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

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

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

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

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

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

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):

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

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,…{

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}

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)}

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

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

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)

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

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 i = 1,j = 4, j – i=3  ζ 2,1 ) 2) 21543

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 ζ 2,1 ) 2)  ζ 1,2 ) 1) 25413

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

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 i = 2, j = 5, j – i=3  ζ 2,1 ) 3 ) 25341

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 ζ 2,1 ) 3 )  ζ 1,2 ) 2 ) 53241

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

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

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

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

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):

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):

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

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)

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

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

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)

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

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

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 ?