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.


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