Two Sum II - O(n log n) Time Complexity in Java

Input: nums = [2, 7, 11, 15], target = 9
Output: [0, 1]
Explanation: nums[0] + nums[1] = 2 + 7 = 9

Your algorithm should run in O(n log n) time and use O(n) extra space.

Understanding the Problem

The core challenge of this problem is to find two distinct indices in the array such that the sum of the elements at these indices equals the target value. This problem is significant in various applications such as financial transactions, where you need to find pairs of transactions that sum up to a specific amount.

Potential pitfalls include assuming that there are multiple solutions or using the same index twice, which is not allowed.

Approach

To solve this problem, we can use a hash map to store the difference between the target and each element as we iterate through the array. This allows us to check in constant time whether the complement of the current element exists in the hash map.

Let's discuss a naive solution first:

Naive Solution

The naive approach involves using two nested loops to check all possible pairs of elements. This solution has a time complexity of O(n^2), which is not optimal for large arrays.

Optimized Solution

An optimized solution involves using a hash map to store the indices of the elements as we iterate through the array. This allows us to find the complement of the current element in constant time, resulting in an overall time complexity of O(n).

Algorithm

Here is a step-by-step breakdown of the optimized algorithm:

1. Initialize an empty hash map to store the indices of the elements.
2. Iterate through the array.
3. For each element, calculate its complement by subtracting the element from the target.
4. Check if the complement exists in the hash map.
5. If it exists, return the indices of the current element and its complement.
6. If it does not exist, add the current element and its index to the hash map.
7. If no solution is found, return [-1, -1].

Code Implementation

import java.util.HashMap;

public class TwoSum {
    public int[] twoSum(int[] nums, int target) {
        // Create a hash map to store the indices of the elements
        HashMap<Integer, Integer> map = new HashMap<>();
        
        // Iterate through the array
        for (int i = 0; i < nums.length; i++) {
            // Calculate the complement
            int complement = target - nums[i];
            
            // Check if the complement exists in the hash map
            if (map.containsKey(complement)) {
                // Return the indices of the current element and its complement
                return new int[] { map.get(complement), i };
            }
            
            // Add the current element and its index to the hash map
            map.put(nums[i], i);
        }
        
        // If no solution is found, return [-1, -1]
        return new int[] { -1, -1 };
    }
}

Complexity Analysis

The time complexity of this solution is O(n) because we iterate through the array once. The space complexity is also O(n) because we store the indices of the elements in a hash map.

Edge Cases

Potential edge cases include:

• An empty array: The function should return [-1, -1].
• An array with one element: The function should return [-1, -1].
• No two elements sum up to the target: The function should return [-1, -1].

Testing

To test the solution comprehensively, consider the following test cases:

• Simple case: nums = [2, 7, 11, 15], target = 9, expected output = [0, 1]
• No solution: nums = [1, 2, 3], target = 6, expected output = [-1, -1]
• Negative numbers: nums = [-1, -2, -3, -4], target = -6, expected output = [1, 3]
• Multiple pairs: nums = [1, 3, 3, 4], target = 6, expected output = [1, 2]

Thinking and Problem-Solving Tips

When approaching such problems, consider the following tips:

• Understand the problem requirements and constraints thoroughly.
• Think about different data structures that can help optimize the solution.
• Break down the problem into smaller steps and solve each step incrementally.
• Practice solving similar problems to improve your problem-solving skills.

Conclusion

In this blog post, we discussed the Two Sum II problem and provided a detailed solution using a hash map to achieve an optimal time complexity of O(n). We also covered edge cases, testing, and tips for approaching such problems. Understanding and solving such problems is crucial for improving your algorithmic thinking and coding skills.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode Two Sum Problem
• GeeksforGeeks Two Sum Problem
• Coding Ninjas Two Sum Problem