# 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} }

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…

## References

SHOWING 1-10 OF 19 REFERENCES

### Instance-Optimal Geometric Algorithms

- Computer Science, Mathematics2009 50th Annual IEEE Symposium on Foundations of Computer Science
- 2009

For 2-d convex hulls, it is proved that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound, and a new algorithm is proposed.

### Adaptive Algorithms for Constructing Convex Hulls and Triangulations of Polygonal Chains

- Computer Science, MathematicsSWAT
- 2002

We study some fundamental computational geometry problems with the goal to exploit structure in input data that is given as a sequence C= (p1, p2, ..., pn) of points that are "almost sorted" in the…

### The Ultimate Planar Convex Hull Algorithm?

- Computer ScienceSIAM J. Comput.
- 1986

We present a new planar convex hull algorithm with worst case time complexity $O(n \log H)$ where $n$ is the size of the input set and $H$ is the size of the output set, i.e. the number of vertices…

### On-Line Construction of the Convex Hull of a Simple Polyline

- Computer ScienceInf. Process. Lett.
- 1987

### Gaussian elimination is not optimal

- Mathematics
- 1969

t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical…

### Closest-point problems

- Computer Science16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
- 1975

The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.

### Sorting and Searching in Multisets

- MathematicsSIAM J. Comput.
- 1976

A lower bound on finding the mode of a multiset as a function of the actual multiplicity is given, and it is demonstrated that the bound can be achieved to within a multiplicative constant.

### On compressing permutations and adaptive sorting

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2013

### On computing Voronoi diagrams for sorted point sets

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 1995

We show that the Voronoi diagram of a finite sequence of points in the plane which gives sorted order of the points with respect to two perpendicular directions can be computed in linear time. In…

### Computing Dirichlet Tessellations in the Plane

- Computer ScienceComput. J.
- 1978

A recursive algorithm for computing the Dirichlet tessellation in a highly efficient way is described, and the problems which arise in its implementation are discussed.