Finding all triplets in an array whose sum is less than or equal to a given sum is a classic problem in computer science that blends elements of algorithm design and mathematical analysis. This task involves efficiently searching through combinations of array elements to identify triplets that fit the desired constraint. This article explores the problem in detail, considering both a naive approach and a more optimized solution using sorting and a two-pointer technique.

Problem Statement

Given an array of integers and a target sum, the goal is to find all unique triplets such that the sum of each triplet is less than or equal to the target sum.

Naive Approach

Explanation

The naive approach involves three nested loops. The outer loop fixes the first element of the triplet, and the two inner loops traverse through the potential pairs to check whether their sum meets the criteria.

Pseudocode

• Calculate the sum of elements at indices `i`, `left`, and `right`.
• If the sum is less than or equal to the target sum, include all possible pairs from `left` to `right` as they form valid triplets with `arr[i]` (due to the sorted nature).
• Move the left pointer if successful; otherwise, decrement the right pointer to test smaller sums.
• Sorted Array: [1, 3, 4, 5, 7]
• Output Triplets:
  • (1, 3, 4)
  • (1, 3, 5)
  • (1, 3, 7)
  • (1, 4, 5)
  • (3, 4, 5)
• Empty Array: Return none as no triplets can be formed.
• Less than Three Elements: Directly return none.
• All Elements Greater than Target: Return none as no valid triplets exist.
• Triplet with Exact Sum: Modify the condition to find an exact sum.
• Triplet with Maximum Sum: Use a similar approach but maximize the sum for the closest possible to the target without exceeding it.