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 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)
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.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.