Flow Computations on Imprecise Terrains

TL;DR

This paper investigates water flow computation on imprecise terrains, analyzing two models with different flow constraints, establishing computational complexity results, and providing efficient algorithms for watershed determination.

Contribution

It introduces a new complexity result for flow on polyhedral terrains and offers efficient algorithms for watershed computation on graph-based models.

Findings

Deciding watershed containment is NP-hard for the surface flow model.

An O(n log n) algorithm computes minimal and maximal watersheds on general graphs.

An O(n) algorithm computes watersheds efficiently on grid models.

Abstract

We study the computation of the flow of water on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time.