Exploring the Two Sum Problem: Two-Pointer Approach and Complexity Analysis.

Given an array of integers nums?and an integer target, return indices of the two numbers such that they add up to target.

You may assume that each input would have exactly one solution, and you may not use the same element twice.

You can return the answer in any order.

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Example 1:

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

Example 2:

Input: nums = [3,2,4], target = 6
Output: [1,2]
        

Example 3:

Input: nums = [3,3], target = 6
Output: [0,1]        

The Two Sum problem is a common coding challenge that involves finding two numbers in an array that add up to a specific target. While a hash map-based solution is often used for this problem, another effective method is the two-pointer approach. This approach is particularly useful when the input array is sorted or can be sorted. Let's examine the solution steps for the two-pointer approach and analyze its time and space complexity.

Solution Steps with Two Pointers

1. Sort the Array: If the input array is not already sorted, the first step is to sort it. This is essential for the two-pointer technique to work effectively.
2. Initialize Two Pointers: Two pointers are initialized, one at the beginning of the array (left) and the other at the end (right).
3. Iterate and Find the Pair: The main part of the algorithm is iterating through the array with these two pointers. In each iteration, the sum of the elements pointed by left and right is compared to the target.If the sum equals the target, the indices (or values) of these elements are the required answer.If the sum is less than the target, the left pointer is moved one step to the right (to increase the sum).If the sum is greater than the target, the right pointer is moved one step to the left (to decrease the sum).
4. Repeat Until a Solution is Found: This process is repeated until the two pointers meet or the target sum is found.
5. Return the Result: Once the pair that sums up to the target is found, the indices (or the values, depending on the problem's requirement) of these elements are returned.

Time Complexity

1. Sorting: The initial sorting of the array has a time complexity of O(n log n), where n is the number of elements in the array.
2. Two-Pointer Iteration: After sorting, the algorithm iterates through the array once with the two pointers. This iteration is O(n), as each element is looked at most once.

Combining these steps, the overall time complexity of the two-pointer approach to the Two Sum problem is O(n log n) due to the sorting step.

Space Complexity

1. Sorting: The space complexity for sorting can vary based on the algorithm used. For example, if an in-place sorting algorithm like Heapsort is used, the space complexity can be O(1). However, for algorithms like Mergesort, it could be O(n).
2. Pointers: The two pointers themselves use constant space, O(1).

In scenarios where in-place sorting is used, the space complexity is O(1). Otherwise, it could be O(n) due to the space required for sorting.

Conclusion

The two-pointer approach for the Two Sum problem is a viable alternative, especially for sorted arrays. It offers an O(n log n) time complexity mainly due to the sorting step and a potential O(1) space complexity if in-place sorting is used. This method is a useful tool in a programmer's arsenal, particularly when dealing with sorted data and aiming to minimize space usage.