Recursive tilings and space-filling curves with little fragmentation

This paper defines the Arrwwid number of a recursive tiling (or space-filling curve) as the smallest number a such that any ball Q can be covered by a tiles (or curve fragments) with total volume O(volume(Q)). Recursive tilings and space-filling curves with low Arrwwid numbers can be applied to optimize disk, memory or server access patterns when processing sets of points in Rd. This paper presents recursive tilings and space-filling curves with optimal Arrwwid numbers. For d = 3, we see that regular cube tilings and space-filling curves cannot have optimal Arrwwid number, and we see how to construct alternatives with better Arrwwid numbers

Results on geometric networks and data structures

This thesis discusses four problems in computational geometry. In traditional colored range-searching problems, one wants to store a set of n objects with m distinct colors for the following queries: report all colors such that there is at least one object of that color intersecting the query range. Such an object, however, could be an `outlier' in its color class. We consider a variant of this problem where one has to report only those colors such that at least a fraction t of the objects of that color intersects the query range, for some parameter t. Our main results are on an approximate version of this problem, where we are also allowed to report those colors for which a fraction (1-epsilon)t intersects the query range, for some fixed epsilon > 0. We present efficient data structures for such queries with orthogonal query ranges in sets of colored points, and for point stabbing queries in sets of colored rectangles. A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. R-trees are box-trees with nodes of high degree. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. The geometric minimum-diameter spanning tree (MDST) of a set of n points is a tree that spans the set and minimizes the Euclidian length of the longest path in the tree. So far, the MDST can only be found in slightly subcubic time. We give two fast approximation schemes for the MDST, i.e. factor-(1+epsilon) approximation algorithms. One algorithm uses a grid and takes time O*(1/epsilon^(5 2/3) + n), where the O*-notation hides terms of type O(log^O(1) 1/epsilon). The other uses the well-separated pair decomposition and takes O(1/epsilon^3 n + (1/epsilon) n log n) time. A combination of the two approaches runs in O*(1/epsilon^5 + n) time. The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the L1-distance. We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain the following results: for arbitrary graphs, the problem is NP-hard; for trees, we can solve the problem by linear programming on O(n^2) variables and constraints; for paths, we can solve the problem in time O(n^3 log n); for rectangles sorted vertically along a path, the problem can be solved in O(n^2) time

Implicit flow routing on terrains with applications to surface networks and drainage structures

Flow-related structures on terrains are defined in terms of paths of steepest descent (or ascent). A steepest descent path on a polyhedral terrain T with n vertices can have T(n^2) complexity. The watershed of a point p --- the set of points on T whose paths of steepest descent reach p --- can have complexity T(n^3). We present a technique for tracing a collection of n paths of steepest descent on T implicitly in O(n logn) time. We then derive O(n log n) time algorithms for: (i) computing for each local minimum p of T the triangles contained in the watershed of p and (ii) computing the surface network graph of T. We also present an O(n^2) time algorithm that computes the watershed area for each local minimum of T

I/O-efficient map overlay and point location in low-density subdivisions

We present improved and simplified i/o-efficient algorithms for two problems on planar low-density subdivisions, namely map overlay and point location. More precisely, we show how to preprocess a lowdensity subdivision with n edges in O(sort(n)) i/o’s into a compressed linear quadtree such that one can: (i) compute the overlay of two such preprocessed subdivisions in O(scan(n)) i/o’s, where n is the total number of edges in the two subdivisions, (ii) answer a single point location query in O(logB n) i/o’s and k batched point location queries in O(scan(n) + sort(k)) i/o’s. For the special case where the subdivision is a fat triangulation, we show how to obtain the same bounds with an ordinary (uncompressed) quadtree, and we show how to make the structure fully dynamic using O(logB n) i/o’s per update. Our algorithms and data structures improve on the previous best known bounds for general subdivisions both in the number of i/o’s and storage usage, they are significantly simpler, and several of our algorithms are cache-oblivious

I/O-efficient map overlay and point location in low-density subdivisions

We present improved and simplified i/o-efficient algorithms for two problems on planar low-density subdivisions, namely map overlay and point location. More precisely, we show how to preprocess a lowdensity subdivision with n edges in O(sort(n)) i/o’s into a compressed linear quadtree such that one can: (i) compute the overlay of two such preprocessed subdivisions in O(scan(n)) i/o’s, where n is the total number of edges in the two subdivisions, (ii) answer a single point location query in O(logB n) i/o’s and k batched point location queries in O(scan(n) + sort(k)) i/o’s. For the special case where the subdivision is a fat triangulation, we show how to obtain the same bounds with an ordinary (uncompressed) quadtree, and we show how to make the structure fully dynamic using O(logB n) i/o’s per update. Our algorithms and data structures improve on the previous best known bounds for general subdivisions both in the number of i/o’s and storage usage, they are significantly simpler, and several of our algorithms are cache-oblivious

On IO-efficient viewshed algorithms and their accuracy

Given a terrain T and a point v, the viewshed or visibility map of v is the set of points in T that are visible from v. To decide whether a point p is visible one needs to interpolate the elevation of the terrain along the line-of-sight (LOS) vp. Existing viewshed algorithms differ widely in which and how many points they chose to interpolate, how many lines-of-sight they consider, and how they interpolate the terrain. These choices crucially affect the running time and accuracy of the algorithms. In this paper our goal was to obtain an IO-efficient algorithm that computes the viewshed on a grid terrain with as much accuracy as possible given the resolution of the data. We describe two algorithms which are based on computing and merging horizons, and we prove that the complexity of horizons on a grid of n points is O(n), improving on the general O(na(n)) bound on triangulated terrains. Our finding is that, in practice, horizons on grids are significantly smaller than their theoretical worst case bound, which makes horizon-based approaches very fast. To measure the differences between viewsheds computed with various algorithms we implement an error metric that averages differences over a large number of viewsheds computed from a set of viewpoints with topological significance, like valleys and ridges. Using this metric we compare our current approach, Van Kreveld's model used in our previous work [7], the algorithm of Ferreira et al. [6], and the viewshed module r.los in the open source GIS GRASS

Topologically safe curved schematization

Traditionally schematized maps make extensive use of curves. However, automated methods for schematization are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematization that allows a choice of quality measures and curve types. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs

Kennisbasisthema: Ketens en Agrologistiek

Wageningen UR voert een meerjarig onderzoeksprogramma uit, getiteld Ketens en Agrologistiek. Deze projectbundel geeft informatie over projecten in dit thema en van elk project is een flyer opgenome