Longest Bitonic Subsequence

Competitive Coding/Dynamic Programming/Longest Bitonic Subsequence/Longest Bitonic Subsequence Readme.md

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+Problem :-
+=======
+To find the longest Bitonic Subsequence(a bitonic subsequence is a sequence which strictly increases first then strictly decrases) in a given array of numbers.
+
+Input :-
+-----
+The size of an array(n) and the contents of the array(arr).
+
+Output :-
+------
+A single integer printing the maximum length of the bitonic subsequence present in the given array.
+
+#### Language : `C++`
+
+#### Algorithm Paradigm : `Dynamic Programming`
+
+#### Time Complexity : `O(N^2)`
+
+#### Space Complexity : `O(N)`
+
+Working :-
+-------
+This problem is an application of the longest increasing subsequence problem also solved using dynamic programming.
+
+For any `arr[i]` we are finding the longest increasing subsequence from left to right keeping the `arr[i]` the last element(lets call it `a`).Then we are finding the longest increasing subsequence from right to left keeping `arr[i]` the last element(lets call it `b`).Then we are finding `a+b-1` (subtracting `1` because `arr[i]` has been counted twice) which gives us the length of the bitonic subsequence with `arr[i]` as its peak element.
+We are repeating above process for all the elements in the array and finding the max value of all `a+b-1` which gives us the length of the longest bitonic subsequence.
+
+By using dynamic programming we are bringing the time complexity down from exponential(as in case of brute force solution) to polynomial.

Competitive Coding/Dynamic Programming/Longest Bitonic Subsequence/Longest Bitonic Subsequence.cpp

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+//C++ program to find the longest bitonic sequence.
+//A bitonic sequence is a sequence which increases first then decreases.
+#include<bits/stdc++.h>
+
+using namespace std;
+
+int main()
+{
+	int n;//n would store the size of the array.
+	cout<<"Enter the length of the array : ";
+	cin>>n;//Inputting the size of the array.
+	cout<<endl;
+
+	int arr[n];//arr array would store the input data given by the user.
+	for(int i=0;i<n;i++)
+	{
+		cout<<"Enter the value of element "<<i+1<<" : ";
+		cin>>arr[i];//Inputting the values in the array.
+	}
+
+	/*temp1[i] would store the longest increasing subsequence from left to right if arr[i] was the last element of the 
+	  subsequene.*/
+	int temp1[n];
+	for(int i=0;i<n;i++)
+		temp1[i]=1;//Initializing all values of temp1 with 1.
+
+	for(int i=1;i<n;i++)//Implementing the algorithm to find temp1[i].
+	{
+		for(int j=0;j<i;j++)
+		{
+			if(arr[i]>arr[j])
+			temp1[i]=max(temp1[i],temp1[j]+1);
+		}
+	}
+
+	/*temp2[i] would store the longest increasing subsequence from right to left if arr[i] was the last element of the 
+	  subsequence*/
+	int temp2[n];
+	for(int i=0;i<n;i++)
+		temp2[i]=1;//Initializing all values of temp2 with 1.
+
+	for(int i=n-2;i>=0;i--)//Implementing the algorithm to find temp2[i].
+	{
+		for(int j=n-1;j>i;j--)
+		{
+			if(arr[i]>arr[j])
+				temp2[i]=max(temp2[i],temp2[j]+1);
+		}
+	}
+
+	int max=-1;//max would store the maximum length of the longest bitonic subsequence.
+	for(int i=0;i<n;i++)//Implementing algorithm to find max.
+	{
+		if(max<temp1[i]+temp2[i]-1)
+			max=temp1[i]+temp2[i]-1;
+	}	
+
+	cout<<endl;
+
+	cout<<"The longest Bitonic Subsequence is of length : "<<max<<endl;//Printing the longest length of such sequence.
+
+	return 0;
+}