Problem Description
You are given an even integer n. You initially have a permutation perm of size n where perm[i] == i (0-indexed). In one operation, you will create a new array arr, and for each i:
- If i % 2 == 0, then arr[i] = perm[i / 2].
- If i % 2 == 1, then arr[i] = perm[n / 2 + (i - 1) / 2].
You will then assign arr to perm. Return the minimum non-zero number of operations you need to perform on perm to return the permutation to its initial value.
Key Insights
- The transformation of the permutation can be viewed as a series of index swaps.
- Each operation effectively rearranges the indices of the permutation.
- The number of operations required to return to the initial state relates to the cycles formed by the index transformations.
Space and Time Complexity
Time Complexity: O(log(n)) - The number of operations is logarithmic in relation to n due to the cycle structure. Space Complexity: O(1) - Only a constant amount of space is used for variables.
Solution
The problem can be solved by analyzing how the indices of the permutation change with each operation. Each index can be tracked to see how many operations it takes to return to its original position. This can be done using a simulation of the operations and counting until the permutation returns to its initial state.
To implement this, we can create a loop that simulates the operations on the permutation until it matches the original state, while counting the number of operations performed.