Average number of flips in pancake sorting

TL;DR

This paper analyzes the average number of flips needed to sort burnt and unburnt pancake stacks, providing algorithms with bounds, new lower bounds, and exact values for small stacks, advancing understanding of pancake sorting complexity.

Contribution

It introduces new algorithms with proven average flip counts, establishes lower bounds, and determines exact flip counts for small stacks, improving theoretical understanding of pancake sorting.

Findings

Average flips for burnt pancakes: 7n/4+O(1).

Average flips for unburnt pancakes: at most 17n/12+O(1).

Exact flip counts for stacks up to size 19.

Abstract

We are given a stack of pancakes of different sizes and the only allowed operation is to take several pancakes from top and flip them. The unburnt version requires the pancakes to be sorted by their sizes at the end, while in the burnt version they additionally need to be oriented burnt-side down. We present an algorithm with the average number of flips, needed to sort a stack of n burnt pancakes, equal to 7n/4+O(1) and a randomized algorithm for the unburnt version with at most 17n/12+O(1) flips on average. In addition, we show that in the burnt version, the average number of flips of any algorithm is at least n+\Omega(n/log n) and conjecture that some algorithm can reach n+\Theta(n/log n). We also slightly increase the lower bound on g(n), the minimum number of flips needed to sort the worst stack of n burnt pancakes. This bound, together with the upper bound found by Heydari and Sudborough in 1997, gives the exact number of flips to sort the previously conjectured worst stack -I_n for n=3 mod 4 and n>=15. Finally we present exact values of f(n) up to n=19 and of g(n) up to n=17 and disprove a conjecture of Cohen and Blum by showing that the burnt stack -I_{15} is not the worst one for n=15.