Introduction to algorithms
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Digital Elevation Models and TIN Algorithms
Algorithmic Foundations of Geographic Information Systems, this book originated from the CISM Advanced School on the Algorithmic Foundations of Geographic Information Systems
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Connecting several locations, for example towns by roads, is a typical network planning problem. A first approach would be to connect pairs of nodes (network language term for location) with a straight edge. The result is a connected network not considering the total network length and the average trip length. The construction of new junctions, so called Steiner Points, for reducing the total network length and thereby the global costs is more or less known. On the other hand slopes, obstacles or other unsteady costs can not be taken into account during construction. Therefore an independent and homogeneous representation for costs and network planning is necessary.Our approach distinguishes between topographical and additional costs, for example nature reserves and restricted areas. The proposed representation of the topography in a digital terrain model (DTM) is a constrained delaunay triangulated irregular network (TIN). The initial problem of connecting different locations is at the first sight reduced to the shortest path problem, taking the extra attributes into account. But these algorithms are not suitable to connect more than two locations.In this paper we will show how to use the information of a slightly extended Dijkstra Shortest-path algorithm to classify the nodes of the TIN and identify candidates for Steiner Points. Assuming that each geomorphologically closed area can be represented by one Steiner Point, the presented algorithm leads to an optimal network. Steiner Points, passes and the shortest paths between the latter constitute the basis for a network, which provides cheapest connections with regard to topographical costs.To avoid exhaustive searching, geomorphology of the DTM is used as a powerful heuristic. It leads to a partitioning of the space and as a result to a divide- and conquer strategy.