Problem Link - Maximize Disjoint Pair Sum in Greedy Algorithms
Problem Statement:
We are given an array of integers and need to find the maximum possible sum of disjoint pairs such that the difference between the two numbers in each pair is strictly less than a given value D. The pairs must be disjoint, meaning no number can be part of more than one pair.
Approach:
-
Sort the Array: First, sort the array to facilitate pairing adjacent elements. Sorting ensures that smaller differences appear between consecutive numbers, making it easier to find pairs that satisfy the condition of having a difference less than
D. -
Form Pairs from Largest to Smallest:
- After sorting, start forming pairs from the largest elements. This is because if we take the largest numbers first, we maximize the sum of pairs while ensuring the difference constraint is satisfied.
- Traverse the array from the largest element downwards. For each element, if it can form a valid pair with the next smaller element (i.e., the difference is less than
D), add the sum of the pair to the total sum and skip the next element (since it’s already part of a pair).
Complexity:
- Time Complexity: Sorting the array takes
O(N log N). Forming pairs by iterating through the array takesO(N). The overall complexity isO(N log N). - Space Complexity:
O(1).