A Constant Optimization of the Binary Indexed Tree Query Operation

TL;DR

This paper introduces a modified Binary Indexed Tree that leverages Zeckendorf's Theorem to reduce the constant factor in query operations, achieving similar logarithmic complexity with fewer computations.

Contribution

The paper presents a new data structure similar to BIT that uses Fibonacci properties to optimize query operation constants.

Findings

Achieves $O( ext{log} n)$ query complexity with fewer computations.

Uses Zeckendorf's Theorem to optimize the data structure.

Reduces the constant factor in BIT query operations.

Abstract

There are several data structures which can calculate the prefix sums of an array efficiently, while handling point updates on the array, such as Segment Trees and Binary Indexed Trees (BIT). Both these data structures can handle the these two operations (query and update) in $O(\log{n})$ time. In this paper, we present a data structure similar to the BIT, but with an even smaller constant. To do this, we use Zeckendorf's Theorem, a property of the Fibonacci sequence of numbers. The new data structure achieves the same complexity of $O(\log{n})$, but requires about $\log_{\phi^{2}} n$ computations for the Query Operation as opposed to the $\log_{2} n$ computations required for a BIT Query Operation in the worst case.