Almost-sure asymptotic for the number of heaps inside a random sequence

TL;DR

This paper proves that the minimum number of heaps needed to sort a random sequence converges almost surely and in L1, refining previous logarithmic growth results.

Contribution

It establishes almost-sure and L1 convergence for the number of heaps, improving upon prior expected value results.

Findings

Number of heaps grows logarithmically on average

Almost-sure convergence of heap count

L1 convergence of heap count

Abstract

We study the minimum number of heaps required to sort a random sequence using a generalization of Istrate and Bonchis's algorithm (2015). In a previous paper, the authors proved that the expected number of heaps grows logarithmically. In this note, we improve on the previous result by establishing the almost-sure and L 1 convergence.