A tight bound on the worst-case number of comparisons for Floyd's heap construction algorithm

TL;DR

This paper establishes a precise upper bound on the maximum comparisons needed by Floyd's heap construction algorithm, linking it to the binary representation of the input size.

Contribution

It provides a tight bound on the worst-case comparisons for Floyd's heap construction, expressed in terms of binary representation properties.

Findings

Maximum comparisons are at most 2n - 2μ(n) - σ(n).

The bound is tight and depends on the binary form of n.

The analysis improves understanding of the algorithm's efficiency.

Abstract

In this paper a tight bound on the worst-case number of comparisons for Floyd's well known heap construction algorithm, is derived. It is shown that at most 2n-2{\mu}(n)-{\sigma}(n) comparisons are executed in the worst case, where {\mu}(n) is the number of ones and {\sigma}(n) is the number of zeros after the last one in the binary representation of the number of keys n.