C++ Program for Zeckendorf\’s Theorem (Non-Neighbouring Fibonacci Representation)

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.

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 

CPP

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

Output:

Non-neighbouring Fibonacci Representation of 30 is 
21 8 1

Time Complexity: O(n)

Auxiliary Space: O(1)

Please refer complete article on Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation) for more details!