# Synergistic Computation of Planar Maxima and Convex Hull

## TL;DR

This paper introduces synergistic algorithms that simultaneously leverage input order and structure to efficiently compute planar Maxima and Convex Hulls, outperforming methods that use only one feature.

## Contribution

It extends synergistic computation techniques to Maxima Set and Convex Hull problems, providing the first adaptive algorithms for merging these structures.

## Key findings

- Improved algorithms for Maxima Set and Convex Hull computation.
- First adaptive algorithms for merging Maxima and Convex Hulls.
- Enhanced performance by exploiting both input order and structure.

## Abstract

Refinements of the worst case complexity over instances of fixed input size consider the input order or the input structure, but rarely both at the same time. Barbay et al. [2016] described ``synergistic'' solutions on multisets, which take advantage of the input order and the input structure, such as to asymptotically outperform any comparable solution which takes advantage only of one of those features. We consider the extension of their results to the computation of the \textsc{Maxima Set} and the \textsc{Convex Hull} of a set of planar points. After revisiting and improving previous approaches taking advantage only of the input order or of the input structure, we describe synergistic solutions taking optimally advantage of various notions of the input order and input structure in the plane. As intermediate results, we describe and analyze the first adaptive algorithms for \textsc{Merging Maxima} and \textsc{Merging Convex Hulls}.