Point location

Abstract: A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.

Abstract: I BASICS 1 Quick-sort and Search Quick-sort Another view of quick-sort Randomized binary trees Skip lists 2 What Is Computational Geometry? Range queries Arrangements Trapezoidal decompositions Convex polytopes Voronoi diagrams Hidden surface removal Numerical precision and degeneracies Early deterministic algorithms Deterministic vs randomized algorithms The model of randomness 3 Incremental Algorithms Trapezoidal decompositions Convex polytopes Voronoi diagrams Configuration spaces Tail estimates 4 Dynamic Algorithms trapezoidal decompositions Voronoi diagrams History and configuration spaces Rebuilding history Deletions in history Dynamic shuffling 5 Random Sampling Configuration spaces with bounded valence Top-down sampling Bottom-up sampling Dynamic sampling Average conflict size More dynamic algorithms Range spaces and E-nets Comparisons II APPLICATIONS 6 Arrangements of Hyperplanes Incremental construction Zone Theorem Canonical triangulations Point location and ray shooting Point location and range queries 7 Convex Polytopes Linear Programming The number of faces Incremental construction The expected structural and conflict change Dynamic maintenance Voronoi diagrams Search problems Levels and Voronoi diagrams of order k 8 Range Search Orthogonal intersection search Nonintersecting segments in the plane Dynamic point location Simplex range search Half-space range queries Decomposable search problems Parametric search 9 Computer Graphics Hidden surface removal Binary Space Partitions Moving viewpoint 10 How Crucial Is Randomness? Pseudo-random sources Derandomization Appendix: Tail Estimates Chernoff's technique Chebychev's technique Bibliography Index

Abstract: Point location, often known in graphics as “hit detection,” is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let $\mathcal{S}$ denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of m edges. In this paper we exhibit a substantial refinement of the technique of Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594–606] for locating a point in $\mathcal{S}$ based on separating chains. The new data structure, called a layered dag, can be built in $O(m)$ time, uses $O(m)$ storage, and makes possible point location in $O(\log m)$ time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.

Abstract: A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rather complicated. We propose a simple O(log n)-query-time, O(n)-space solution, using persistent search trees. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed. We develop a persistent form of binary search tree that supports insertions and deletions in the present and queries in the past. The time per query or update is O(log m), where m is the total number of updates, and the space needed is O(1) per update. Our planar point location algorithm is an immediate application of this data structure. The structure also provides an alternative to Chazelle's "hive graph" structure, which has a variety of applications in geometric retrieval.

Abstract: This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(nlog^*n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(nlog^*n+klogn). As a by-product our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to ''coinflips'' generated by the algorithm and are not based on assumptions about the geometric distribution of the input.