Fibonacci search technique

This article is about the programming algorithm. For the technique for finding extremum of a mathematical function, see Golden section search.

In computer science, the Fibonacci search technique is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers.[1] Compared to binary search where the sorted array is divided into two equal-sized parts, one of which is examined further, Fibonacci search divides the array into two parts that have sizes that are consecutive Fibonacci numbers. On average, this leads to about 4% more comparisons to be executed,[2] but it has the advantage that one only needs addition and subtraction to calculate the indices of the accessed array elements, while classical binary search needs bit-shift (see Bitwise operation), division or multiplication,[1] operations that were less common at the time Fibonacci search was first published. Fibonacci search has an average- and worst-case complexity of O(log n) (see Big O notation).

The Fibonacci sequence has the property that a number is the sum of its two predecessors. Therefore the sequence can be computed by repeated addition. The ratio of two consecutive numbers approaches the Golden ratio, 1.618... Binary search works by dividing the seek area in equal parts (1:1). Fibonacci search can divide it into parts approaching 1:1.618 while using the simpler operations.

If the elements being searched have non-uniform access memory storage (i. e., the time needed to access a storage location varies depending on the location accessed), the Fibonacci search may have the advantage over binary search in slightly reducing the average time needed to access a storage location. If the machine executing the search has a direct mapped CPU cache, binary search may lead to more cache misses because the elements that are accessed often tend to gather in only a few cache lines; this is mitigated by splitting the array in parts that do not tend to be powers of two. If the data is stored on a magnetic tape where seek time depends on the current head position, a tradeoff between longer seek time and more comparisons may lead to a search algorithm that is skewed similarly to Fibonacci search.

Fibonacci search is derived from Golden section search, an algorithm by Jack Kiefer (1953) to search for the maximum or minimum of a unimodal function in an interval.[3]

  1. ^ a b Ferguson, David E. (1960). "Fibonaccian searching". Communications of the ACM. 3 (12): 648. doi:10.1145/367487.367496. S2CID 7982182. Note that the running time analysis is this article is flawed, as pointed out by Overholt in 1972 (published 1973).
  2. ^ Overholt, K. J. (1973). "Efficiency of the Fibonacci search method". BIT Numerical Mathematics. 13 (1): 92–96. doi:10.1007/BF01933527. S2CID 120681132.
  3. ^ Kiefer, J. (1953). "Sequential minimax search for a maximum". Proceedings of the American Mathematical Society. 4 (3): 502–506. doi:10.1090/S0002-9939-1953-0055639-3.