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Count Fibonacci numbers in given range in O(Log n) time and O(1) space

  • Difficulty Level : Medium
  • Last Updated : 28 Jun, 2021

Given a range, count Fibonacci numbers in given range. First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 141, .. 
Example : 

Input: low = 10, high = 100
Output: 5
There are five Fibonacci numbers in given
range, the numbers are 13, 21, 34, 55 and 89.

Input: low = 10, high = 20
Output: 1
There is only one Fibonacci Number, 13.

Input: low = 0, high = 1
Output: 3
Fibonacci numbers are 0, 1 and 1

We strongly recommend you to minimize your browser and try this yourself first. 
A Brute Force Solution is to one by one find all Fibonacci Numbers and count all Fibonacci numbers in given range
An Efficient Solution is to use previous Fibonacci Number to generate next using simple Fibonacci formula that fn = fn-1 + fn-2.
 

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C++




// C++ program to count Fibonacci numbers in given range
#include <bits/stdc++.h>
using namespace std;
 
// Returns count of fibonacci numbers in [low, high]
int countFibs(int low, int high)
{
    // Initialize first three Fibonacci Numbers
    int f1 = 0, f2 = 1, f3 = 1;
 
    // Count fibonacci numbers in given range
    int result = 0;
 
    while (f1 <= high)
    {
        if (f1 >= low)
        result++;
        f1 = f2;
        f2 = f3;
        f3 = f1 + f2;
    }
 
    return result;
}
 
// Driver program
int main()
{
int low = 10, high = 100;
cout << "Count of Fibonacci Numbers is "
        << countFibs(low, high);
return 0;
}

Java




// Java program to count Fibonacci numbers in given range
import java.io.*;
 
class GFG {
 
    // Returns count of fibonacci numbers in [low, high]
    public static int countFibs(int low, int high)
    {
        // Initialize first three Fibonacci Numbers
        int f1 = 0, f2 = 1, f3 = 1;
 
        // Count fibonacci numbers in given range
        int result = 0;
 
        while (f1 <= high) {
            if (f1 >= low)
                result++;
            f1 = f2;
            f2 = f3;
            f3 = f1 + f2;
        }
 
        return result;
    }
 
    // Driver program
    public static void main(String[] args)
    {
        int low = 10, high = 100;
        System.out.println("Count of Fibonacci Numbers is "
                           + countFibs(low, high));
    }
}
 
// This code is contributed by RohitOberoi.

Python3




# Python3 program to count Fibonacci
# numbers in given range
 
# Returns count of fibonacci
# numbers in [low, high]
def countFibs(low, high):
     
    # Initialize first three
    # Fibonacci Numbers
    f1, f2, f3 = 0, 1, 1
 
    # Count fibonacci numbers in
    # given range
    result = 0
 
    while (f1 <= high):
        if (f1 >= low):
            result += 1
        f1 = f2
        f2 = f3
        f3 = f1 + f2
 
    return result
 
# Driver Code
low, high = 10, 100
print("Count of Fibonacci Numbers is",
                 countFibs(low, high))
 
# This code is contributed
# by mohit kumar

C#




// C# program to count Fibonacci
// numbers in given range
using System;
 
public class GFG
{
     
    // Returns count of fibonacci
    // numbers in [low, high]
    static int countFibs(int low,
                        int high)
    {
         
        // Initialize first three
        // Fibonacci Numbers
        int f1 = 0, f2 = 1, f3 = 1;
     
        // Count fibonacci numbers
        // in given range
        int result = 0;
     
        while (f1 <= high)
        {
            if (f1 >= low)
            result++;
            f1 = f2;
            f2 = f3;
            f3 = f1 + f2;
        }
     
        return result;
    }
     
    // Driver Code
    public static void Main(String []args)
    {
        int low = 10, high = 100;
        Console.WriteLine("Count of Fibonacci Numbers is "
                        + countFibs(low, high));
    }
}
     
// This code is contributed by Sam007.

PHP




<?php
// PHP program to count
// Fibonacci numbers in
// given range
 
// Returns count of fibonacci
// numbers in [low, high]
function countFibs($low, $high)
{
    // Initialize first
    // three Fibonacci Numbers
    $f1 = 0; $f2 = 1; $f3 = 1;
 
    // Count fibonacci
    // numbers in given range
    $result = 0;
 
    while ($f1 <= $high)
    {
        if ($f1 >= $low)
        $result++;
        $f1 = $f2;
        $f2 = $f3;
        $f3 = $f1 + $f2;
    }
 
    return $result;
}
 
// Driver Code
$low = 10; $high = 100;
echo "Count of Fibonacci Numbers is ",
               countFibs($low, $high);
 
// This code is contributed by nitin mittal.
?>

Javascript




<script>
 
// JavaScript program to count Fibonacci
// numbers in given range
 
    // Returns count of fibonacci
    // numbers in [low, high]
    function countFibs(low, high)
    {
           
        // Initialize first three
        // Fibonacci Numbers
        let f1 = 0, f2 = 1, f3 = 1;
       
        // Count fibonacci numbers
        // in given range
        let result = 0;
       
        while (f1 <= high)
        {
            if (f1 >= low)
            result++;
            f1 = f2;
            f2 = f3;
            f3 = f1 + f2;
        }
       
        return result;
    }
 
// Driver program
 
        let low = 10, high = 100;
        document.write("Count of Fibonacci Numbers is "
                           + countFibs(low, high));
 
// This code is contributed by susmitakundugoaldanga.
</script>

Output : 

Count of Fibonacci Numbers is 5

Time Complexity Analysis: 
Consider the that Fibonacci Numbers can be written as below 
fib(n)=\left [ \frac {1}{\sqrt{5}}\left ( \frac {1+\sqrt{5}}{2} \right )^n \right ]\sim c*1.62^n
for n\sim c*1.62^n we make O(n')comparisons,we,thus,needO(log(n))comparisons.
So the value of Fibonacci numbers grow exponentially. It means that the while loop grows exponentially till it reaches ‘high’. So the loop runs O(Log (high)) times. 
One solution could be directly use above formula to find count of Fibonacci Numbers, but that is not practically feasible (See this for details).
Auxiliary Space: O(1)
This article is contributed by Sudhanshu Gupta. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
 




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