Trekking in the Alps without freezing or getting tired

Abstract

For a polyhedral terrain F with n vertices, the concept of height level map is defined. This concept has several useful properties for paths that have certain height restrictions. The height level map is used to store F, such that for any two query points, one can decide whether there exists a path on F between the two points whose height decreases monotonically. More generally, one can compute the minimum height difference along any path between the two points. It is also possible to decide, given two query points and a height, whether there is a path that stays below this height. Although the height level map has quadratic worst case complexity, it is stored implicitly using only linear storage. The query time for all the above queries is O(log n), and the structure can be built in O(n log n) time. A path with the desired property can also be reported in additional time that is linear in the description size of the path.

This research was performed when the second author visited the first author at Utrecht University. The research of the first author is supported by the Dutch Organization for Scientific Research (N.W.O.) and by ESPRIT Basic Research Action 7141 (project ALCOM II: Algorithms and Complexity). The research of the second author is supported by an NSERC international fellowship.