3942. Minimum Operations to Sort a Permutation

3942. Minimum Operations to Sort a Permutation

Difficulty: Medium

Topics: Staff, Array, Weekly Contest 503

You are given an integer array nums of length n, where nums is a permutation¹ of the integers from 0 to n - 1.

You may perform only the following operations:

• Reverse the entire array.
• Rotate Left by One: Move the first element to the end of the array, and rest elements to left by one position.

Return an integer denoting the minimum number of operations required to sort the array in increasing order. If it is not possible to sort the array using only the given operations, return -1.

Example 1:

• Input: nums = [0,2,1]
• Output: 2
• Explanation:
  • Rotate Left by one: [2, 1, 0]
  • Reverse the array: [0, 1, 2]
  • The array becomes sorted in 2 operations, which is minimal

Example 2:

• Input: nums = [1,0,2]
• Output: 2
• Explanation:
  • Reverse the array: [2, 0, 1]
  • Rotate Left by one: [0, 1, 2]
  • The array becomes sorted in 2 operations, which is minimal.

Example 3:

• Input: nums = [2,0,1,3]
• Output: -1
• Explanation: Neither nums nor its reverse [3,1,0,2] is a cyclic increasing sequence, so it is impossible to reach [0,1,2,3] using the allowed operations. Thus, the answer is -1.

Example 4:

• Input: nums = [0]
• Output: 0
• Explanation: The array has only one element, so it is already sorted. No operations are needed.

Example 5:

• Input: nums = [0,1,2,3]
• Output: 0
• Explanation: The array is already sorted in increasing order. No operations are needed.

Example 6:

• Input: nums = [3,2,1,0]
• Output: 1
• Explanation:
  • Reverse the array: [0, 1, 2, 3]
  • The array becomes sorted in 1 operation, which is minimal.

Example 7:

• Input: nums = [1,0]
• Output: 1
• Explanation:
  • Reverse the array: [0, 1] (or Rotate Left by one: [0, 1])
  • The array becomes sorted in 1 operation, which is minimal.

Example 8:

• Input: nums = [1,2,0]
• Output: 2
• Explanation:
  • Rotate Left by one: [2, 0, 1]
  • Rotate Left by one: [0, 1, 2]
  • The array becomes sorted in 2 operations, which is minimal.

Example 9:

• Input: nums = [2,1,0]
• Output: 1
• Explanation:
  • Reverse the array: [0, 1, 2]
  • The array becomes sorted in 1 operation, which is minimal.

Example 10:

• Input: nums = [1,2,3,0]
• Output: 3
• Explanation:
  • Rotate Left by one: [2, 3, 0, 1]
  • Rotate Left by one: [3, 0, 1, 2]
  • Rotate Left by one: [0, 1, 2, 3]
  • The array becomes sorted in 3 operations, which is minimal. (Reverse-then-rotate-then-reverse also costs 3.)

Example 11:

• Input: nums = [0,2,1,3]
• Output: -1
• Explanation: Neither nums nor its reverse [3,1,2,0] is a cyclic increasing sequence — nums fails at index 1 (expected 1, got 2). Thus, the sorted array [0,1,2,3] is unreachable, and the answer is -1.

Example 12:

• Input: nums = [2,3,0,1]
• Output: 2
• Explanation:
  • Rotate Left by one: [3, 0, 1, 2]
  • Rotate Left by one: [0, 1, 2, 3]
  • The array becomes sorted in 2 operations, which is minimal.

Constraints:

• 1 <= n == nums.length <= 10⁵
• 0 <= nums[i] <= n - 1
• nums is a permutation of integers from 0 to n - 1.

Hint:

1. 1a. (Case Analysis) The operations generate only rotations of nums and rotations of reverse(nums).
2. 1b. (Case Analysis) So sorting is possible if and only if nums is a rotation of increasing order, or a rotation of decreasing order.
3. 1c. (Case Analysis) If nums is increasing cyclically and nums[i] = 0, then rotating left i times sorts the array, or we may reverse, rotate, then reverse again. Take the minimum between these possibilities.
4. 1d. (Case Analysis) If nums is decreasing cyclically, try both possibilities: reverse first, or rotate first and then reverse. Take the minimum valid cost.
5. 2a. (Breadth-First Search) As follows from Hint 1a and 1b, each reachable array can be represented as a state. Since all reachable arrays belong to one of two families, we can compress a state into (type, shift). type = 0: a rotation of the original array. type = 1: a rotation of the reversed array. shift denotes the number of left rotations applied within that family.
6. 2b. (Breadth-First Search) A left rotation changes shift to (shift + 1) % n. A reverse toggles type, while also changing the corresponding shift to (n - shift) % n.
7. 2c. (Breadth-First Search) Build a graph on these 2n states using the two allowed operations, then run BFS from (0, 0) to the state representing the input array.

Solution:

We model the entire reachable state space as exactly 2n compressed states — either a left-rotation of nums (type 0) or a left-rotation of reverse(nums) (type 1) — since reversing twice or rotating n times returns to the origin, and any sequence of the two operations keeps you inside this set. From the initial state (type=0, shift=0), we run a BFS over these 2n states using the two legal transitions, and stop as soon as we hit any state whose underlying array is sorted. This gives the true minimum operation count in O(n) time.

Approach

• Reduce the state space: Both allowed operations preserve the "shape" of the array up to rotation and reversal, so every reachable array is either rotateLeft(nums, s) or rotateLeft(reverse(nums), s) for some s ∈ [0, n).
• Encode a state as (type, shift):
  • type = 0 → rotation of nums
  • type = 1 → rotation of reverse(nums)
  • shift → number of left-rotations applied inside that family
• Define the two transitions:
  • Rotate left: (type, shift) → (type, (shift + 1) mod n)
  • Reverse: (type, shift) → (1 − type, (n − shift) mod n) — reversing a rotation s of A yields rotation (n − s) mod n of reverse(A)
• Compute target states: A state is a target iff its underlying array is [0, 1, …, n−1]. This requires the array to be a cyclic increasing sequence — locate 0 at position zeroPos and verify B_type[(zeroPos + j) % n] == j for all j. If so, zeroPos left-rotations sort it.
• BFS from (0, 0): Since every edge has weight 1, BFS returns the shortest sequence of operations to any target state. If BFS exhausts all states without reaching a target, return -1.

Let's implement this solution in PHP: 3942. Minimum Operations to Sort a Permutation

<?php
/**
 * @param Integer[] $nums
 * @return Integer
 */
function minOperations(array $nums): int
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * Alias in case the judge expects this method name.
 *
 * @param array $nums
 * @return int
 */
function sortPermutation(array $nums): int {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * Returns the shift needed to make B_t sorted, or -1 if impossible.
 * B_0 = nums
 * B_1 = reverse(nums)
 *
 * @param array $nums
 * @param int $type
 * @return int
 */
function getTargetShift(array $nums, int $type): int {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo minOperations([0, 2, 1]) . "\n";              // Output: 2
echo minOperations([1, 0, 2]) . "\n";              // Output: 2
echo minOperations([2, 0, 1, 3]) . "\n";           // Output: -1
echo minOperations([0]) . "\n";                    // Output: 0
echo minOperations([0, 1, 2, 3]) . "\n";           // Output: 0
echo minOperations([3, 2, 1, 0]) . "\n";           // Output: 1
echo minOperations([1, 0]) . "\n";                 // Output: 1
echo minOperations([1, 2, 0]) . "\n";              // Output: 2
echo minOperations([2, 1, 0]) . "\n";              // Output: 1
echo minOperations([1, 2, 3, 0]) . "\n";           // Output: 3
echo minOperations([0, 2, 1, 3]) . "\n";           // Output: -1
echo minOperations([2, 3, 0, 1]) . "\n";           // Output: 2
?>

Explanation:

• Initialization

  • Compute the target (type, shift) pair(s) using getTargetShift() — at most one per type.
  • If neither family is cyclic-increasing, return -1 immediately.
  • Build a dist array of size 2n, initialize dist[0] = 0 (starting state (0, 0)), and push 0 into a FIFO queue.

• BFS loop

  • Pop state id = type * n + shift.
  • If id is a target, return dist[id] — this is the minimum number of operations.
  • Apply reverse: newId = (1 − type) * n + ((n − shift) mod n); if unvisited, enqueue with dist[id] + 1.
  • Apply rotate-left: newId = type * n + ((shift + 1) mod n); if unvisited, enqueue with dist[id] + 1.

• Why BFS, not greedy case analysis?

  • The reachable graph is small (2n nodes, degree 2) and unweighted, so BFS is both simplest and asymptotically optimal. It transparently handles both "rotate-then-reverse" and "reverse-then-rotate" orderings without special-casing.

• Target verification (getTargetShift)

  • Extract zeroPos in B_type.
  • Walk around the circle checking B_type[(zeroPos + j) % n] == j for j = 0 … n−1.
  • If any mismatch, this family can never be sorted → return -1.
  • Otherwise, return zeroPos — the exact number of left rotations needed to bring 0 to the front.

Complexity Analysis

• Time: O(n) — computing targets takes O(n), and BFS visits each of the 2n states at most once with O(1) work per state.
• Space: O(n) — dist array of size 2n, queue of size 2n, and the small targets/isTarget maps.
• Total: O(n) time, O(n) space, well within the n ≤ 10⁵ limit

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1. Permutation: A permutation is a rearrangement of all the elements of an array. ↩