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!