Region-Based Approximation of Probability Distributions (for Visibility Between Imprecise Points Among Obstacles)

Let p and q be two imprecise points, given as probability density functions on R 2 , and let O be a set of disjoint polygonal obstacles in R 2 . We study the problem of approximating the probability that p and q can see each other; i.e., that the segment connecting p and q does not cross any obstacl...

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Bibliographic Details
Published in:Algorithmica Vol. 81; no. 7; pp. 2682 - 2715
Main Authors: Buchin, Kevin, Kostitsyna, Irina, Löffler, Maarten, Silveira, Rodrigo I.
Format: Journal Article Publication
Language:English
Published: New York Springer US 01.07.2019
Springer Nature B.V
Subjects:
35 Partial differential equations
35R Miscellaneous topics involving partial differential equations
60 Probability theory and stochastic processes
60G Stochastic processes
Algorithm Analysis and Problem Complexity
Algorithms
Anàlisi matemàtica
Barriers
Classificació AMS
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Differential equations, Partial
Equacions en derivades parcials
Estadística matemàtica
Gaussian distribution
Matemàtiques i estadística
Mathematics of Computing
Polygons
Probability density functions
Probability distribution
Processos estocàstics
Stochastic processes
Theory of Computation
Visibility
Visibility in polygonal domains
Àrees temàtiques de la UPC
Gaussian distribution
Visibility in polygonal domains
Probability distribution
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Let p and q be two imprecise points, given as probability density functions on R 2 , and let O be a set of disjoint polygonal obstacles in R 2 . We study the problem of approximating the probability that p and q can see each other; i.e., that the segment connecting p and q does not cross any obstacle in O . To solve this problem, we first approximate each density function by a weighted set of polygons. Then we focus on computing the visibility between two points inside two of such polygons, where we can assume that the points are drawn uniformly at random. We show how this problem can be solved exactly in O ( ( n + m ) 2 ) time, where n and m are the total complexities of the two polygons and the set of obstacles, respectively. Using this as a subroutine, we show that the probability that p and q can see each other amidst a set of obstacles of total complexity m can be approximated within error ε in O ( 1 / ε 3 + m 2 / ε 2 ) time.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00551-2