Given an array of integers. A subsequence of arr[] is called Bitonic if it is first increasing, then decreasing.

Examples:

Input : arr[] = {1, 15, 51, 45, 33, 100, 12, 18, 9} Output : 194 Explanation : Bi-tonic Sub-sequence are : {1, 51, 9} or {1, 50, 100, 18, 9} or {1, 15, 51, 100, 18, 9} or {1, 15, 45, 100, 12, 9} or {1, 15, 45, 100, 18, 9} .. so on Maximum sum Bi-tonic sub-sequence is 1 + 15 + 51 + 100 + 18 + 9 = 194 Input : arr[] = {80, 60, 30, 40, 20, 10} Output : 210

This problem is a variation of standard Longest Increasing Subsequence (LIS) problem and longest Bitonic Sub-sequence.

We construct two arrays MSIBS[] and MSDBS[]. MSIBS[i] stores the sum of the Increasing subsequence ending with arr[i]. MSDBS[i] stores the sum of the Decreasing subsequence starting from arr[i]. Finally, we need to return the max sum of MSIBS[i] + MSDBS[i] – Arr[i].

Below is the implementation of above idea

## C/C++

// C++ program to find maximum sun of bi-tonic sub-sequence #include<bits/stdc++.h> using namespace std; // Function return maximum sum of Bi-tonic sub-sequence int MaxSumBS(int arr[] , int n ) { int max_sum = INT_MIN; // MSIBS[i] ==> Maximum sum Increasing Bi-tonic // subsequence ending with arr[i] // MSDBS[i] ==> Maximum sum Decreasing Bi-tonic // subsequence starting with arr[i] // Initialize MSDBS and MSIBS values as arr[i] for // all indexes int MSIBS[n], MSDBS[n]; for (int i = 0 ; i<n ; i++ ) { MSDBS[i] = arr[i]; MSIBS[i] = arr[i]; } // Compute MSIBS values from left to right */ for (int i = 1; i < n; i++) for (int j = 0; j < i; j++) if (arr[i] > arr[j] && MSIBS[i] < MSIBS[j] + arr[i]) MSIBS[i] = MSIBS[j] + arr[i]; // Compute MSDBS values from right to left for (int i = n-2; i >= 0; i--) for (int j = n-1; j > i; j--) if (arr[i] > arr[j] && MSDBS[i] < MSDBS[j] + arr[i]) MSDBS[i] = MSDBS[j] + arr[i]; // Find the maximum value of MSIBS[i] + MSDBS[i] - arr[i] for (int i = 0 ; i <n ; i++) max_sum = max(max_sum, (MSDBS[i] + MSIBS[i] - arr[i])); // return max sum of bi-tonic sub-sequence return max_sum; } // Driver program int main() { int arr[]= {1 , 15 ,51 ,45 ,33 ,100 ,12 ,18 ,9}; int n = sizeof(arr)/sizeof(arr[0]); cout << "Maximum Sum : " << MaxSumBS(arr, n); return 0; }

## Python

# Dynamic Programming implementation of maximum sum of bitonic subsequence # Function return maximum sum of Bi-tonic sub-sequence def max_sum(arr, n): # MSIBS[i] ==> Maximum sum Increasing Bi-tonic # subsequence ending with arr[i] # MSDBS[i] ==> Maximum sum Decreasing Bi-tonic # subsequence starting with arr[i] # allocate memory for MSIBS and initialize it to arr[i] for # all indexes MSIBS = arr[:] # Compute MSIBS values from left to right for i in range(n): for j in range(0, i): if arr[i] > arr[j] and MSIBS[i] < MSIBS[j] + arr[i]: MSIBS[i] = MSIBS[j] + arr[i] # allocate memory for MSDBS and initialize it to arr[i] for # all indexes MSDBS = arr[:] # Compute MSDBS values from right to left for i in range(1, n + 1): for j in range(1, i): if arr[-i] > arr[-j] and MSDBS[-i] < MSDBS[-j] + arr[-i]: MSDBS[-i] = MSDBS[-j] + arr[-i] max_sum = float("-Inf") # Find the maximum value of MSIBS[i] + MSDBS[i] - arr[i] for i ,j ,k in zip(MSIBS, MSDBS, arr): max_sum = max(max_sum, i + j - k) # return max sum of bi-tonic sub-sequence return max_sum # Driver program to test the above function def main(): arr = [1 , 15 ,51 ,45 ,33 ,100 ,12 ,18 ,9] n = len(arr) print max_sum(arr, n) if __name__ == '__main__': main() #This code is contributed by Neelam Yadav

Output:

194

Time complexity : O(n^{2})

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