Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation)

Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of distinct non-neighbouring Fibonacci numbers. Two Fibonacci numbers are neighbours if they one come after other in Fibonacci Sequence (0, 1, 1, 2, 3, 5, ..). For example, 3 and 5 are neighbours, but 2 and 5 are not.

Given a number, find a representation of number as sum of non-consecutive Fibonacci numbers.

Examples:

Input:  n = 10
Output: 8 2
8 and 2 are two non-consecutive Fibonacci Numbers
and sum of them is 10.

Input:  n = 30
Output: 21 8 1
21, 8 and 1 are non-consecutive Fibonacci Numbers
and sum of them is 30.

We strongly recommend you to minimize your browser and try this yourself first.
The idea is to use Greedy Algorithm. 

1) Let n be input number

2) While n >= 0
     a) Find the greatest Fibonacci Number smaller than n.
        Let this number be 'f'.  Print 'f'
     b) n = n - f 

C++

// C++ program for Zeckendorf's theorem. It finds
// representation of n as sum of
// non-neighbouring Fibonacci Numbers.
#include <bits/stdc++.h>
using namespace std;

// Returns the greatest Fibonacci Number smaller than
// or equal to n.
int nearestSmallerEqFib(int n)
{
    // Corner cases
    if (n == 0 || n == 1)
        return n;

    // Find the greatest Fibonacci Number smaller
    // than n.
    int f1 = 0, f2 = 1, f3 = 1;
    while (f3 <= n)
    {
        f1 = f2;
        f2 = f3;
        f3 = f1 + f2;
    }
    return f2;
}

// Prints Fibonacci Representation of n using
// greedy algorithm
void printFibRepresntation(int n)
{
    while (n > 0)
    {
        // Find the greates Fibonacci Number smaller
        // than or equal to n
        int f = nearestSmallerEqFib(n);

        // Print the found fibonacci number
        cout << f << " ";

        // Reduce n
        n = n - f;
    }
}

// Driver code
int main()
{
    int n = 30;
    cout << "Non-neighbouring Fibonacci Representation of "
         << n << " is \n";
    printFibRepresntation(n);
    return 0;
}

Java

// Java program for Zeckendorf's theorem. It finds
// representation of n as sum of non-neighbouring
// Fibonacci Numbers.
class GFG {
    public static int nearestSmallerEqFib(int n)
    {
        // Corner cases
        if (n == 0 || n == 1)
            return n;

        // Find the greatest Fibonacci Number smaller
        // than n.
        int f1 = 0, f2 = 1, f3 = 1;
        while (f3 <= n) {
            f1 = f2;
            f2 = f3;
            f3 = f1 + f2;
        }
        return f2;
    }

    // Prints Fibonacci Representation of n using
    // greedy algorithm
    public static void printFibRepresntation(int n)
    {
        while (n > 0) {
            // Find the greates Fibonacci Number smaller
            // than or equal to n
            int f = nearestSmallerEqFib(n);

            // Print the found fibonacci number
            System.out.print(f + " ");

            // Reduce n
            n = n - f;
        }
    }

    // Driver method to test
    public static void main(String[] args)
    {
        int n = 30;
        System.out.println("Non-neighbouring Fibonacci "
                           + " Representation of " + n + " is");

        printFibRepresntation(n);
    }
}

// Code Contributed by Mohit Gupta_OMG

Python3

# Python program for Zeckendorf's theorem. It finds
# representation of n as sum of non-neighbouring
# Fibonacci Numbers.

# Returns the greatest Fibonacci Number smaller than
# or equal to n.
def nearestSmallerEqFib(n):

    # Corner cases
    if (n == 0 or n == 1):
        return n

    # Finds the greatest Fibonacci Number smaller
    # than n.
    f1, f2, f3 = 0, 1, 1
    while (f3 <= n):
        f1 = f2;
        f2 = f3;
        f3 = f1 + f2;
    return f2;


# Prints Fibonacci Representation of n using
# greedy algorithm
def printFibRepresntation(n):

    while (n>0):

        # Find the greates Fibonacci Number smaller
        # than or equal to n
        f = nearestSmallerEqFib(n);

        # Print the found fibonacci number
        print (f,end=" ")

        # Reduce n
        n = n-f

# Driver code test above functions
n = 30
print ("Non-neighbouring Fibonacci Representation of", n, "is")
printFibRepresntation(n)

C#

// C# program for Zeckendorf's theorem.
// It finds the representation of n as
// sum of non-neighbouring  Fibonacci
// Numbers.
using System;

class GFG {
    public static int nearestSmallerEqFib(int n)
    {
        // Corner cases
        if (n == 0 || n == 1)
            return n;

        // Find the greatest Fibonacci
        // Number smaller than n.
        int f1 = 0, f2 = 1, f3 = 1;
        while (f3 <= n) {
            f1 = f2;
            f2 = f3;
            f3 = f1 + f2;
        }
        return f2;
    }

    // Prints Fibonacci Representation
    // of n using greedy algorithm
    public static void printFibRepresntation(int n)
    {
        while (n > 0) {
            // Find the greates Fibonacci
            // Number smallerthan or equal
            // to n
            int f = nearestSmallerEqFib(n);

            // Print the found fibonacci number
            Console.Write(f + " ");

            // Reduce n
            n = n - f;
        }
    }

    // Driver method
    public static void Main()
    {
        int n = 40;
        Console.WriteLine("Non-neighbouring Fibonacci "
                          + " Representation of " + n + " is");

        printFibRepresntation(n);
    }
}

// Code Contributed by vt_m

PHP

<?php
// PHP program for Zeckendorf's theorem.
// It finds representation of n as sum
// of non-neighbouring Fibonacci Numbers.

// Returns the greatest Fibonacci
// Number smaller than or equal
// to n.
function nearestSmallerEqFib($n)
{

    // Corner cases
    if ($n == 0 || $n==1)
    return $n;

    // Find the greatest Fibonacci
    // Number smaller than n.
    $f1 = 0;
    $f2 = 1;
    $f3 = 1;
    while ($f3 <= $n)
    {
        $f1 = $f2;
        $f2 = $f3;
        $f3 = $f1 + $f2;
    }
    return $f2;
}

// Prints Fibonacci Representation
// of n using greedy algorithm
function printFibRepresntation($n)
{
    while ($n > 0)
    {

        // Find the greates Fibonacci
        // Number smaller than or
        // equal to n
        $f = nearestSmallerEqFib($n);

        // Print the found
        // fibonacci number
        echo $f, " ";

        // Reduce n
        $n = $n - $f;
    }
}

    // Driver Code
    $n = 30;
    echo "Non-neighbouring Fibonacci Representation of ",
                                            $n, " is \n";
    printFibRepresntation($n);

// This code is contributed by ajit
?>

Javascript

<script>
    // Javascript program for Zeckendorf's theorem.
// It finds representation of n as sum
// of non-neighbouring Fibonacci Numbers.

// Returns the greatest Fibonacci
// Number smaller than or equal
// to n.
function nearestSmallerEqFib(n)
{

    // Corner cases
    if (n == 0 || n==1)
    return n;

    // Find the greatest Fibonacci
    // Number smaller than n.
    let f1 = 0;
    let f2 = 1;
    let f3 = 1;
    while (f3 <= n)
    {
        f1 = f2;
        f2 = f3;
        f3 = f1 + f2;
    }
    return f2;
}

// Prints Fibonacci Representation
// of n using greedy algorithm
function printFibRepresntation(n)
{
    while (n > 0)
    {

        // Find the greates Fibonacci
        // Number smaller than or
        // equal to n
        let f = nearestSmallerEqFib(n);

        // Print the found
        // fibonacci number
        document.write(f, " ");

        // Reduce n
        n = n - f;
    }
}

    // Driver Code
    let n = 30;
    document.write("Non-neighbouring Fibonacci Representation of " +
                                            n + " is <br>");
    printFibRepresntation(n);

// This code is contributed by _saurabh_jaiswal
</script>

Output

Non-neighbouring Fibonacci Representation of 30 is 
21 8 1 

Time Complexity:  O(N*LogN)
Auxiliary Space: O(1)

How does above Greedy Algorithm work?
Let the greatest Fibonacci number smaller than or equal to ‘n’ be fib(i) [i’th Fibonacci Number]. 
Then n – fib(i) will have its own representation as sum of non-neighbouring Fibonacci numbers.
All we want to make sure is that there is no neighbouring problem. By induction, n-fib(i) does not have neighbouring problem, then the only way n could have a neighbouring problem is if n-fib(i) uses fib(i-1) in its representation. 
So all we have to further prove is that n-fib(i) does not use fib(i-1) in its representation
Let us prove it using contradiction. If n-fib(i) = fib(i-1) + fib(i-x) +…, then fib(i) cannot be the closest smallest Fibonacci number to n, since fib(i) + fib(i-1) itself is fib(i+1). 
So if n-fib(i) contains fib(i-1) in its representation then fib(i+1) would be closer smaller fib number to n, contradicting our assumption that fib(i) is the closest smaller fib number to n.

Can this representation be useful?
Like Binary Representation. This can be an alternate representation to represent positive numbers. One important observation about this representation is, number of 1’s in the Fibonacci representation tends to be much less than the number of 1’s in the binary representation. Hence if in any application where it is more costly to store a 1 than to store a 0, it would make sense to use the fibonacci representation.
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