Abstract
Given a simple polygon P of n vertices, the watchman route problem asks for a shortest (closed) route inside P such that each point in the interior of P can be seen from at least one point along the route. We present a simple, linear-time algorithm for computing a watchman route of length at most 2 times that of the shortest watchman route. The best known algorithm for computing a shortest watchman route takes O(n 4 log n) time, which is too complicated to be suitable in practice.
This paper also involves an optimal O(n) time algorithm for computing the set of so-called essential cuts, which are the line segments inside the polygon P such that any route visiting them is a watchman route. It solves an intriguing open problem by improving the previous O(n log n) time result, and is thus of interest in its own right.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alsuwaiyel, M.H., Lee, D.T.: Finding an approximate minimum-link visibility path inside a simple polygon. Inform. Process. Lett. 55, 75–79 (1995)
Carlsson, S., Jonsson, H., Nilsson, B.J.: Approximating the shortest watchman route in a simple polygon, Technical report, Lund University, Sweden (1997)
Carlsson, S., Jonsson, H., Nilsson, B.J.: Finding the shortest watchman route in a simple polygon. Discrete Comput. Geom. 22, 377–402 (1999)
Chazelle, B., Guibas, L.: Visibility and intersection problem in plane geometry. Discrete Comput. Geom. 4, 551–581 (1989)
Das, G., Heffernan, P.J., Narasimhan, G.: LR-visibility in polygons. Comput. Geom. Theory Appl. 7, 37–57 (1997)
Dror, M., Efrat, A., Lubiw, A., Mitchell, J.S.B.: Touring a sequence of polygons. In: Proc. 35th Annu. ACM Sympos. Theory Comput., pp. 473–482 (2003)
Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.: Linear time algorithms for visibility and shortest path problems inside simple triangulated polygons. Algorithmica 2, 209–233 (1987)
Heffernan, P.J.: An optimal algorithm for the two-guard problem. Int. J. Comput. Geom. Appl. 6, 15–44 (1996)
O’Rourke, J.: Computational Geometry in C. Cambridge University Press, Cambridge (1993)
Tan, X.: Fast computation of shortest watchman routes in simple polygons. IPL 77, 27–33 (2001)
Tan, X.: Approximation algorithms for the watchman route and zoo-keeper’s problems. Discrete Applied Math. 136, 363–376 (2004)
Tan, X., Hirata, T., Inagaki, Y.: Corrigendum to an incremental algorithm for constructing shortest watchman routes. Int. J. Comput. Geom. Appl. 9, 319–323 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tan, X. (2006). Linear-Time 2-Approximation Algorithm for the Watchman Route Problem. In: Cai, JY., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2006. Lecture Notes in Computer Science, vol 3959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11750321_17
Download citation
DOI: https://doi.org/10.1007/11750321_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34021-8
Online ISBN: 978-3-540-34022-5
eBook Packages: Computer ScienceComputer Science (R0)