Fibonacci Search

Fibonacci Search is a comparison-based technique that uses Fibonacci numbers to search an element in a sorted array.

Similarities with Binary Search:

1. Works for sorted arrays
2. A Divide and Conquer Algorithm.
3. Has Log n time complexity.

Differences with Binary Search:

1. Fibonacci Search divides given array in unequal parts
2. Binary Search uses division operator to divide range. Fibonacci Search doesn’t use /, but uses + and -. The division operator may be costly on some CPUs.
3. Fibonacci Search examines relatively closer elements in subsequent steps. So when input array is big that cannot fit in CPU cache or even in RAM, Fibonacci Search can be useful.

Background:
Fibonacci Numbers are recursively defined as F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1. First few Fibonacci Numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Observations:
Below observation is used for range elimination, and hence for the O(log(n)) complexity.
F(n – 2) ≈ (1/3)*F(n) and F(n – 1) ≈ (2/3)*F(n).

Approach:
Let the searched element be x.
The idea is to first find the smallest Fibonacci number that is greater than or equal to the length of given array. Let the found Fibonacci number be fib (m’th Fibonacci number). We use (m-2)’th Fibonacci number as the index (If it is a valid index). Let (m-2)’th Fibonacci Number be i, we compare arr[i] with x, if x is same, we return i. Else if x is greater, we recur for subarray after i, else we recur for subarray before i.
Below is the complete algorithm.
Let arr[0..n-1] be the input array and element to be searched be x.

1. Find the smallest Fibonacci Number greater than or equal to n. Let this number be fibM [m’th Fibonacci Number]. Let the two Fibonacci numbers preceding it be fibMm1 [(m-1)’th Fibonacci Number] and fibMm2 [(m-2)’th Fibonacci Number].
2. While the array has elements to be inspected:
   1. Compare x with the last element of the range covered by fibMm2.
   2. If x matches, return index.
   3. Else If x is less than the element, move the three Fibonacci variables two Fibonacci down, indicating elimination of approximately rear two-third of the remaining array.
   4. Else x is greater than the element, move the three Fibonacci variables one Fibonacci down. Reset offset to index. Together these indicate elimination of approximately front one-third of the remaining array.
3. Since there might be a single element remaining for comparison, check if fibMm1 is 1. If Yes, compare x with that remaining element. If match, return index.

Worst case time complexity O(log n) Best case time complexity O(1)
Average case time complexity O(log n) Average space complexity O(1)

CODE

from bisect import bisect_left 
  
# Returns index of x if present,  else  
# returns -1  
def fibMonaccianSearch(arr, x, n): 
      
    # Initialize fibonacci numbers  
    fibMMm2 = 0 # (m-2)'th Fibonacci No. 
    fibMMm1 = 1 # (m-1)'th Fibonacci No. 
    fibM = fibMMm2 + fibMMm1 # m'th Fibonacci 
  
    # fibM is going to store the smallest  
    # Fibonacci Number greater than or equal to n  
    while (fibM < n): 
        fibMMm2 = fibMMm1 
        fibMMm1 = fibM 
        fibM = fibMMm2 + fibMMm1 
  
    # Marks the eliminated range from front 
    offset = -1; 
  
    # while there are elements to be inspected. 
    # Note that we compare arr[fibMm2] with x. 
    # When fibM becomes 1, fibMm2 becomes 0  
    while (fibM > 1): 
          
        # Check if fibMm2 is a valid location 
        i = min(offset+fibMMm2, n-1) 
  
        # If x is greater than the value at  
        # index fibMm2, cut the subarray array  
        # from offset to i  
        if (arr[i] < x): 
            fibM = fibMMm1 
            fibMMm1 = fibMMm2 
            fibMMm2 = fibM - fibMMm1 
            offset = i 
  
        # If x is greater than the value at  
        # index fibMm2, cut the subarray  
        # after i+1 
        elif (arr[i] > x): 
            fibM = fibMMm2 
            fibMMm1 = fibMMm1 - fibMMm2 
            fibMMm2 = fibM - fibMMm1 
  
        # element found. return index  
        else : 
            return i 
  
   
 # comparing the last element with x */ 
    if(fibMMm1 and arr[offset+1] == x): 
        return offset+1; 
  
    # element not found. return -1  
    return -1
  
# Driver Code 
arr = [10, 22, 35, 40, 45, 50, 
       80, 82, 85, 90, 100] 
n = len(arr) 
x = 85
print("Found at index:",fibMonaccianSearch(arr, x, n)) 

Output: Found at index: 8