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

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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 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; 
} 

Java

// 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 

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 

C#

// 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 

PHP

<?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. 
?> 

Output:

Maximum Sum : 194

Time complexity : O(n²)

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