04. Count All Triplets with Given Sum in Sorted Array

The problem can be found at the following link: Problem Link

Problem Description

You are given a sorted array arr[] of size n and an integer target. Your task is to count the number of triplets (i, j, k) such that:

• i < j < k

• arr[i] + arr[j] + arr[k] == target

Examples:

Input: arr[] = [-3, -1, -1, 0, 1, 2], target = -2 Output: 4 Explanation: The triplets are:

• arr[0] + arr[3] + arr[4] = -3 + 0 + 1 = -2

• arr[0] + arr[1] + arr[5] = -3 + (-1) + 2 = -2

• arr[0] + arr[2] + arr[5] = -3 + (-1) + 2 = -2

• arr[1] + arr[2] + arr[3] = -1 + (-1) + 0 = -2

Input: arr[] = [-2, 0, 1, 1, 5], target = 1 Output: 0 Explanation: No triplet adds up to 1.

Constraints:

• $3 ≤ arr.size() ≤ 10^3$

• $-10^5 ≤ arr[i], target ≤ 10^5$

My Approach

1. Two-Pointer Technique in a Sorted Array: To efficiently solve the problem, we use the two-pointer approach in a sorted array:

   • Iterate through the array, treating each element as the first element of a potential triplet.

   • Use two pointers (left and right) to find the other two elements that satisfy the required sum.

   • If the current sum matches the target, count all unique combinations while handling duplicates.

2. Handling Duplicates:

   • If arr[left] == arr[right], count all combinations formed by the elements between left and right.

   • If duplicates exist in either pointer range, skip them while counting.

3. Steps:

   • Fix the first element (arr[i]) and use two pointers for the remaining subarray.

   • Calculate the sum of the triplet. If it equals the target:

     • Count the combinations and adjust pointers to skip duplicates.

   • If the sum is smaller than the target, move the left pointer.

   • If the sum is larger, move the right pointer.

Time and Auxiliary Space Complexity

• Expected Time Complexity: O(n²), where n is the size of the array. The outer loop iterates through each element, and the two-pointer traversal for each element takes O(n) time in the worst case.

• Expected Auxiliary Space Complexity: O(1), as we only use a constant amount of additional space for variables.

Code (C++)

class Solution {
public:
    int countTriplets(vector<int>& arr, int target) {
        int n = arr.size(), res = 0;
        for (int i = 0; i < n - 2; ++i) {
            int left = i + 1, right = n - 1;
            while (left < right) {
                int sum = arr[i] + arr[left] + arr[right];
                if (sum < target) {
                    ++left;
                } else if (sum > target) {
                    --right;
                } else {
                    if (arr[left] == arr[right]) {
                        int count = right - left + 1;
                        res += (count * (count - 1)) / 2;
                        break;
                    } else {
                        int cnt1 = 1, cnt2 = 1;
                        while (left + 1 < right && arr[left] == arr[left + 1]) ++left, ++cnt1;
                        while (right - 1 > left && arr[right] == arr[right - 1]) --right, ++cnt2;
                        res += cnt1 * cnt2;
                        ++left;
                        --right;
                    }
                }
            }
        }
        return res;
    }
};

Code (Java)

class Solution {
    public int countTriplets(int[] arr, int target) {
        int n = arr.length, res = 0;
        for (int i = 0; i < n - 2; i++) {
            int left = i + 1, right = n - 1;
            while (left < right) {
                int sum = arr[i] + arr[left] + arr[right];
                if (sum < target) {
                    left++;
                } else if (sum > target) {
                    right--;
                } else {
                    if (arr[left] == arr[right]) {
                        int count = right - left + 1;
                        res += count * (count - 1) / 2;
                        break;
                    } else {
                        int cnt1 = 1, cnt2 = 1;
                        while (left + 1 < right && arr[left] == arr[left + 1]) {
                            left++;
                            cnt1++;
                        }
                        while (right - 1 > left && arr[right] == arr[right - 1]) {
                            right--;
                            cnt2++;
                        }
                        res += cnt1 * cnt2;
                        left++;
                        right--;
                    }
                }
            }
        }
        return res;
    }
}

Code (Python)

class Solution:
    def countTriplets(self, arr, target):
        n, res = len(arr), 0
        for i in range(n - 2):
            left, right = i + 1, n - 1
            while left < right:
                sum_triplet = arr[i] + arr[left] + arr[right]
                if sum_triplet < target:
                    left += 1
                elif sum_triplet > target:
                    right -= 1
                else:
                    if arr[left] == arr[right]:
                        count = right - left + 1
                        res += count * (count - 1) // 2
                        break
                    cnt1, cnt2 = 1, 1
                    while left + 1 < right and arr[left] == arr[left + 1]:
                        left += 1
                        cnt1 += 1
                    while right - 1 > left and arr[right] == arr[right - 1]:
                        right -= 1
                        cnt2 += 1
                    res += cnt1 * cnt2
                    left += 1
                    right -= 1
        return res

Contribution and Support

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