Removing Local Extrema from Imprecise Terrains

TL;DR

This paper investigates the computational complexity of removing local extrema from imprecise terrains, showing efficient solutions for removing only minima or maxima, but NP-hardness for removing both simultaneously.

Contribution

It proves that removing both minima and maxima simultaneously from imprecise terrains is NP-hard, and that even simplified versions are computationally intractable.

Findings

Removing only minima or maxima can be done in O(n log n) time.

Removing both minima and maxima simultaneously is NP-hard.

Even with only two height levels, the problem remains NP-hard.

Abstract

In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n vertices. Interestingly, however, removing both the minima and maxima simultaneously is NP-hard, and is even hard to approximate within a factor of O(log log n) unless P=NP. Moreover, we show that even a simplified version of the problem where vertices can have only two different heights is already NP-hard, a result we obtain by proving hardness of a special case of 2-Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graph-algorithms community.