Two Sum : Check if a pair with given sum exists in Array

Problem Statement: Given an array of integers arr[] and an integer target.

1st variant: Return YES if there exist two numbers such that their sum is equal to the target. Otherwise, return NO.

2nd variant: Return indices of the two numbers such that their sum is equal to the target. Otherwise, we will return {-1, -1}.

Note: You are not allowed to use the same element twice. Example: If the target is equal to 6 and num[1] = 3, then nums[1] + nums[1] = target is not a solution.

Examples:

Example 1:
Input Format: N = 5, arr[] = {2,6,5,8,11}, target = 14
Result: YES (for 1st variant)
       [1, 3] (for 2nd variant)
Explanation: arr[1] + arr[3] = 14. So, the answer is “YES” for the first variant and [1, 3] for 2nd variant.

Example 2:
Input Format: N = 5, arr[] = {2,6,5,8,11}, target = 15
Result: NO (for 1st variant)
	[-1, -1] (for 2nd variant)
Explanation: There exist no such two numbers whose sum is equal to the target.

Disclaimer: Don’t jump directly to the solution, try it out yourself first.

Problem Link.

Naive Approach(Brute-force approach):

Intuition: For each element of the given array, we will try to search for another element such that its sum is equal to the target. If such two numbers exist, we will return the indices or “YES” accordingly.

Approach:

First, we will use a loop(say i) to select the indices of the array one by one.
For every index i, we will traverse through the remaining array using another loop(say j) to find the other number such that the sum is equal to the target (i.e. arr[i] + arr[j] = target).

Observation: In every iteration, if the inner loop starts from index 0, we will be checking the same pair of numbers multiple times. For example, in iteration 1, for i = 0, we will check for the pair arr[0] and arr[1]. Again in iteration 2, for i = 1, we will check arr[1] and arr[0]. So, to eliminate these same pairs, we will start the inner loop from i+1.

Dry Run: Given array, nums = [2,1,3,4], target = 4

Using the naive approach, we first select one number and then find the second one.

For index 0, element= 2,
Then, we iterate through indices 1 to 3 to check if target – x, i.e. 4 – 2 = 2 exists. 2 does not exist from index 1 to 3, we move to the next index.

For index 1, element=1,
Then, we iterate through indices 2 to 3 to find if target – x, i.e. 4 – 1 = 3 exists. 3 exists at index 2, so we store the indices 1 and 2, break the loop, and return the indices.

Code Variant 1:

C++ Code

#include <bits/stdc++.h>
using namespace std;

string twoSum(int n, vector<int> &arr, int target) {
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            if (arr[i] + arr[j] == target) return "YES";
        }
    }
    return "NO";
}

int main()
{
    int n = 5;
    vector<int> arr = {2, 6, 5, 8, 11};
    int target = 14;
    string ans = twoSum(n, arr, target);
    cout << "This is the answer for variant 1: " << ans << endl;
    return 0;
}

Output: This is the answer for variant 1: YES

Time Complexity: O(N²), where N = size of the array.
Reason: There are two loops(i.e. nested) each running for approximately N times.

Space Complexity: O(1) as we are not using any extra space.

Java Code

import java.util.*;

public class tUf {
    public static String twoSum(int n, int []arr, int target) {
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                if (arr[i] + arr[j] == target) return "YES";
            }
        }
        return "NO";
    }

    public static void main(String args[]) {
        int n = 5;
        int[] arr = {2, 6, 5, 8, 11};
        int target = 14;
        String ans = twoSum(n, arr, target);
        System.out.println("This is the answer for variant 1: " + ans);
    }

}

Output: This is the answer for variant 1: YES

Time Complexity: O(N²), where N = size of the array.
Reason: There are two loops(i.e. nested) each running for approximately N times.

Space Complexity: O(1) as we are not using any extra space.

Code Variant 2:

C++ Code

#include <bits/stdc++.h>
using namespace std;

vector<int> twoSum(int n, vector<int> &arr, int target) {
    vector<int> ans;
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            if (arr[i] + arr[j] == target) {
                ans.push_back(i);
                ans.push_back(j);
                return ans;
            }
        }
    }
    return { -1, -1};
}

int main()
{
    int n = 5;
    vector<int> arr = {2, 6, 5, 8, 11};
    int target = 14;
    vector<int> ans = twoSum(n, arr, target);
    cout << "This is the answer for variant 2: [" << ans[0] << ", "
         << ans[1] << "]" << endl;
    return 0;
}

Output: This is the answer for variant 2: [1, 3]

Time Complexity: O(N²), where N = size of the array.
Reason: There are two loops(i.e. nested) each running for approximately N times.

Space Complexity: O(1) as we are not using any extra space.

Java Code

import java.util.*;

public class Main {
    public static int[] twoSum(int n, int []arr, int target) {
        int[] ans = new int[2];
        ans[0] = ans[1] = -1;
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                if (arr[i] + arr[j] == target) {
                    ans[0] = i;
                    ans[1] = j;
                    return ans;
                }
            }
        }
        return ans;
    }

    public static void main(String args[]) {
        int n = 5;
        int[] arr = {2, 6, 5, 8, 11};
        int target = 14;
        int[] ans = twoSum(n, arr, target);
        System.out.println("This is the answer for variant 2: [" + ans[0] + ", "
                           + ans[1] + "]");
    }

}

Output: This is the answer for variant 2: [1, 3]

Time Complexity: O(N²), where N = size of the array.
Reason: There are two loops(i.e. nested) each running for approximately N times.

Space Complexity: O(1) as we are not using any extra space.

Better Approach(using Hashing):

Intuition: Basically, in the previous approach we selected one element and then searched for the other one using a loop. Here instead of using a loop, we will use the HashMap to check if the other element i.e. target-(selected element) exists. Thus we can trim down the time complexity of the problem.

And for the second variant, we will store the element along will its index in the HashMap. Thus we can easily retrieve the index of the other element i.e. target-(selected element) without iterating the array.

Approach:

The steps are as follows:

We will select the element of the array one by one using a loop(say i).
Then we will check if the other required element(i.e. target-arr[i]) exists in the hashMap.
1. If that element exists, then we will return “YES” for the first variant or we will return the current index i.e. i, and the index of the element found using map i.e. mp[target-arr[i]].
2. If that element does not exist, then we will just store the current element in the hashMap along with its index. Because in the future, the current element might be a part of our answer.
Finally, if we are out of the loop, that means there is no such pair whose sum is equal to the target. In this case, we will return either “NO” or {-1, -1} as per the variant of the question.

Dry Run: Given array, nums = [2,3,1,4], target = 4

Note: Here x denotes the currently selected element.

For index 0, x = 2, and currently map is empty.
We try to find if target – x = 4 – 2 = 2 is present in the map or not.
For now, 2 does not exist on the map.
And we store the index of element 2. i.e., mp[2] = 0,

For index 1, x = 3
We try to find if target – x = 4  – 3 = 1 is present in the map or not.
For now, 1 does not exist on the map.
And we store the index of element 3. i.e., mp[3] = 1,

For index 2, x = 1 
We try to find if target – i = 4  – 1 = 3 is present in the map or not. 3 exists in the map, so we store index 2 and the value stored for key 3 in the map and break the loop. And return [1,2].

Code for Variant 1:

C++ Code

#include <bits/stdc++.h>
using namespace std;

string twoSum(int n, vector<int> &arr, int target) {
    unordered_map<int, int> mpp;
    for (int i = 0; i < n; i++) {
        int num = arr[i];
        int moreNeeded = target - num;
        if (mpp.find(moreNeeded) != mpp.end()) {
            return "YES";
        }
        mpp[num] = i;
    }
    return "NO";
}

int main()
{
    int n = 5;
    vector<int> arr = {2, 6, 5, 8, 11};
    int target = 14;
    string ans = twoSum(n, arr, target);
    cout << "This is the answer for variant 1: " << ans << endl;
    return 0;
}

Output: This is the answer for variant 1: YES

Time Complexity: O(N), where N = size of the array.
Reason: The loop runs N times in the worst case and searching in a hashmap takes O(1) generally. So the time complexity is O(N).

Note: In the worst case(which rarely happens), the unordered_map takes O(N) to find an element. In that case, the time complexity will be O(N²). If we use map instead of unordered_map, the time complexity will be O(N* logN) as the map data structure takes logN time to find an element.

Space Complexity: O(N) as we use the map data structure.

Note: We have optimized this problem enough. But if in the interview, we are not allowed to use the map data structure, then we should move on to the following approach i.e. two pointer approach. This approach will have the same time complexity as the better approach.

Java Code

import java.util.*;

public class tUf {
    public static String twoSum(int n, int []arr, int target) {
        HashMap<Integer, Integer> mpp = new HashMap<>();
        for (int i = 0; i < n; i++) {
            int num = arr[i];
            int moreNeeded = target - num;
            if (mpp.containsKey(moreNeeded)) {
                return "YES";
            }

            mpp.put(arr[i], i);
        }
        return "NO";
    }

    public static void main(String args[]) {
        int n = 5;
        int[] arr = {2, 6, 5, 8, 11};
        int target = 14;
        String ans = twoSum(n, arr, target);
        System.out.println("This is the answer for variant 1: " + ans);
    }

}

Output: This is the answer for variant 1: YES

Time Complexity: O(N), where N = size of the array.
Reason: The loop runs N times in the worst case and searching in a hashmap takes O(1) generally. So the time complexity is O(N).

Space Complexity: O(N) as we use the map data structure.

Code for Variant 2:

C++ Code

#include <bits/stdc++.h>
using namespace std;

vector<int> twoSum(int n, vector<int> &arr, int target) {
    unordered_map<int, int> mpp;
    for (int i = 0; i < n; i++) {
        int num = arr[i];
        int moreNeeded = target - num;
        if (mpp.find(moreNeeded) != mpp.end()) {
            return {mpp[moreNeeded], i};
        }
        mpp[num] = i;
    }
    return { -1, -1};
}

int main()
{
    int n = 5;
    vector<int> arr = {2, 6, 5, 8, 11};
    int target = 14;
    vector<int> ans = twoSum(n, arr, target);
    cout << "This is the answer for variant 2: [" << ans[0] << ", "
         << ans[1] << "]" << endl;
    return 0;
}

Output: This is the answer for variant 2: [1, 3]

Time Complexity: O(N), where N = size of the array.
Reason: The loop runs N times in the worst case and searching in a hashmap takes O(1) generally. So the time complexity is O(N).

Space Complexity: O(N) as we use the map data structure.

Java Code

import java.util.*;

public class tUf {
    public static int[] twoSum(int n, int []arr, int target) {
        int[] ans = new int[2];
        ans[0] = ans[1] = -1;
        HashMap<Integer, Integer> mpp = new HashMap<>();
        for (int i = 0; i < n; i++) {
            int num = arr[i];
            int moreNeeded = target - num;
            if (mpp.containsKey(moreNeeded)) {
                ans[0] = mpp.get(moreNeeded);
                ans[1] = i;
                return ans;
            }

            mpp.put(arr[i], i);
        }
        return ans;
    }

    public static void main(String args[]) {
        int n = 5;
        int[] arr = {2, 6, 5, 8, 11};
        int target = 14;
        int[] ans = twoSum(n, arr, target);
        System.out.println("This is the answer for variant 2: [" + ans[0] + ", "
                           + ans[1] + "]");

    }

}

Output: This is the answer for variant 2: [1, 3]

Time Complexity: O(N), where N = size of the array.
Reason: The loop runs N times in the worst case and searching in a hashmap takes O(1) generally. So the time complexity is O(N).

Space Complexity: O(N) as we use the map data structure.

Optimized Approach(using two-pointer):

Intuition: In this approach, we will first sort the array and will try to choose the numbers in a greedy way.

We will keep a left pointer at the first index and a right pointer at the last index. Now until left < right, we will check the sum of arr[left] and arr[right]. Now if the sum < target, we need bigger numbers and so we will increment the left pointer. But if sum > target, we need to consider lesser numbers and so we will decrement the right pointer.

If sum == target we will return either “YES” or the indices as per the question.
But if the left crosses the right pointer, we will return “NO” or {-1, -1}.

Approach:

The steps are the following:

We will sort the given array first.
Now, we will take two pointers i.e. left, which points to the first index, and right, which points to the last index.
Now using a loop we will check the sum of arr[left] and arr[right] until left < right.
1. If arr[left] + arr[right] > sum, we will decrement the right pointer.
2. If arr[left] + arr[right] < sum, we will increment the left pointer.
3. If arr[left] + arr[right] == sum, we will return the result.
Finally, if no results are found we will return “No” or {-1, -1}.

Dry Run: Given array, nums = [2,1,3,4], target = 4

First, we sort the array. So nums after sorting are [1,2,3,4]

We take two-pointers, left and right. The left points to index 0 and the right points to index 3.

Now we check if nums[left] + nums[right] == target. In this case, they don’t sum up, and nums[left] + nums[right] > target so that we will reduce right by 1.

Now, left = 0, right=2

Here, nums[left] + nums[right] == 1 + 3 == 4, which is the required target, so we will return the result.

Code:

C++ Code

#include <bits/stdc++.h>
using namespace std;

string twoSum(int n, vector<int> &arr, int target) {
    sort(arr.begin(), arr.end());
    int left = 0, right = n - 1;
    while (left < right) {
        int sum = arr[left] + arr[right];
        if (sum == target) {
            return "YES";
        }
        else if (sum < target) left++;
        else right--;
    }
    return "NO";
}

int main()
{
    int n = 5;
    vector<int> arr = {2, 6, 5, 8, 11};
    int target = 14;
    string ans = twoSum(n, arr, target);
    cout << "This is the answer for variant 1: " << ans << endl;
    return 0;
}

Output: This is the answer for variant 1: YES

Note: For variant 2, we can store the elements of the array along with its index in a new array. Then the rest of the code will be similar. And while returning, we need to return the stored indices instead of returning “YES”. But for this variant, the recommended approach is approach 2 i.e. hashing approach.

Time Complexity: O(N) + O(N*logN), where N = size of the array.
Reason: The loop will run at most N times. And sorting the array will take N*logN time complexity.

Space Complexity: O(1) as we are not using any extra space.

Note: Here we are distorting the given array. So, if we need to consider this change, the space complexity will be O(N).

Python Code

import java.util.*;

public class Main {
    public static String twoSum(int n, int []arr, int target) {
        Arrays.sort(arr);
        int left = 0, right = n - 1;
        while (left < right) {
            int sum = arr[left] + arr[right];
            if (sum == target) {
                return "YES";
            } else if (sum < target) left++;
            else right--;
        }
        return "NO";
    }

    public static void main(String args[]) {
        int n = 5;
        int[] arr = {2, 6, 5, 8, 11};
        int target = 14;
        String ans = twoSum(n, arr, target);
        System.out.println("This is the answer for variant 1: " + ans);

    }

}

Output: This is the answer for variant 1: YES

Time Complexity: O(N) + O(N*logN), where N = size of the array.
Reason: The loop will run at most N times. And sorting the array will take N*logN time complexity.

Space Complexity: O(1) as we are not using any extra space.

Note: Here we are distorting the given array. So, if we need to consider this change, the space complexity will be O(N).

Special thanks to Pranendu Bikash Pradhan and Sudip Ghosh and KRITIDIPTA GHOSH for contributing to this article on takeUforward. If you also wish to share your knowledge with the takeUforward fam, please check out this article.