Maximum sum Bi-tonic Sub-sequence

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++

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// C++ program to find maximum sum 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;
}

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// java program to find maximum 


// sum of bi-tonic sub-sequence 


import java.io.*; 


  


class GFG { 


  


    // Function return maximum sum 


    // of Bi-tonic sub-sequence 


    static int MaxSumBS(int arr[], int n) 


    { 


        int max_sum = Integer.MIN_VALUE; 


  


        // 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[] = new int[n]; 


        int MSDBS[] = new int[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 = Math.max(max_sum, (MSDBS[i] + MSIBS[i] - arr[i])); 


  


        // return max sum of bi-tonic 


        // sub-sequence 


        return max_sum; 


    } 


  


    // Driver program 


    public static void main(String[] args) 


    { 


        int arr[] = { 1, 15, 51, 45, 33, 100, 12, 18, 9 }; 


        int n = arr.length; 


        System.out.println("Maximum Sum : " + MaxSumBS(arr, n)); 


    } 


} 


  


// This code is contributed by vt_m 



















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Python














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# 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 



















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// C# program to find maximum 


// sun of bi-tonic sub-sequence 


using System; 


  


class GFG { 


  


    // Function return maximum sum 


    // of Bi-tonic sub-sequence 


    static int MaxSumBS(int[] arr, int n) 


    { 


        int max_sum = int.MinValue; 


  


        // 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 = new int[n]; 


        int[] MSDBS = new int[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 = Math.Max(max_sum, (MSDBS[i] + MSIBS[i] - arr[i])); 


  


        // return max sum of bi-tonic 


        // sub-sequence 


        return max_sum; 


    } 


  


    // Driver program 


    public static void Main() 


    { 


        int[] arr = { 1, 15, 51, 45, 33, 100, 12, 18, 9 }; 


        int n = arr.Length; 


        Console.WriteLine("Maximum Sum : " + MaxSumBS(arr, n)); 


    } 


} 


  


// This code is contributed by vt_m 



















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<?php 


// PHP program to find maximum 


// sun of bi-tonic sub-sequence 


function MaxSumBS($arr, $n) 


{ 


    $max_sum = PHP_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 


    $MSIBS = array(); 


    $MSDBS = array(); 


    for ($i = 0; $i < $n; $i++)  


    { 


        $MSDBS[$i] = $arr[$i]; 


        $MSIBS[$i] = $arr[$i]; 


    } 


  


    // Compute MSIBS values 


    // from left to right */ 


    for ($i = 1; $i < $n; $i++) 


        for ($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 ($i = $n - 2; $i >= 0; $i--) 


        for ($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 ($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 Code 


$arr = array(1, 15, 51, 45, 33,  


             100, 12, 18, 9); 


$n = count($arr); 


echo "Maximum Sum : " ,  


      MaxSumBS($arr, $n); 


  


// This code is contributed 


// by shiv_bhakt. 


?> 



















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Output:


Maximum Sum : 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.


Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



          
          
          
          

            

    

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