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Pronađite indeks zadanog fibonaccijevog broja u konstantnom vremenu

Dato nam je a Fibonaccijev broj . Prvih nekoliko Fibonaccijevih brojeva je 0 1 1 2 3 5 8 13 21 34 55 89 144 ..... 
Moramo pronaći indeks zadanog Fibonaccijevog broja, tj. kao što je Fibonaccijev broj 8 na indeksu 6. 

Primjeri:  

Input : 13  
Output : 7  
Input : 34  
Output : 9

Metoda 1 (jednostavna)  : Jednostavan pristup je pronaći Fibonaccijeve brojeve do zadanih Fibonaccijevih brojeva i brojati broj izvedenih ponavljanja.



C++ 
// A simple C++ program to find index of given // Fibonacci number. #include   int findIndex(int n) {  // if Fibonacci number is less than 2  // its index will be same as number  if (n <= 1)  return n;  int a = 0 b = 1 c = 1;  int res = 1;  // iterate until generated fibonacci number   // is less than given fibonacci number  while (c < n)  {  c = a + b;    // res keeps track of number of generated   // fibonacci number  res++;  a = b;  b = c;  }  return res; } // Driver program to test above function int main() {  int result = findIndex(21);  printf('%dn' result); } // This code is contributed by Saket Kumar
Java 
// A simple Java program to find index of  // given Fibonacci number. import java.io.*; class GFG {    static int findIndex(int n)  {    // if Fibonacci number is less   // than 2 its index will be  // same as number  if (n <= 1)  return n;    int a = 0 b = 1 c = 1;  int res = 1;    // iterate until generated fibonacci  // number is less than given   // fibonacci number  while (c < n)  {  c = a + b;    // res keeps track of number of  // generated fibonacci number  res++;  a = b;  b = c;  }    return res;  }    // Driver program to test above function  public static void main (String[] args)   {  int result = findIndex(21);  System.out.println( result);  } } // This code is contributed by anuj_67.
Python3 
# A simple Python 3 program to find  # index of given Fibonacci number. def findIndex(n) : # if Fibonacci number is less than 2 # its index will be same as number if (n <= 1) : return n a = 0 b = 1 c = 1 res = 1 # iterate until generated fibonacci number  # is less than given fibonacci number while (c < n) : c = a + b # res keeps track of number of  # generated fibonacci number res = res + 1 a = b b = c return res # Driver program to test above function result = findIndex(21) print(result) # this code is contributed by Nikita Tiwari
C# 
// A simple C# program to  // find index of given  // Fibonacci number. using System; class GFG  {  static int findIndex(int n)  {    // if Fibonacci number   // is less than 2 its   // index will be same   // as number  if (n <= 1)  return n;    int a = 0 b = 1 c = 1;  int res = 1;    // iterate until generated   // fibonacci number is less   // than given fibonacci number  while (c < n)  {  c = a + b;    // res keeps track of   // number of generated  // fibonacci number  res++;  a = b;  b = c;  }    return res;  }    // Driver Code  public static void Main ()   {  int result = findIndex(21);  Console.WriteLine(result);  } } // This code is contributed // by anuj_67.
JavaScript 
<script> // A simple Javascript program to  // find index of given  // Fibonacci number. function findIndex(n) {    // If Fibonacci number   // is less than 2 its   // index will be same   // as number  if (n <= 1)  return n;    let a = 0 b = 1 c = 1;  let res = 1;    // Iterate until generated   // fibonacci number is less   // than given fibonacci number  while (c < n)  {  c = a + b;    // res keeps track of   // number of generated  // fibonacci number  res++;  a = b;  b = c;  }  return res; } // Driver code let result = findIndex(21); document.write(result); // This code is contributed by decode2207  </script>
PHP 
 // A simple PHP program to  // find index of given // Fibonacci number. function findIndex($n) { // if Fibonacci number // is less than 2 // its index will be  // same as number if ($n <= 1) return $n; $a = 0; $b = 1; $c = 1; $res = 1; // iterate until generated  // fibonacci number  // is less than given // fibonacci number while ($c < $n) { $c = $a + $b; // res keeps track of  // number of generated  // fibonacci number $res++; $a = $b; $b = $c; } return $res; } // Driver Code $result = findIndex(21); echo($result); // This code is contributed by Ajit. ?>

Izlaz 
8

Metoda 2 (na temelju formule)
Ali ovdje smo morali generirati sve Fibonaccijeve brojeve do navedenog Fibonaccijevog broja. Ali postoji brzo rješenje za ovaj problem. Da vidimo kako! Imajte na umu da je izračunavanje dnevnika broja operacija O(1) na većini platformi.


Fibonaccijev broj je opisan kao 
F n = 1 / sqrt(5) (pow(an) - pow(bn)) gdje 
a = 1/2 (1 + sqrt(5)) i b = 1/2 (1 - sqrt(5))

Zanemarujući pow(b n) koji je vrlo mali zbog velike vrijednosti n dobivamo 
n = okruglo { 2,078087 * log(Fn) + 1,672276 }  
gdje krug znači zaokružiti na najbliži cijeli broj.

ako inače bash

U nastavku je implementacija gore navedene ideje. 

C++ 
// C++ program to find index of given Fibonacci // number #include   int findIndex(int n) {  float fibo = 2.078087 * log(n) + 1.672276;  // returning rounded off value of index  return round(fibo); } // Driver program to test above function int main() {  int n = 55;  printf('%dn' findIndex(n)); }
Java 
// A simple Java program to find index of given // Fibonacci number public class Fibonacci {  static int findIndex(int n)  {  float fibo = 2.078087F * (float) Math.log(n) + 1.672276F;  // returning rounded off value of index  return Math.round(fibo);  }  public static void main(String[] args)  {  int result = findIndex(55);  System.out.println(result);  } }
Python3 
# Python 3 program to find index of given Fibonacci # number import math def findIndex(n) : fibo = 2.078087 * math.log(n) + 1.672276 # returning rounded off value of index return round(fibo) # Driver program to test above function n = 21 print(findIndex(n)) # This code is contributed by Nikita Tiwari.
C# 
// A simple C# program to find  // index of given Fibonacci number using System; class Fibonacci { static int findIndex(int n) {  float fibo = 2.078087F * (float) Math.Log(n) +  1.672276F;  // returning rounded off value of index  return (int)(Math.Round(fibo)); }  // Driver code  public static void Main()  {  int result = findIndex(55);  Console.Write(result);  } } // This code is contributed by nitin mittal
JavaScript 
<script> // A simple Javascript program to find  // index of given Fibonacci number function findIndex(n) {  var fibo = 2.078087 * parseFloat(Math.log(n)) + 1.672276;    // Returning rounded off value of index  return Math.round(fibo); } // Driver code var result = findIndex(55); document.write(result); // This code is contributed by Ankita saini </script>
PHP 
 // PHP program to find index // of given Fibonacci Number function findIndex($n) { $fibo = 2.078087 * log($n) + 1.672276; // returning rounded off // value of index return round($fibo); } // Driver code $n = 55; echo(findIndex($n)); // This code is contributed by Ajit. ?>

Izlaz 
10

Vremenska složenost : O(1)
Pomoćni prostor : O(1)

Pristup:

možemo riješiti ovaj problem pomoću formule za n-ti Fibonaccijev broj koja je:

F(n) = (pow((1+sqrt(5))/2 n) - pow((1-sqrt(5))/2 n)) / sqrt(5)

Pomoću ove formule možemo izvesti indeks danog Fibonaccijevog broja. Možemo ponavljati vrijednosti n i izračunati odgovarajući Fibonaccijev broj koristeći gornju formulu dok ne pronađemo Fibonaccijev broj koji je veći ili jednak zadanom broju. U ovom trenutku možemo vratiti indeks Fibonaccijevog broja koji odgovara zadanom broju.

U nastavku je implementacija gornjeg pristupa:

C++ 
#include    #include  using namespace std; int findIndex(int n) {  double phi = (1 + sqrt(5)) / 2;  int index = round(log(n * sqrt(5) + 0.5) / log(phi));  int fib = round((pow(phi index) - pow(1 - phi index)) / sqrt(5));  if (fib == n)  return index;  else  return -1; // n is not a Fibonacci number } int main() {  int n = 34;  int index = findIndex(n);  cout << 'The index of ' << n << ' is ' << index << endl;  return 0; }
Java 
//Java code for the above approach import java.util.*; public class FibonacciIndex {  public static int findIndex(int n) {  double phi = (1 + Math.sqrt(5)) / 2;  int index = (int) Math.round(Math.log(n * Math.sqrt(5) + 0.5) / Math.log(phi));  int fib = (int) Math.round((Math.pow(phi index) - Math.pow(1 - phi index)) / Math.sqrt(5));  if (fib == n)  return index;  else  return -1; // n is not a Fibonacci number  }  public static void main(String[] args) {  int n = 34;  int index = findIndex(n);  System.out.println('The index of ' + n + ' is ' + index);  } }
Python3 
import math def find_index(n): phi = (1 + math.sqrt(5)) / 2 index = round(math.log(n * math.sqrt(5) + 0.5) / math.log(phi)) fib = round((math.pow(phi index) - math.pow(1 - phi index)) / math.sqrt(5)) if fib == n: return index else: return -1 # n is not a Fibonacci number def main(): n = 34 index = find_index(n) print(f'The index of {n} is {index}') if __name__ == '__main__': main()
C# 
using System; class Program {  // Function to find the index of a number in the Fibonacci sequence  static int FindIndex(int n)  {  double phi = (1 + Math.Sqrt(5)) / 2; // Golden ratio    // Calculate the index using the formula for Fibonacci numbers  int index = (int)Math.Round(Math.Log(n * Math.Sqrt(5) + 0.5) / Math.Log(phi));    // Calculate the Fibonacci number at the found index  int fib = (int)Math.Round((Math.Pow(phi index) - Math.Pow(1 - phi index)) / Math.Sqrt(5));    // Check if the calculated Fibonacci number is equal to n  if (fib == n)  return index;  else  return -1; // n is not a Fibonacci number  }  static void Main()  {  int n = 34;  int index = FindIndex(n);  Console.WriteLine('The index of ' + n + ' is ' + index);  } }
JavaScript 
// Function to find the index of a number in the Fibonacci sequence function findIndex(n) {  const phi = (1 + Math.sqrt(5)) / 2;  const index = Math.round(Math.log(n * Math.sqrt(5) + 0.5) / Math.log(phi));  const fib = Math.round((Math.pow(phi index) - Math.pow(1 - phi index)) / Math.sqrt(5));  if (fib === n) {  return index;  } else {  return -1; // n is not a Fibonacci number  } } // Main function to test the findIndex function function main() {  const n = 34;  const index = findIndex(n);  console.log('The index of ' + n + ' is ' + index); } main();

Izlaz 
The index of 34 is 9

Vremenska složenost: O(1) jer uključuje samo nekoliko aritmetičkih operacija.
Složenost prostora: O(1) jer koristi samo konstantnu količinu memorije za pohranjivanje varijabli.

Ovaj je članak pridonio Aditya Kumar .

 

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