1865. Finding Pairs With a Certain Sum

1865. Finding Pairs With a Certain Sum

Difficulty: Medium

Topics: Array, Hash Table, Design

You are given two integer arrays nums1 and nums2. You are tasked to implement a data structure that supports queries of two types:

• Add a positive integer to an element of a given index in the array nums2.
• Count the number of pairs (i, j) such that nums1[i] + nums2[j] equals a given value (0 <= i < nums1.length and 0 <= j < nums2.length).

Implement the FindSumPairs class:

• FindSumPairs(int[] nums1, int[] nums2) Initializes the FindSumPairs object with two integer arrays nums1 and nums2.
• void add(int index, int val) Adds val to nums2[index], i.e., apply nums2[index] += val.
• int count(int tot) Returns the number of pairs (i, j) such that nums1[i] + nums2[j] == tot.

Example 1:

• Input:

  ["FindSumPairs", "count", "add", "count", "count", "add", "add", "count"]
  [[[1, 1, 2, 2, 2, 3], [1, 4, 5, 2, 5, 4]], [7], [3, 2], [8], [4], [0, 1], [1, 1], [7]]

• Output: [null, 8, null, 2, 1, null, null, 11]
• Explanation:

  FindSumPairs findSumPairs = new FindSumPairs([1, 1, 2, 2, 2, 3], [1, 4, 5, 2, 5, 4]);
  findSumPairs.count(7);  // return 8; pairs (2,2), (3,2), (4,2), (2,4), (3,4), (4,4) make 2 + 5 and pairs (5,1), (5,5) make 3 + 4
  findSumPairs.add(3, 2); // now nums2 = [1,4,5,4,5,4]
  findSumPairs.count(8);  // return 2; pairs (5,2), (5,4) make 3 + 5
  findSumPairs.count(4);  // return 1; pair (5,0) makes 3 + 1
  findSumPairs.add(0, 1); // now nums2 = [2,4,5,4,5,4]
  findSumPairs.add(1, 1); // now nums2 = [2,5,5,4,5,4]
  findSumPairs.count(7);  // return 11; pairs (2,1), (2,2), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,4) make 2 + 5 and pairs (5,3), (5,5) make 3 + 4

Constraints:

• 1 <= nums1.length <= 1000
• 1 <= nums2.length <= 105
• 1 <= nums1[i] <= 109
• 1 <= nums2[i] <= 105
• 0 <= index < nums2.length
• 1 <= val <= 105
• 1 <= tot <= 109
• At most 1000 calls are made to add and count each.

Hint:

1. The length of nums1 is small in comparison to that of nums2
2. If we iterate over elements of nums1 we just need to find the count of tot - element for all elements in nums1

Solution:

We need to efficiently handle two operations on two given arrays: adding a value to an element in the second array and counting the number of pairs from both arrays that sum to a given value. The challenge lies in optimizing these operations, especially the count operation, given the constraints.

Approach

1. Initialization:

   • Store the given arrays nums1 and nums2.
   • Create a frequency map (hash table) for nums2 to keep track of how many times each number appears. This allows efficient updates and lookups.

2. Add Operation:

   • When adding a value val to an element at a specific index in nums2, first retrieve the old value at that index.
   • Compute the new value by adding val to the old value.
   • Update the frequency map by decrementing the count of the old value (and removing it if the count drops to zero) and incrementing the count of the new value.

3. Count Operation:

   • For each element x in nums1, calculate the required value tot - x that, when paired with x, sums to tot.
   • If the required value is less than 1, skip it since nums2 contains only positive integers.
   • Use the frequency map to quickly check how many times the required value appears in nums2 and accumulate this count.
   • Return the total count of valid pairs.

Let's implement this solution in PHP: 1865. Finding Pairs With a Certain Sum

<?php
class FindSumPairs {
    /**
     * @param Integer[] $nums1
     * @param Integer[] $nums2
     */
    function __construct($nums1, $nums2) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
    }

    /**
     * @param Integer $index
     * @param Integer $val
     * @return NULL
     */
    function add($index, $val) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */        
    }

    /**
     * @param Integer $tot
     * @return Integer
     */
    function count($tot) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
    }
}

/////// 🔽 Example usage (same as problem) 🔽 ///////

$commands = ["FindSumPairs", "count", "add", "count", "count", "add", "add", "count"];
$params = [
    [[1, 1, 2, 2, 2, 3], [1, 4, 5, 2, 5, 4]],
    [7],
    [3, 2],
    [8],
    [4],
    [0, 1],
    [1, 1],
    [7]
];

$results = array();
$object = null;

foreach ($commands as $i => $cmd) {
    if ($cmd == "FindSumPairs") {
        $object = new FindSumPairs($params[$i][0], $params[$i][1]);
        $results[] = null;
    } elseif ($cmd == "add") {
        $object->add($params[$i][0], $params[$i][1]);
        $results[] = null;
    } elseif ($cmd == "count") {
        $results[] = $object->count($params[$i][0]);
    }
}

print_r($results);
?>

Explanation:

• Initialization: The constructor initializes the two arrays and builds a frequency map for nums2 to count occurrences of each element.
• Add Operation: The add method updates an element in nums2 by adding a specified value. It adjusts the frequency map by decrementing the count of the old value and incrementing the count of the new value.
• Count Operation: The count method iterates over each element in nums1, calculates the complementary value needed in nums2 to sum to the target value, and uses the frequency map to quickly determine how many such values exist in nums2. This ensures efficient counting without scanning nums2 each time.

This approach efficiently handles both operations by leveraging the frequency map, reducing the time complexity for the count operation from O(n*m) to O(n), where n is the length of nums1 and m is the length of nums2. The add operation remains O(1) per call due to hash table operations. This optimization is crucial given the problem constraints.

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