(\alpha, \beta) Fibonacci Search

TL;DR

This paper reveals the underlying principles of Fibonacci search, presenting a generalized procedure that optimally adapts to variable comparison costs, enhancing understanding of its efficiency.

Contribution

It introduces a generalized Fibonacci search method that aligns with the optimal decision tree for search problems with outcome-dependent costs.

Findings

The generalized Fibonacci procedure matches the implicit optimal decision tree.

Fibonacci search can be extended to handle variable comparison costs.

The approach clarifies the 'magic' behind Fibonacci search's effectiveness.

Abstract

Knuth [12, Page 417] states that "the (program of the) Fibonaccian search technique looks very mysterious at first glance" and that "it seems to work by magic". In this work, we show that there is even more magic in Fibonaccian (or else Fibonacci) search. We present a generalized Fibonacci procedure that follows perfectly the implicit optimal decision tree for search problems where the cost of each comparison depends on its outcome.