Maximum sum Bi-tonic Sub-sequence

2.7

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(n2)

This article is contributed by Nishant_Singh (Pintu). 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@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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