What is the Two Sum Problem in DSA.

🧩 What is the Two Sum Problem?

The Two Sum Problem is a classic Data Structures and Algorithms question you’ll see in coding interviews.

A list (array) of numbers.
A target number.

Find two different numbers in the array that add up exactly to the target number, and return their positions (indices).

Example:

Array:  [2, 7, 11, 15]  
Target: 9  
Answer: [0, 1]  // Because 2 + 7 = 9

In this example:

0 is the position of 2 in the array.
1 is the position of 7 in the array.

📜 Problem Statement in Java

When solving the problem in Java, we usually write it as:

public int[] twoSum(int[] nums, int target)

Where:

nums: the given array of numbers.
target: the sum we are trying to find.
Output: an array with two numbers, which are the indices of the elements that add up to target.

🐌 Brute Force Approach

What is Brute Force?

Brute force means trying every possible combination until you find the answer. It’s like checking every pair of shoes in a shop until you find the perfect match.

How It Works for Two Sum:

Start at the first number in the array.
Pair it with every other number that comes after it.
Check if their sum equals the target.
If yes - return their positions (indices).
If no - move to the next number and repeat.

Why It’s Simple but Slow:

If you have a small array, this is fine.
If you have a large array (like 10,000 numbers), it will take too long because you’re checking every possible pair.

Time and Space Complexity:

Time Complexity: O(n²): If you double the size of the array, the time it takes grows a lot.
Space Complexity: O(1): No extra storage is needed.

Example Brute Force:

public int[] twoSumBruteForce(int[] nums, int target) {
    for (int i = 0; i < nums.length; i++) {
        for (int j = i + 1; j < nums.length; j++) {
            if (nums[i] + nums[j] == target) {
                return new int[] { i, j };
            }
        }
    }
    return new int[] {}; // If no solution found
}

⚡ Optimal Solution Using Hash Map

What is a Hash Map?

A hash map is like a smart dictionary where:

You have a key (like a word).
You have a value (like the meaning of the word).

You can look up any key instantly, without searching through everything.

How It Works for Two Sum:

Start with an empty hash map.
Go through the array one number at a time.
For the current number, find out what number would complete the target:

complement = target - currentNumber

Check if this complement is already in the hash map:
- If yes: we have found our answer (return the stored index and the current index).
- If no: store the current number in the hash map with its index.

Why It’s Fast:

You only pass through the array once.
Each lookup in the hash map takes constant time (O(1)).

Time and Space Complexity:

Time Complexity: O(n) Much faster than O(n²).
Space Complexity: O(n) Uses extra memory to store seen numbers.

Example Optimal Solution:

import java.util.*;

public int[] twoSum(int[] nums, int target) {
    Map<Integer, Integer> map = new HashMap<>();
    for (int i = 0; i < nums.length; i++) {
        int complement = target - nums[i];
        if (map.containsKey(complement)) {
            return new int[] { map.get(complement), i };
        }
        map.put(nums[i], i);
    }
    return new int[] {}; // If no solution found
}

🧠 Why Interviewers Ask the Two Sum Question

Tests your basics: Can you loop through arrays?
Checks your problem-solving: Can you think of faster ways?
Introduces optimization: Can you improve from O(n²) to O(n)?
Leads to harder problems: Two Sum is the first step to learning other “pair sum” problems.

📚 Summary

The brute force approach checks all possible pairs and is easy to understand, but it’s slow for large arrays. The optimal solution uses a hash map to solve the problem in a single pass, making it much faster. By learning this problem, you not only understand arrays and hash maps better but also prepare yourself for more advanced problems in coding interviews.