Given an array of positive integers of size n. Find the maximum sum of triplet( ai + aj + ak ) such that 0 <= i < j < k < n and ai < aj < ak.
Input: a = 2 5 3 1 4 9 Output: 16 Explanation: All possible triplets are:- 2 3 4 => sum = 9 2 5 9 => sum = 16 2 3 9 => sum = 14 3 4 9 => sum = 16 1 4 9 => sum = 14 Maximum sum = 16
Simple Approach is to traverse for every triplet with three nested ‘for loops’ and find update the sum of all triplets one by one. Time complexity of this approach is O(n3) which is not sufficient for larger value of ‘n’.
Better approach is to make further optimization in above approach. Instead of traversing through every triplets with three nested loops, we can traverse through two nested loops. While traversing through each number(assume as middle element(aj)), find maximum number(ai) smaller than aj preceding it and maximum number(ak) greater than aj beyond it. Now after that, update the maximum answer with calculated sum of ai + aj + ak
Time complexity: O(n2)
Auxiliary space: O(1)
Best and efficient approach is use the concept of maximum suffix-array and binary search.
- For finding maximum number greater number greater than given number beyond it, we can maintain a maximum suffix-array array such that for any number(suffixi) it would contain maximum number from index i, i+1, … n-1. Suffix array can be calculated in O(n) time.
- For finding maximum number smaller than the given number preceding it, we can maintain a sorted list of numbers before a given number such we can simply perform a binary search to find a number which is just smaller than the given number. In C++ language, we can perform this by using set associative container of STL library.
Time complexity: O(n*log(n))
Auxiliary space: O(n)
This article is contributed by Shubham Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeksorg. See your article appearing on the GeeksforGeeks main page and help other Geeks
- Find triplets in an array whose AND is maximum
- Maximum value of XOR among all triplets of an array
- Find number of triplets in array such that a[i]>a[j]>a[k] and i<j<k
- Find all triplets in a sorted array that forms Geometric Progression
- Find all triplets with zero sum
- Given an array arr, find the maximum j - i such that arr[j] > arr[i]
- Find Maximum value of abs(i - j) * min(arr[i], arr[j]) in an array arr
- Triplets in array with absolute difference less than k
- Find maximum xor of k elements in an array
- Find the maximum element in the array other than Ai
- Find Maximum XOR value of a sub-array of size k
- Number of triplets in array having subarray xor equal
- Print all triplets in sorted array that form AP
- Count of triplets in an array that satisfy the given conditions
- Find a pair (n,r) in an integer array such that value of nCr is maximum