We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas—convex hulls, intersections, searching, proximity, and combinatorial optimizations—are discussed. Seven algorithmic techniques—incremental construction, plane-sweep, locus, divide-and-conquer, geometric transformation, prune-and-search, and dynamization—are each illustrated with an example. A collection of problem transformations to establish lower bounds for geo-metric problems in the algebraic computation/decision model is also included.

Computational Geometry&#8212;A Survey

We provide two algorithms for solving the following problem: Given a rectangle containing n points, compute the largest-area and the largest-perimeter subrectangles with sides parallel to the given rectangle that lie within this rectangle and that do not contain any points in their interior. For finding the largest-area empty rectangle, the first algorithm takes O(n log3n) time and O(n) memory space and it simplifies the algorithm given by Chazelle, Drysdale and Lee which takes O(n log3n) time but O(n log n) storage. The second algorithm for computing the largest-area empty rectangle is more complicated but it only takes O(n log2n) time and O(n) memory space. The two algorithms for computing the largest-area rectangle can be modified to compute the largest-perimeter rectangle in O(n log2n) and O(n log n) time, respectively. Since O(n log n) is a lower bound on time for computing the largest-perimeter empty rectangle, the second algorithm for computing such a rectangle is optimal within a multiplicative constant.

Fast algorithms for computing the largest empty rectangle

A technique for image segmentation using shape-directed covers is described and applied to the fully automatic analysis of complex printed-page layouts. The structure of the background (white space) is analyzed, assisted by an enumeration of all maximal white rectangles. For this enumeration, the most computationally expensive step, an algorithm has been developed that, aside from a sort, achieves an expected runtime linear in the number of black connected components. The crucial engineering decision is the specification of a partial order on white rectangles to express domain-specific knowledge of preferred shapes and sizes. This order determines a sequence of partial covers of the background, and thus, a sequence of nested page segmentations. In experimental trials on Manhattan layouts, good segmentations often occur early in this sequence, using a simple and uniform shape-direction rule. This is a global-to-local strategy, which for some tasks is superior to strategies currently emphasized in the literature, including bottom-up and top-down. >

Image segmentation by shape-directed covers

A rectangleA and a setS ofn points inA are given. We present a new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior. The computational complexity of the presented algorithm isO(n logn + s), where s is the number of possible restricted rectangles considered. Moreover, the expected performance isO(n · logn).

A new algorithm for the largest empty rectangle problem

We show the existence of e-nets of size O(1/e log log 1/e) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in reals3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of e-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of sizeO(1/e log log log 1/e) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

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W.L.Hsu

Papers

On Maximum Empty Rectangle Problem