Carving Polytopes with Saws in 3D

TL;DR

This paper explores algorithms for carving 3D polytopes with different cutting tools, providing efficient deterministic and randomized methods for half-plane cuts and a polynomial-time algorithm for sweep cuts.

Contribution

It introduces new algorithms with improved time complexities for carving polytopes using half-plane and sweep cuts.

Findings

Deterministic algorithm for half-plane cuts in O(n^2) time

Randomized algorithm for half-plane cuts in O(n^{3/2+ε}) expected time

Algorithm for sweep cuts in O(n^5) time

Abstract

We investigate the problem of carving an $n$-face triangulated three-dimensional polytope using a tool to make cuts modelled by either a half-plane or sweeps from an infinite ray. In the case of half-planes cuts, we present a deterministic algorithm running in $O(n^2)$ time and a randomized algorithm running in $O(n^{3/2+\varepsilon})$ expected time for any $\varepsilon>0$. In the case of cuts defined by sweeps of infinite rays, we present an algorithm running in $O(n^5)$ time.