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Minimum Number of Operations to Reinitialize a Permutation

Number: 1935

Difficulty: Medium

Paid? No

Companies: Google

Problem Description

Given an even integer n, you start with a permutation perm where perm[i] == i for 0 ≤ i < n. In one operation, a new array arr is formed such that:

  • For even indices i, arr[i] = perm[i/2].
  • For odd indices i, arr[i] = perm[n/2 + (i-1)/2]. The permutation perm is then updated to be arr. The goal is to determine the minimum (non-zero) number of these operations needed for perm to become the initial permutation again.

Key Insights

  • The operation defines a permutation on the indices. The total number of operations needed is essentially the order of this permutation transform.
  • Since every operation is deterministic and reversible, repeating the process will eventually cycle back to the original permutation.
  • Simulation (or finding the cycle length mathematically) is a straightforward approach given the small constraints.
  • Even though theoretically one could compute cycle lengths for each individual index and take the least common multiple, simulation suffices for n up to 1000.

Space and Time Complexity

Time Complexity: O(n^2) in the worst case (each operation takes O(n) and in worst-case the cycle length is O(n)). Space Complexity: O(n) to store the permutation arrays.

Solution

We simulate the permutation transformation starting from the identity permutation. In each operation:

  1. Create a new array based on the given rules.
  2. Update the permutation.
  3. Check if the new permutation matches the original. Repeat until the original permutation is reached again. This approach leverages simple array manipulation and comparison, guaranteeing correctness within the constraint limits.

Code Solutions

def reinitializePermutation(n):
    # Create the original identity permutation
    original = list(range(n))
    perm = original.copy()
    count = 0
    # Repeat operations until perm returns to original
    while True:
        count += 1
        new_perm = [0] * n
        # Apply the transformation rules for each index
        for i in range(n):
            if i % 2 == 0:
                new_perm[i] = perm[i // 2]
            else:
                new_perm[i] = perm[n // 2 + (i - 1) // 2]
        perm = new_perm
        # Check if we have returned to the identity permutation
        if perm == original:
            return count

# Example usage:
print(reinitializePermutation(2))  # Expected output: 1
print(reinitializePermutation(4))  # Expected output: 2
print(reinitializePermutation(6))  # Expected output: 4
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