3761. Minimum Absolute Distance Between Mirror Pairs

Difficulty: Medium

Topics: Staff, Array, Hash Table, Math, Weekly Contest 478

You are given an integer array nums.

A mirror pair is a pair of indices (i, j) such that:

0 <= i < j < nums.length, and
reverse(nums[i]) == nums[j], where reverse(x) denotes the integer formed by reversing the digits of x. Leading zeros are omitted after reversing, for example reverse(120) = 21.

Return the minimum absolute distance between the indices of any mirror pair. The absolute distance between indices i and j is abs(i - j).

If no mirror pair exists, return -1.

Example 1:

Input: nums = [12,21,45,33,54]
Output: 1
Explanation:
- The mirror pairs are:
  - (0, 1) since reverse(nums[0]) = reverse(12) = 21 = nums[1], giving an absolute distance abs(0 - 1) = 1.
  - (2, 4) since reverse(nums[2]) = reverse(45) = 54 = nums[4], giving an absolute distance abs(2 - 4) = 2.
- The minimum absolute distance among all pairs is 1.

Example 2:

Input: nums = [120,21]
Output: 1
Explanation:
- There is only one mirror pair (0, 1) since reverse(nums[0]) = reverse(120) = 21 = nums[1].
- The minimum absolute distance is 1.

Example 3:

Input: nums = [21,120]
Output: -1
Explanation: There are no mirror pairs in the array.

Example 4:

Input: nums = [1,10,100,1,10]
Output: 1

Constraints:

1 <= nums.length <= 10⁵
1 <= nums[i] <= 10⁹

Hint:

Scan left to right with a hash map: for each nums[i], if the map contains key nums[i] then set ans = min(ans, i - map[nums[i]]).
Store/update the current index under key reverse(nums[i]), so future matches use the most recent index.

Solution:

This solution finds the minimum absolute distance between indices of mirror pairs in an array. A mirror pair consists of two numbers where one is the digit-reversed version of the other. The algorithm uses a single pass with a hash map to track the most recent index of each number's reverse, efficiently finding the smallest index gap in O(n) time.

Approach:

Single-pass traversal with hash map storage
Reverse number computation for each array element
Look for existing matches before storing current index
Track minimum distance by comparing current index with previously stored index of the same number
Store current index under the reversed number key for future matches

Let's implement this solution in PHP: 3761. Minimum Absolute Distance Between Mirror Pairs

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

// Test cases
echo minAbsoluteDistanceMirrorPairs([12, 21, 45, 33, 54]) . "\n";           // Output: 1
echo minAbsoluteDistanceMirrorPairs([120, 21]) . "\n";                      // Output: 1
echo minAbsoluteDistanceMirrorPairs([21, 120]) . "\n";                      // Output: -1
echo minAbsoluteDistanceMirrorPairs([1,10,100,1,10]) . "\n";                // Output: 1
?>

Explanation:

Key Insight: When scanning left to right, if we encounter a number that matches a previously stored reversed number, they form a mirror pair. The reverse of the previous number equals the current number, and vice versa.
Hash Map Strategy: Store each number's latest index under the key equal to its reverse. This ensures that when we later see a number that matches a previously stored reverse, we can calculate the distance immediately.
Distance Calculation: Since we always store the latest index for each reversed number, we minimize the distance for each potential pair. Using abs(i - j) simplifies to i - j because i is always greater than j during forward traversal.
Edge Cases:
- Numbers with trailing zeros (e.g., 120 → 21) are handled correctly by converting to integer after reversal
- Single-element arrays or arrays with no mirror pairs return -1
- Multiple occurrences of the same number are handled by overwriting the map entry, which actually helps find smaller gaps

Complexity

Time Complexity: O(n) - single pass through the array with O(1) operations per element (reversal of digits is O(log₁₀(num)) which is effectively constant as numbers ≤ 10⁹ have at most 10 digits)
Space Complexity: O(n) - in worst case, the hash map stores a unique entry for each distinct reversed number in the array

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