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!
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