Refining the Analysis of Divide and Conquer: How and When

TL;DR

This paper explores refined complexity analyses of divide-and-conquer algorithms, introducing entropy-based measures to better understand their performance on instances with varying difficulty levels.

Contribution

It extends entropy-based analysis to a broader class of divide-and-conquer algorithms, providing a more nuanced understanding of their complexity.

Findings

Refined analysis applies to sorting and convex hull algorithms.

Complexity depends on instance difficulty measured by entropy.

Potential for adaptive algorithms based on instance structure.

Abstract

Divide-and-conquer is a central paradigm for the design of algorithms, through which some fundamental computational problems, such as sorting arrays and computing convex hulls, are solved in optimal time within $\Theta(n\log{n})$ in the worst case over instances of size $n$. A finer analysis of those problems yields complexities within $O(n(1 + \mathcal{H}(n_1, \dots, n_k))) \subseteq O(n(1{+}\log{k})) \subseteq O(n\log{n})$ in the worst case over all instances of size $n$ composed of $k$ "easy" fragments of respective sizes $n_1, \dots, n_k$ summing to $n$, where the entropy function $\mathcal{H}(n_1, \dots, n_k) = \sum_{i=1}^k{\frac{n_i}{n}}\log{\frac{n}{n_i}}$ measures the "difficulty" of the instance. We consider whether such refined analysis can be applied to other algorithms based on divide-and-conquer, such as polynomial multiplication, input-order adaptive computation of convex hulls in 2D and 3D, and computation of Delaunay triangulations.