The Geometry of Tree-Based Sorting

TL;DR

This paper establishes a formal connection between sorting algorithms and binary search tree models, introducing the geometric interpretation and a new complexity measure called the log-interleave bound, with implications for parallel sorting efficiency.

Contribution

It demonstrates that many sorting algorithms are equivalent in cost to offline BST algorithms and introduces the log-interleave bound as a new measure of permutation complexity.

Findings

Log-interleave bound closely approximates BST lower bounds.

Parallel sorting algorithm with polylogarithmic span based on log-interleave.

Existence of offline BST algorithms near the log-interleave bound for any permutation.

Abstract

We study the connections between sorting and the binary search tree (BST) model, with an aim towards showing that the fields are connected more deeply than is currently appreciated. While any BST can be used to sort by inserting the keys one-by-one, this is a very limited relationship and importantly says nothing about parallel sorting. We show what we believe to be the first formal relationship between the BST model and sorting. Namely, we show that a large class of sorting algorithms, which includes mergesort, quicksort, insertion sort, and almost every instance-optimal sorting algorithm, are equivalent in cost to offline BST algorithms. Our main theoretical tool is the geometric interpretation of the BST model introduced by Demaine et al., which finds an equivalence between searches on a BST and point sets in the plane satisfying a certain property. To give an example of the utility of our approach, we introduce the log-interleave bound, a measure of the information-theoretic complexity of a permutation $\pi$, which is within a $\lg \lg n$ multiplicative factor of a known lower bound in the BST model; we also devise a parallel sorting algorithm with polylogarithmic span that sorts a permutation $\pi$ using comparisons proportional to its log-interleave bound. Our aforementioned result on sorting and offline BST algorithms can be used to show existence of an offline BST algorithm whose cost is within a constant factor of the log-interleave bound of any permutation $\pi$.