# Find maximum sum of triplets in an array such than i < j < k and a[i] < a[j] < a[k]

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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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

## CPP

 `// C++ program to find maximum triplet sum ` `#include ` `using` `namespace` `std; ` ` `  `// Function to calculate maximum triplet sum ` `int` `maxTripletSum(``int` `arr[], ``int` `n) ` `{ ` `    ``// Initialize the answer ` `    ``int` `ans = 0; ` ` `  `    ``for` `(``int` `i = 1; i < n - 1; ++i) { ` `        ``int` `max1 = 0, max2 = 0; ` ` `  `        ``// find maximum value(less than arr[i]) ` `        ``// from i+1 to n-1 ` `        ``for` `(``int` `j = 0; j < i; ++j) ` `            ``if` `(arr[j] < arr[i]) ` `                ``max1 = max(max1, arr[j]); ` ` `  `        ``// find maximum value(greater than arr[i]) ` `        ``// from i+1 to n-1 ` `        ``for` `(``int` `j = i + 1; j < n; ++j) ` `            ``if` `(arr[j] > arr[i]) ` `                ``max2 = max(max2, arr[j]); ` ` `  `        ``// store maximum answer ` `        ``ans = max(ans, max1 + arr[i] + max2); ` `    ``} ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `arr[] = { 2, 5, 3, 1, 4, 9 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``cout << maxTripletSum(arr, n); ` `    ``return` `0; ` `} `

## Java

 `// Java program to find maximum triplet sum ` ` `  `import` `java.io.*; ` `import` `java.math.*; ` ` `  `class` `GFG { ` ` `  `    ``// Function to calculate maximum triplet sum ` `    ``static` `int` `maxTripletSum(``int` `arr[], ``int` `n) ` `    ``{ ` `        ``// Initialize the answer ` `        ``int` `ans = ``0``; ` ` `  `        ``for` `(``int` `i = ``1``; i < n - ``1``; ++i) { ` `            ``int` `max1 = ``0``, max2 = ``0``; ` ` `  `            ``// find maximum value(less than arr[i]) ` `            ``// from i+1 to n-1 ` `            ``for` `(``int` `j = ``0``; j < i; ++j) ` `                ``if` `(arr[j] < arr[i]) ` `                    ``max1 = Math.max(max1, arr[j]); ` ` `  `            ``// find maximum value(greater than arr[i]) ` `            ``// from i+1 to n-1 ` `            ``for` `(``int` `j = i + ``1``; j < n; ++j) ` `                ``if` `(arr[j] > arr[i]) ` `                    ``max2 = Math.max(max2, arr[j]); ` ` `  `            ``// store maximum answer ` `            ``ans = Math.max(ans, max1 + arr[i] + max2); ` `        ``} ` ` `  `        ``return` `ans; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `arr[] = { ``2``, ``5``, ``3``, ``1``, ``4``, ``9` `}; ` `        ``int` `n = arr.length; ` `        ``System.out.println(maxTripletSum(arr, n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Nikita Tiwari. `

## Python3

 `# Python 3 program to ` `# find maximum triplet sum ` ` `  `# Function to calculate ` `# maximum triplet sum ` `def` `maxTripletSum(arr, n) : ` ` `  `    ``# Initialize the answer ` `    ``ans ``=` `0` `  `  `    ``for` `i ``in` `range``(``1``, (n ``-` `1``)) : ` `        ``max1 ``=` `0` `        ``max2 ``=` `0` `  `  `        ``# find maximum value(less than arr[i]) ` `        ``# from i + 1 to n-1 ` `        ``for` `j ``in` `range``(``0``, i) : ` `            ``if` `(arr[j] < arr[i]) : ` `                ``max1 ``=` `max``(max1, arr[j]) ` `  `  `        ``# find maximum value(greater than arr[i]) ` `        ``# from i + 1 to n-1 ` `        ``for` `j ``in` `range``((i ``+` `1``), n) : ` `            ``if` `(arr[j] > arr[i]) : ` `                ``max2 ``=` `max``(max2, arr[j]) ` `  `  `        ``# store maximum answer ` `        ``ans ``=` `max``(ans, max1 ``+` `arr[i] ``+` `max2) ` `  `  `    ``return` `ans ` ` `  ` `  `# Driver code ` ` `  `arr ``=` `[ ``2``, ``5``, ``3``, ``1``, ``4``, ``9` `] ` `n ``=` `len``(arr) ` `print``(maxTripletSum(arr, n)) ` ` `  ` `  `# This code is contributed ` `# by Nikita Tiwari. `

## C#

 `// C# program to find maximum triplet sum ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// Function to calculate maximum triplet sum ` `    ``static` `int` `maxTripletSum(``int``[] arr, ``int` `n) ` `    ``{ ` `         `  `        ``// Initialize the answer ` `        ``int` `ans = 0; ` ` `  `        ``for` `(``int` `i = 1; i < n - 1; ++i) ` `        ``{ ` `            ``int` `max1 = 0, max2 = 0; ` ` `  `            ``// find maximum value(less than  ` `            ``// arr[i]) from i+1 to n-1 ` `            ``for` `(``int` `j = 0; j < i; ++j) ` `                ``if` `(arr[j] < arr[i]) ` `                    ``max1 = Math.Max(max1, arr[j]); ` ` `  `            ``// find maximum value(greater than ` `            ``// arr[i]) from i+1 to n-1 ` `            ``for` `(``int` `j = i + 1; j < n; ++j) ` `                ``if` `(arr[j] > arr[i]) ` `                    ``max2 = Math.Max(max2, arr[j]); ` ` `  `            ``// store maximum answer ` `            ``ans = Math.Max(ans, max1 + arr[i] + max2); ` `        ``} ` ` `  `        ``return` `ans; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `         `  `        ``int``[] arr = { 2, 5, 3, 1, 4, 9 }; ` `        ``int` `n = arr.Length; ` `         `  `        ``Console.WriteLine(maxTripletSum(arr, n)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` ``\$arr``[``\$i``]) ` `                ``\$max2` `= max(``\$max2``, ``\$arr``[``\$j``]); ` ` `  `        ``// store maximum answer ` `        ``\$ans` `= max(``\$ans``, ``\$max1` `+ ``\$arr``[``\$i``] + ``\$max2``); ` `    ``} ` ` `  `    ``return` `\$ans``; ` `} ` ` `  `    ``// Driver code ` `    ``\$arr` `= ``array``(2, 5, 3, 1, 4, 9); ` `    ``\$n` `= sizeof(``\$arr``); ` `    ``echo` `maxTripletSum(``\$arr``, ``\$n``); ` ` `  `// This code is contributed by nitin mittal. ` `?> `

Output :

```16
```

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.

 `// C++ program to find maximum triplet sum ` `#include ` `using` `namespace` `std; ` ` `  `// Utility function to get the lower last min ` `// value of 'n' ` `int` `getLowValue(set<``int``>& lowValue, ``int``& n) ` `{ ` `    ``auto` `it = lowValue.lower_bound(n); ` ` `  `    ``// Since 'lower_bound' returns the first ` `    ``// iterator greater than 'n', thus we ` `    ``// have to decrement the pointer for ` `    ``// getting the minimum value ` `    ``--it; ` ` `  `    ``return` `(*it); ` `} ` ` `  `// Function to calculate maximum triplet sum ` `int` `maxTripletSum(``int` `arr[], ``int` `n) ` `{ ` `    ``// Initialize suffix-array ` `    ``int` `maxSuffArr[n + 1]; ` ` `  `    ``// Set the last element ` `    ``maxSuffArr[n] = 0; ` ` `  `    ``// Calculate suffix-array containing maximum ` `    ``// value for index i, i+1, i+2, ... n-1 in ` `    ``// arr[i] ` `    ``for` `(``int` `i = n - 1; i >= 0; --i) ` `        ``maxSuffArr[i] = max(maxSuffArr[i + 1], arr[i]); ` ` `  `    ``int` `ans = 0; ` ` `  `    ``// Initialize set container ` `    ``set<``int``> lowValue; ` ` `  `    ``// Insert minimum value for first comparison ` `    ``// in the set ` `    ``lowValue.insert(INT_MIN); ` ` `  `    ``for` `(``int` `i = 0; i < n - 1; ++i) { ` `        ``if` `(maxSuffArr[i + 1] > arr[i]) { ` `            ``ans = max(ans, getLowValue(lowValue, ` `                                       ``arr[i]) ` `                               ``+ arr[i] + maxSuffArr[i + 1]); ` ` `  `            ``// Insert arr[i] in set<> for further ` `            ``// processing ` `            ``lowValue.insert(arr[i]); ` `        ``} ` `    ``} ` `    ``return` `ans; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `arr[] = { 2, 5, 3, 1, 4, 9 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` ` `  `    ``cout << maxTripletSum(arr, n); ` `    ``return` `0; ` `} `

```Output:
16
```

Time complexity: O(n*log(n))
Auxiliary space: O(n)

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Improved By : vt_m, nitin mittal

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