Triplet in a Sorted Array

Find a Triplet That Sum to a Given Value | Comprehensive Guide

The problem of finding a triplet that sums to a given value is a common algorithmic challenge. Given an array of integers, the goal is to identify three distinct elements that, when summed together, equal a specific target value. This problem is widely encountered in coding interviews and competitive programming due to its practical applications and the variety of approaches available for solving it efficiently.

Problem Definition

Given an array of integers and a target value (sum), find three numbers in the array such that their sum equals the target value.

For example, for the array [1, 4, 45, 6, 10, 8] and the target sum 22, the triplet (4, 10, 8) would sum to 22.

Key Approaches to Solve the Problem

1. Brute Force Approach

The simplest solution is to check every combination of three elements in the array and verify if their sum equals the target value. This approach has a time complexity of O(n³) due to the three nested loops required to evaluate all possible triplets.

However, this solution is inefficient for large arrays and can be improved with more optimized methods.

2. Using Sorting and the Two Pointers Technique

A more efficient solution involves sorting the array first and then using the two pointers technique to find the triplet that sums to the target value. This approach reduces the time complexity to O(n²).

Steps for the Two Pointers Approach:

Sort the Array:

• Sort the input array in ascending order. Sorting helps in efficiently applying the two pointers technique.

Fix One Element and Use Two Pointers:

• Fix the first element as the base of the triplet. For the remaining two elements, set two pointers: one at the start of the remaining subarray and the other at the end. Adjust the pointers based on the current sum.

Check the Sum:

• If the sum of the three elements is equal to the target value, return the triplet. If the sum is smaller than the target, move the left pointer to the right (to increase the sum). If the sum is larger than the target, move the right pointer to the left (to decrease the sum).

Example of the Two Pointers Approach

Let’s take an example array [1, 4, 45, 6, 10, 8] and a target sum of 22.

Step 1: Sort the array: [1, 4, 6, 8, 10, 45].

Step 2: Fix the first element as 1, and apply the two pointers technique to the rest of the array:

• Left pointer at 4, right pointer at 45.
• Check the sum: 1 + 4 + 45 = 50. This is greater than 22, so move the right pointer leftward to 10.

Step 3: Check again:

• Sum: 1 + 4 + 10 = 15. This is less than 22, so move the left pointer rightward to 6.

Step 4: Check again:

• Sum: 1 + 6 + 10 = 17. Still less than 22, so move the left pointer rightward to 8.

Step 5: Now, the sum is 1 + 8 + 10 = 19, still less than 22.

Repeat this process by fixing the next element and adjusting the pointers accordingly until you find a triplet that sums to 22, such as (4, 10, 8).

Time Complexity

• Sorting the Array: Sorting takes O(n log n) time.
• Two Pointers for Each Element: For each fixed element, the two pointers scan the rest of the array, resulting in a O(n²) time complexity overall.

Thus, the total time complexity of this approach is O(n²), which is a significant improvement over the brute force method.

Applications

Financial Analysis:

• This algorithm can be used in finance to identify three expenses or transactions that together meet a specific budget or target.

Data Analysis:

• In data analysis, finding triplets that sum to a particular value can help in identifying relationships among three variables or data points.

Game Development:

• In some game algorithms, finding combinations of three items that meet certain criteria can be useful for developing features such as puzzles or resource allocation.

Why Learn This Approach?

Learning how to find a triplet that sums to a given value is a great exercise in improving algorithmic problem-solving skills. By mastering the two pointers technique, you not only solve this problem more efficiently but also gain a tool that is applicable in various other problems related to arrays, sorting, and searching.

Topics Covered:

Problem Definition: Understanding the problem of finding a triplet that sums to a given value.

Approaches: Brute force approach and the optimized two pointers technique.

Applications: Practical uses of the triplet-sum problem in different fields.

For more details and further examples, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/find-a-triplet-that-sum-to-a-given-value/.