζ 2,1 (2+4l) – reversal ↔ 2 – reversal ↔ ζ 1,1 (3+4l) – reversal ↔ 3 – reversal (5+8l) – reversal ↔ 5 – reversal ↔ ζ 2,2 (9+8l) – reversal ↔ 9 – reversal ↔ ζ 2,4, ζ 4,2

9. ζ 1,2, ζ 2,1  4 – reversal : ζ 1,2 (1) : {1,2,3,4,…}  {2,3,1,4,…} ζ 2,1 (2) : {2,3,1,4,…}  {2,4,3,1,…} ζ 1,2 (1) : {2,4,3,1,…}  {4,3,2,1,…{

10. 4 – reversal  ζ 1,2, ζ 2,1 lemma: 4-rev  ζ 2,3, ζ 3,2 {1,2,3,4,5,6,7} → {1,5,4,3,2,6,7} → {3,4,5,1,2,6,7} for ζ 3,2 we simply reverse these operations lemma: ζ 2,3, ζ 3,2  ζ 1,4, ζ 4,1 {1,2,3,4,5,6,7} → {3,4,5,1,2,6,7} → {5, 1,2,3,4,6,7} for ζ 4,1 we simply use ζ 3,2 with same operations lemma: ζ 1,4, ζ 4,1  ζ 1,2, ζ 2,1 {1,2,3,4,5,6,7} → {1,3,4,5,6,2,7} → {1,4,5,6,2,3,7} → {2,1,4,5,6,3,7} → {2,3,1,4,5,6,7}

11. The problem: Given a graph PG(k,n): How many connected components are there? Equiv to: what is the size of any connected component? What is the diameter of each component? Assume n≥k+2 If k=n there are n!/2 components If k=n-1 there are n or 2n components, depending upon parity n=3 {(1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1)} n=4 {(1,2,3,4) (1,4,3,2) (3,2,1,4) (3,4,1,2)}

12. The simple cases How many connected components are in PG(2,n)? 1 component (the graph is connected) How many connected component are in PG(3,n)? there is only a choice of n/2 elements for odd/even places, and therefore components

13. The number of connected components in PG(k,n) K≈0 mod 4 ? K≈5 mod 8 ? K≈1 mod 8 ? K≈2 mod 4 1 k≈3 mod 4

14. Connected components - Sign of permutation (4-rev) The sign of permutation is pair is disordered if i a j Lemma : ζ 1,2, ζ 2,1 do not change the sign of a permutation ζ 2,1 (i)= ζ 1,2 (i) ζ 1,2 (i) x,y,z  y,z,x sign (z-y) (z-x) (y-x) = sign (z-y) (x-z) (x-y)

15. Connected components (4-rev) Lemma : ζ 1,2, ζ 2,1 cannot change the sign of a permutation. The identity permutations has + sign, so permutations with – sign cannot be sorted. Lemma: ζ 2,1 can sort only half of all permutations ζ 2,1  ζ 2m,1 for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for

16. example for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for 1 2 3 4 5 i = 1,j = 4, j – i=3  ζ 2,1 ) 2) 21543

17. example for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for 1 2 3 4 5 ζ 2,1 ) 2)  ζ 1,2 ) 1) 25413

18. example for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for 1 2 3 4 5 25341

19. example for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for 1 2 3 4 5 i = 2, j = 5, j – i=3  ζ 2,1 ) 3 ) 25341

20. example for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for 1 2 3 4 5 ζ 2,1 ) 3 )  ζ 1,2 ) 2 ) 53241

21. example for i=1 to n-2 find j such that a J = i if (j – i) is even, apply ζ j - i,1 ) i ) else apply ζ j – i - 1,1 ) i+1) then ζ 1,2 ) i ) end for 1 2 3 4 5 54321

22. Connected components (4-rev) the i th iteration places a J into i th position, where a J = i at termination, either because they have different signs, using prev lemma we know cannot be transformed into using ζ 1,2 thus, ζ 1,2 divides the permutation group into 2 equal size sub-groups, and ζ 1,2 sorts just half of all permutations

23. The number of connected components in PG(k,n) K≈0 mod 4 2 K≈5 mod 8 K≈1 mod 8 K≈2 mod 4 1 k≈3 mod 4

24. Circular permutations - Notation CPG(k,n) – circular permutation group of size n using k-reversals Each permutation in CPG(n) represents a set of n permutations on PG(n) equivalent under the shift operation {1,2,3,4} = { (1,2,3,4), (2,3,4,1), (3,4,1,2), (4,1,2,3) } Any permutation can be rearranged to exactly n arrangements by shift PG(n) has n! permutations  CPG(n) has n!/n = (n-1)! permutations

25. Notation The Cayley graph is the graph whose vertices are the elements of CPG, with an edge between vertices p and q iff Cayley graph of CPG(3,4): 13242314 24131423 13424312 42131243 12343214 34121432 21343124 34212431 34212431 21343124 32414231 41323142

26. Notation The Cayley graph is the graph whose vertices are the elements of CPG, with an edge between vertices p and q iff Cayley graph of CPG(3,4): 1324 1423 1342 1243 1234 1432

27. Equivalent Transformations in CPG(k,n) All PG(k,n) transformations hold for n > k+2 4l – reversal ↔ 4 – reversal ↔ ζ 1,2, ζ 2,1 (2+4l) – reversal ↔ 2 – reversal ↔ ζ 1,1 (3+4l) – reversal ↔ 3 – reversal (5+8l) – reversal ↔ 5 – reversal ↔ ζ 2,2 (9+8l) – reversal ↔ 9 – reversal ↔ ζ 2,4, ζ 4,2

28. The problem: Given a graph CPG(k,n): How many connected components are there? Equiv to: what is the size of any connected component? What is the diameter of each component? Assume n≥k+2 If k=n or k=n-1 there are (n-1)!/2 components Since all PG(k,n) transformations hold: # Components in CPG(n) ≤ # Components in PG(n)

29. Connected comp. of CPG(k,n) for even k Recall: How many connected components are in PG(2,n)? 1 component (the graph is connected) same in CPG(2,n) – holds for all n

30. Connected comp. of CPG(k,n) for even k & even n CPG(4l,2m) is connected (a single component) Proof: Recall that: 4 – reversals → ς 1,2, ς 2,1 ς 1,2, ς 2,1 sort all permutations to {1,...,2m-1,2m} or {1, …,2m,2m-1} ς 1,2 can sort circular permutation {1, …,2m,2m-1} to {1,...,2m-1,2m} : 1,2,3,4,6,5 → 5,1,2,3,4,6 (shift) 5,1,2,3,4,6 → 1,2,5,3,4,6 → 1,2,3,4,5,6

31. Connected comp. of CPG(k,n) for even k & odd n Recall: 4 – reversals do not change the sign of permutations. If n is odd a shift operation doesn ’ t change the sign x 1, …,x 2m,x 2m+1 → x 2m+1,x 1, …,x 2m 2m = even #(disorders) We can use the algorithm 4l-reversals sorts half of CPG(k,n)

32. Connected comp. of CPG(k,n) for even k So far: n=2 mod 4n=2mn=2m+1k\n 112k=0 mod 4 111k=2 mod 4 2k=5 mod 8 2k=1 mod 8 1k=3 mod 4

33. Diameter of CPG(k,n) bounds Upper bound =O(n 2 /k +nk) Lower bound = Ω(n 2 /k 2 +n)

34. Conclusions & Open problems A complete answer to the connectedness question of the Cayley Graphs for permutations and circular permutations Bounds to the diameter of CPG(k,n) Can we tighten these bounds ? What is the diameter of PG(k,n) ? What happens with signed permutations where each element has 2 possible orientations ? What happens if we allow numerous reversals ?