Enumeration and visibility problems in integer lattices (extended abstract)

SCG '90 Pub Date : 1990-05-01 DOI:10.1145/98524.98582

E. Kranakis, M. Pocchiola

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引用次数: 4

Abstract

We study enumeration and visibility problems in the d-dimensional integer lattice Ldn of d-tuples of integers ≤ n. In the first part of the paper we give several useful enumeration principles and use them to study the asymptotic behavior of the number of straight lines traversing a certain fixed number of lattice vertices of Ldn, the line incidence problem and the edge visibility region. In the second part of the paper we consider an art gallery problem for point obstacles. More specifically we study the camera placement problem for the infinite lattice Ld. A lattice point is visible from a camera C (positioned at a vertex of Ld) if the line segment joining A and C crosses no other lattice vertex. For any given number s ≤ 3d of cameras we determine the position they must occupy in the lattice Ld in order to maximize their visibility.

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整数格的枚举和可见性问题(扩展抽象)

研究了整数≤n的d元组的d维整数格Ldn上的枚举和可见性问题。本文第一部分给出了几种有用的枚举原理，并利用它们研究了经过Ldn上固定数目的格顶点的直线数的渐近性质、线的关联问题和边的可见性区域。在论文的第二部分，我们考虑了一个点障碍物的美术馆问题。更具体地说，我们研究了无限晶格Ld的摄像机放置问题。如果连接A和C的线段没有穿过其他晶格顶点，则从摄像机C(位于Ld的顶点)可以看到一个晶格点。对于任意给定数量s≤3d的摄像机，我们确定它们在晶格Ld中必须占据的位置，以便最大化它们的可见性。

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SCG '90

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