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, 51, 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].
Step-by-step approach of the above idea:
- Start with an array of integers arr and its length n.
- Initialize two arrays MSIBS and MSDBS of length n with the values of arr.
- Compute the values of MSIBS from left to right by iterating through each element i and comparing it with all elements before it. If arr[i] > arr[j] and MSIBS[i] < MSIBS[j] + arr[i], then update MSIBS[i] to MSIBS[j] + arr[i].
- Compute the values of MSDBS from right to left by iterating through each element i and comparing it with all elements after it. If arr[i] > arr[j] and MSDBS[i] < MSDBS[j] + arr[i], then update MSDBS[i] to MSDBS[j] + arr[i].
- Find and return the maximum value of MSIBS[i] + MSDBS[i] – arr[i] for all i.
Pseudocode:
function MaxSumBS(arr, n):
max_sum = INT_MIN
MSIBS = [arr[i] for i in range(n)]
MSDBS = [arr[i] for i in range(n)]
for i from 1 to n-1:
for j from 0 to i-1:
if arr[i] > arr[j] and MSIBS[i] < MSIBS[j] + arr[i]:
MSIBS[i] = MSIBS[j] + arr[i]
for i from n-2 to 0:
for j from n-1 to i+1:
if arr[i] > arr[j] and MSDBS[i] < MSDBS[j] + arr[i]:
MSDBS[i] = MSDBS[j] + arr[i]
for i from 0 to n-1:
max_sum = max(max_sum, MSIBS[i] + MSDBS[i] - arr[i])
return max_sum
Below is the implementation of above idea
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-sequenceint 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 programint 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-sequenceimport 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 |
Python3
# Dynamic Programming implementation of maximum sum of bitonic subsequence # Function return maximum sum of Bi-tonic sub-sequencedef 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 functiondef main(): arr = [1, 15, 51, 45, 33, 100, 12, 18, 9] n = len(arr) print ("Maximum Sum :", max_sum(arr, n))if __name__ == '__main__': main()# This code is contributed by Neelam Yadav |
C#
// C# program to find maximum// sum of bi-tonic sub-sequenceusing 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// sum of bi-tonic sub-sequencefunction 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.?> |
Javascript
<script> // JavaScript program to find maximum // sum of bi-tonic sub-sequence // Function return maximum sum // of Bi-tonic sub-sequence function MaxSumBS(arr, n) { let max_sum = Number.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 let MSIBS = new Array(n); let MSDBS = new Array(n); for (let i = 0; i < n; i++) { MSDBS[i] = arr[i]; MSIBS[i] = arr[i]; } // Compute MSIBS values from left to right */ for (let i = 1; i < n; i++) for (let 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 (let i = n - 2; i >= 0; i--) for (let 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 (let 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; } let arr = [ 1, 15, 51, 45, 33, 100, 12, 18, 9 ]; let n = arr.length; document.write("Maximum Sum : " + MaxSumBS(arr, n)); </script> |
Output
Maximum Sum : 194
Time Complexity: O(n2)
Auxiliary Space: O(n)
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