2364. Count Number of Bad Pairs

2364. Count Number of Bad Pairs

Difficulty: Medium

Topics: Array, Hash Table, Math, Counting

You are given a 0-indexed integer array nums. A pair of indices (i, j) is a bad pair if i < j and j - i != nums[j] - nums[i].

Return the total number of bad pairs in nums.

Example 1:

• Input: nums = [4,1,3,3]
• Output: 5
• Explanation:
  • The pair (0, 1) is a bad pair since 1 - 0 != 1 - 4.
  • The pair (0, 2) is a bad pair since 2 - 0 != 3 - 4, 2 != -1.
  • The pair (0, 3) is a bad pair since 3 - 0 != 3 - 4, 3 != -1.
  • The pair (1, 2) is a bad pair since 2 - 1 != 3 - 1, 1 != 2.
  • The pair (2, 3) is a bad pair since 3 - 2 != 3 - 3, 1 != 0.
  • There are a total of 5 bad pairs, so we return 5.

Example 2:

• Input: nums = [1,2,3,4,5]
• Output: 0
• Explanation: There are no bad pairs.

Constraints:

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

Hint:

1. Would it be easier to count the number of pairs that are not bad pairs?
2. Notice that (j - i != nums[j] - nums[i]) is the same as (nums[i] - i != nums[j] - j).
3. Keep a counter of nums[i] - i. To be efficient, use a HashMap.

Solution:

We need to count the number of bad pairs in a given array. A pair (i, j) is considered bad if i < j and j - i != nums[j] - nums[i]. Directly checking each pair would be inefficient, so we use a mathematical approach to optimize the solution.

Approach

1. Identify Good Pairs: A pair (i, j) is good if j - i == nums[j] - nums[i]. This can be rearranged to nums[i] - i == nums[j] - j. Thus, elements with the same value of (nums[i] - i) form groups of good pairs.
2. Count Good Pairs: For each group of elements with the same (nums[i] - i) value, the number of good pairs is given by the combination formula m*(m-1)/2, where m is the size of the group.
3. Calculate Total Pairs: The total number of pairs (i, j) where i < j is given by n*(n-1)/2, where n is the length of the array.
4. Compute Bad Pairs: Subtract the number of good pairs from the total pairs to get the number of bad pairs.

Let's implement this solution in PHP: 2364. Count Number of Bad Pairs

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

// Example 1
$nums1 = [4, 1, 3, 3];
echo "Output: " . countBadPairs($nums1) . "\n"; // Output: 5

// Example 2
$nums2 = [1, 2, 3, 4, 5];
echo "Output: " . countBadPairs($nums2) . "\n"; // Output: 0
?>

Explanation:

1. Total Pairs Calculation: The total number of pairs (i, j) where i < j is calculated using the formula n*(n-1)/2.
2. Frequency Map: We create a frequency map to count occurrences of each value (nums[i] - i). This helps identify groups of elements that can form good pairs.
3. Good Pairs Calculation: For each group with m elements, the number of good pairs is m*(m-1)/2. Summing these values gives the total number of good pairs.
4. Result Calculation: Subtract the number of good pairs from the total pairs to get the number of bad pairs.

This approach efficiently reduces the problem complexity from O(n^2) to O(n) by leveraging mathematical insights and hash maps for frequency counting.

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