given an array of integers in random order you have to find the minimum number of swaps to convert it to cyclic sorted array

Question

if an array is given in random order , you have to output the minimum number of swaps required to convert into cyclic sorted array.

e.g. array given is 3 5 4 2 1

Answer 1

Assuming:

1. The array elements are the integers 1 to N.

Steps:

1. Allocate a second (histogram) array H of length N.
2. In a single pass through the initial array A, for each index i increment H[ (A[i] - i) % N]
3. In a single pass through H, identify the element R of H with the maximum count
4. Clear H.
5. Identify the number of Orbits in the original array under a rotation by R. (by stepping forward through H until H[i]=0, then stepping through the orbit for which A[i] is a member under the rotation R) setting H[i]=1 for each member of the orbit, and repeating until the element H[N-1] has been processed (counting the nubmer of orbits as you go). This is also O(N) as each element of A is visited exactly once.
6. The required number of swaps = N - Orbits.

This is O(N).

Update: I have updated the algorithm to account for multiple orbits. In processing each orbit, the final swap places two elements instead of just 1.

I suspect that the number of Orbits is invariant under any rotation, which would simplify the algorithm considerbly but would not affect it's complexity, which remains at O(N).