• Corpus ID: 8111215

Refining the Analysis of Divide and Conquer: How and When

@article{Barbay2015RefiningTA,
  title={Refining the Analysis of Divide and Conquer: How and When},
  author={J{\'e}r{\'e}my F{\'e}lix Barbay and Carlos Ochoa and Pablo P{\'e}rez-Lantero},
  journal={ArXiv},
  year={2015},
  volume={abs/1505.02820}
}

• Jérémy Félix Barbay, Carlos Ochoa, P. Pérez-Lantero
• Published 11 May 2015
• Computer Science, Mathematics
• ArXiv

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.