Sinopsis
Compute the area of a union of circles with efficient, practical methods.
This book presents algorithms that scale beyond naive checks, including exact-arithmetic approaches and Monte Carlo techniques, to estimate or compute the union’s area in two dimensions. In clear, accessible terms, you’ll see how the plane is partitioned into circular-arc segments and how to manage these components as you add circles. The text covers both deterministic and probabilistic methods, from straightforward 0(n^2) procedures to linear Monte Carlo ideas, with discussions of time and space efficiency and practical preprocessing ideas such as Voronoi-based queries.
- How to represent the evolving union with arc lists and how to update them efficiently as new circles are added.
- Exact-area strategies alongside Monte Carlo approaches, including error bounds and sampling considerations.
- Concepts like boundary descriptions, Voronoi preprocessing, and probabilistic estimators that scale to many circles.
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Reseña del editor
Excerpt from Very Fast Algorithms for the Area of the Union of Many Circles
The problem of estimating the area of the union of many circles in the plane was first posed by [shamos, We present here an 0(n2) deterministic and an o(u) probabilistic algorithm for solving the problem. The recent work of [karp, Luby, 83] on estimating the failure probability of an n-component system helped us a lot in stating the algorithm and formulating the proof of fast convergence. The probabilistic algorithm can be modified to compute the area of the union of other planar objects (of fixed description) e.g. Triangles or boxes. The time is again o(u). The algorithm can also be extended to compute the volume of the union of n spheres in k dimensions, in time 0(nk). The algorithm has similar time complexity for k-dimensional objects other than spheres, provided that each object has a fixed description (not dependent on n).
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Very Fast Algorithms for the Area of the Union of Many Circles
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Paperback. Condición: New. Print on Demand. This book presents two novel algorithms for calculating the combined area of a group of circles on a two-dimensional plane. The first algorithm, deterministic in nature, has an algorithmic complexity of 0(n2), where n denotes the number of circles. The second algorithm is probabilistic, with a complexity of o(n), making it more efficient for larger sets of circles. The author employs recent developments in the field of reliability estimation to establish the fast convergence of the probabilistic method and extends the algorithm to calculate the volume of n-dimensional spheres with complexity 0(nk). The author situates this work within the broader discourse on estimating the area of unions of geometric shapes, providing a valuable resource for researchers and practitioners in computer science and related fields. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Nº de ref. del artículo: 9781333725426_0
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