Streaming Algorithms for Planar Convex Hulls

TL;DR

This paper introduces simpler, faster streaming algorithms for computing convex hulls of large point sets in 2D, reducing complexity and relying on smaller convex hull computations, with nearly optimal tradeoffs.

Contribution

It presents new streaming and W-stream algorithms for convex hulls that are simpler, faster, and have improved pass complexity compared to previous methods.

Findings

Pass complexity is roughly the square root of the best known bound.

Algorithms rely mainly on computing convex hulls of smaller point sets.

Established a new unconditional lower bound for pass complexity and space tradeoffs.

Abstract

Many classical algorithms are known for computing the convex hull of a set of $n$ point in $\mathbb{R}^2$ using $O(n)$ space. For large point sets, whose size exceeds the size of the working space, these algorithms cannot be directly used. The current best streaming algorithm for computing the convex hull is computationally expensive, because it needs to solve a set of linear programs. In this paper, we propose simpler and faster streaming and W-stream algorithms for computing the convex hull. Our streaming algorithm has small pass complexity, which is roughly a square root of the current best bound, and it is simpler in the sense that our algorithm mainly relies on computing the convex hulls of smaller point sets. Our W-stream algorithms, one of which is deterministic and the other of which is randomized, have nearly-optimal tradeoff between the pass complexity and space usage, as we established by a new unconditional lower bound.