From 8ea50b59a3f6ed86ecdb70b8d7941e16ef0b11d2 Mon Sep 17 00:00:00 2001
From: Naveen Hegde <hegdenaveen1@gmail.com>
Date: Mon, 25 Dec 2017 15:11:31 +0530
Subject: [PATCH] Updated logic of fibonacci.cpp

Implemented an easier to implement and faster algorithm than the already present matrix exponentiation method.
This is faster than matrix exponentiation method as it is a DP form of matrix exponentiation.
---
 Others/fibonacci.cpp | 111 +++++++++++--------------------------------
 1 file changed, 29 insertions(+), 82 deletions(-)

diff --git a/Others/fibonacci.cpp b/Others/fibonacci.cpp
index 2303bf4d8..82b1fd322 100644
--- a/Others/fibonacci.cpp
+++ b/Others/fibonacci.cpp
@@ -1,99 +1,46 @@
+//An efficient way to calculate nth fibonacci number faster and simpler than O(nlogn) method of matrix exponentiation
+//This works by using both recursion and dynamic programming.
+//as 93rd fibonacci exceeds 19 digits, which cannot be stored in a single long long variable, we can only use it till 92nd fibonacci
+//we can use it for 10000th fibonacci etc, if we implement bigintegers.
+//This algorithm works with the fact that nth fibonacci can easily found if we have already found n/2th or (n+1)/2th fibonacci
+//It is a property of fibonacci similar to matrix exponentiation.
+
 #include <iostream>
+#include<cstdio>
 using namespace std;
 
+const long long MAX = 93;
+ 
 
-/*
-	Efficient method for finding nth term in a fibonacci number sequence.
-	Uses Dvide and conquer approach
-	Enter Values from 1
-	Eg-
-	1st term = 0;
-	2nd term = 1;
-	3rd term = 1;
-	.
-	.
-	.
-	.
-*/
-long long a[2][2] = {{1,1},{1,0}};//fibonacci matrix
-long long ans[2][2] = {{1,1},{1,0}};//final ans matrix
-/*
-	Working Principal:
+long long f[MAX] = {0};
+ 
 
-	[F(k+1)	  F(k)]   [1  1]^k
-	|             | = |    |
-	[F(k)	F(k-1)]   [1  0]
-
-	where F(k) is the kth term of the Fibonacci Sequence. 
-
-
-*/
-
-void product(long long b[][2],long long k[][2])
+long long fib(long long n)
 {
-	/*
-		Function for computing product of the two matrices b and k
-		and storing them into the variable ans.
 
-		Implementation :
-		Simple matrix multiplication of two (2X2) matrices.
-	*/
-	long long c[2][2];//temporary stores the answer
-	c[0][0] = b[0][0]*k[0][0]+b[0][1]*k[1][0];
-	c[0][1] = b[0][0]*k[0][1]+b[0][1]*k[1][1];
-	c[1][0] = b[1][0]*k[0][0]+b[1][1]*k[1][0];
-	c[1][1] = b[1][0]*k[0][1]+b[1][1]*k[1][1];
-	ans[0][0] = c[0][0];
-	ans[0][1] = c[0][1];
-	ans[1][0] = c[1][0];
-	ans[1][1] = c[1][1];
-	
-} 
+    if (n == 0)
+        return 0;
+    if (n == 1 || n == 2)
+        return (f[n] = 1);
+ 
 
-void power_rec(long long n)
-{	
-
-	/*
-		Function for calculating A^n(exponent) in a recursive fashion
-
-		Implementation:
-		A^n = { A^(n/2)*A^(n/2) 			if n is even
-			  { A^((n-1)/2)*A^((n-1)/2)*A   if n is odd
-	*/
-	if((n == 1)||(n==0))
-		return;
-	else
-	{
-		if((n%2) == 0)
-		{	
-			power_rec(n/2);
-			product(ans,ans);
-			
-		}	
-		else
-		{
-			power_rec((n-1)/2);	
-			product(ans,ans);
-			product(ans,a);
-			
-		}
-	}
+    if (f[n])
+        return f[n];
+ 
+    long long k = (n%2!=0)? (n+1)/2 : n/2;
+ 
+    f[n] = (n%2!=0)? (fib(k)*fib(k) + fib(k-1)*fib(k-1))
+           : (2*fib(k-1) + fib(k))*fib(k);
+    return f[n];
 }
 
+
 int main()
 {
 	//Main Function
-	cout <<"Enter the value of n\n";
-	long long n;
-	cin >>n;
-	if(n == 1)
+	for(long long i=1;i<93;i++)
 	{
-		cout<<"Ans: 0"<<endl;
-	}
-	else
-	{	
-		power_rec(n-1);
-		cout <<"Ans :"<<ans[0][1]<<endl;
+		cout << i << " th fibonacci number is " << fib(i) << "\n";		
 	}
 	return 0;
 }