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Sweeping Costs of Planar Domains

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Research in Computational Topology

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 13))

Abstract

Let D be a Jordan domain in the plane. We consider a pursuit-evasion, contamination clearing, or sensor sweep problem in which the pursuer at each point in time is modeled by a continuous curve, called the sensor curve. Both time and space are continuous, and the intruders are invisible to the pursuer. Given D, what is the shortest length of a sensor curve necessary to provide a sweep of domain D, so that no continuously-moving intruder in D can avoid being hit by the curve? We define this length to be the sweeping cost of D. We provide an analytic formula for the sweeping cost of any Jordan domain in terms of the geodesic Fréchet distance between two curves on the boundary of D with non-equal winding numbers. As a consequence, we show that the sweeping cost of any convex domain is equal to its width, and that a convex domain of unit area with maximal sweeping cost is the equilateral triangle.

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Notes

  1. 1.

    Our definition is similar to the graph-based definition in [3, Definition 2.1].

  2. 2.

    This is not quite precise if ∂D contains a straight line segment of nonzero length parallel to v (there are at most two such segments). In this case, pick an arbitrary point on each line segment parallel to v; each such point will be either the starting point or the ending point for both α and β.

References

  1. H. Adams, G. Carlsson, Evasion paths in mobile sensor networks. Int. J. Robot. Res. 34(1), 90–104 (2015)

    Article  Google Scholar 

  2. L. Alonso, A.S. Goldstein, E.M. Reingold, “Lion and man”: upper and lower bounds. ORSA J. Comput. 4, 447–452 (1992)

    Article  MathSciNet  Google Scholar 

  3. B. Alspach, Searching and sweeping graphs: a brief survey. Le matematiche 59, 5–37 (2006)

    MathSciNet  MATH  Google Scholar 

  4. M. Berger, Geometry I (Springer Science & Business Media, New York, 2009)

    MATH  Google Scholar 

  5. F. Berger, A. Gilbers, A. Grüne, R. Klein, How many lions are needed to clear a grid? Algorithms 2(3), 1069–1086 (2009)

    Article  MathSciNet  Google Scholar 

  6. A. Beveridge, Y. Cai, Pursuit-evasion in a two-dimensional domain. ARS Mat. Contemp. 13, 187–206 (2017)

    MathSciNet  MATH  Google Scholar 

  7. D. Bienstock, Graph searching, path-width, tree-width and related problems (a survey). DIMACS Ser. Discret. Math. Theor. Comput. Sci. 5, 33–49 (1991)

    Article  MathSciNet  Google Scholar 

  8. R.L. Bishop, The intrinsic geometry of a Jordan domain, International Electronic Journal of Geometry, 1, 33–39 (2018)

    MathSciNet  MATH  Google Scholar 

  9. R.D. Bourgin, P.L. Renz, Shortest paths in simply connected regions in \(\mathbb {R}^2\). Adv. Math. 76(2), 260–295 (1989)

    Google Scholar 

  10. D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, vol. 33 (American Mathematical Society, Providence, 2001)

    MATH  Google Scholar 

  11. A. Cerdán, Comparing the relative volume with the relative inradius and the relative width. J. Inequal. Appl. 2006(1), 1–8 (2006)

    Article  MathSciNet  Google Scholar 

  12. E.W. Chambers, Y. Wang, Measuring similarity between curves on 2-manifolds via homotopy area, in Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry (ACM, New York, 2013), pp. 425–434

    Google Scholar 

  13. E.W. Chambers, D. Letscher, T. Ju, L. Liu, Isotopic Fréchet distance, in CCCG (2011)

    Google Scholar 

  14. J.-C. Chin, Y. Dong, W.-K. Hon, C.Y.-T. Ma, D.K.Y. Yau, Detection of intelligent mobile target in a mobile sensor network. IEEE/ACM Trans. Netw. 18(1), 41–52 (2010)

    Article  Google Scholar 

  15. T.H. Chung, G.A. Hollinger, V. Isler, Search and pursuit-evasion in mobile robotics. Auton. Robot. 31(4), 299–316 (2011)

    Article  Google Scholar 

  16. D.L. Cohn, Measure Theory, vol. 165 (Springer, Berlin, 1980)

    Book  Google Scholar 

  17. J. Cortes, S. Martinez, T. Karatas, F. Bullo, Coverage control for mobile sensing networks, in Proceedings of the IEEE International Conference on Robotics and Automation, vol. 2 (2002), pp. 1327–1332

    Google Scholar 

  18. V. de Silva, R. Ghrist, Coordinate-free coverage in sensor networks with controlled boundaries via homology. Int. J. Robot. Res. 25(12), 1205–1222 (2006)

    Article  Google Scholar 

  19. A. Efrat, L.J. Guibas, S. Har-Peled, J.S.B. Mitchell, T.M. Murali, New similarity measures between polylines with applications to morphing and polygon sweeping. Discret. Comput. Geom. 28(4), 535–569 (2002)

    Article  MathSciNet  Google Scholar 

  20. L. Esposito, V. Ferone, B. Kawohl, C. Nitsch, C. Trombetti, The longest shortest fence and sharp Poincaré–Sobolev inequalities. Arch. Ration. Mech. Anal. 206(3), 821–851 (2012)

    Article  MathSciNet  Google Scholar 

  21. B.T. Fasy, S. Karakoc, C. Wenk, On minimum area homotopies of normal curves in the plane. arXiv:1707.02251 (2017, Preprint)

    Google Scholar 

  22. M.E. Houle, G.T. Toussaint, Computing the width of a set. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 761–765 (1988)

    Article  Google Scholar 

  23. I. Johnstone, D. Siegmund, On Hotelling’s formula for the volume of tubes and Naiman’s inequality. Ann. Stat. 17, 184–194 (1989)

    Article  MathSciNet  Google Scholar 

  24. J.R. Kline, Separation theorems and their relation to recent developments in analysis situs. Bull. Am. Math. Soc. 34(2), 155–192 (1928)

    Article  MathSciNet  Google Scholar 

  25. B. Liu, P. Brass, O. Dousse, P. Nain, D. Towsley, Mobility improves coverage of sensor networks, in Proceedings of the 6th ACM International Symposium on Mobile Ad hoc Networking and Computing (ACM, New York, 2005), pp. 300–308

    Google Scholar 

  26. Z. Nie, On the minimum area of null homotopies of curves traced twice. arXiv:1412.0101 (2014, Preprint)

    Google Scholar 

  27. J. Pál, Ein minimumproblem für ovale. Math. Ann. 83(3), 311–319 (1921)

    Article  MathSciNet  Google Scholar 

  28. T.D. Parsons, Pursuit-evasion in a graph, in Theory and Applications of Graphs (Springer, Berlin, 1978), pp. 426–441

    Book  Google Scholar 

  29. G. Pólya, Mathematics and Plausible Reasoning: Volume 1: Induction and Analogy in Mathematics (Oxford University Press, Oxford, 1965)

    Google Scholar 

  30. E. Sriraghavendra, K. Karthik, C. Bhattacharyya, Fréchet distance based approach for searching online handwritten documents, in Ninth International Conference on Document Analysis and Recognition, vol. 1 (2007), pp. 461–465

    Google Scholar 

  31. G.T. Toussaint, Solving geometric problems with the rotating calipers, in Proceedings of IEEE Melecon, vol. 83 (1983), p. A10

    Google Scholar 

  32. C. Wenk, A.F. Cook IV, Geodesic Fréchet distance inside a simple polygon. ACM Trans. Algorithms 7(1), 9 (2010)

    Google Scholar 

  33. L. Zoretti, Sur les fonctions analytiques uniformes. J. Math. pures appl 1, 9–11 (1905)

    MATH  Google Scholar 

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Acknowledgements

We would like to thank Clayton Shonkwiler for pointing us to the references [11, 27].

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Authors and Affiliations

  1. Colorado State University, Department of Mathematics, Fort Collins, CO, USA

    Brooks Adams, Henry Adams & Colin Roberts

Authors
  1. Brooks Adams
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  2. Henry Adams
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  3. Colin Roberts
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Correspondence to Henry Adams .

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Editors and Affiliations

  1. Department of Computer Science, St Louis University, St Louis, Missouri, USA

    Erin Wolf Chambers

  2. Gianforte School of Computing, and Department of Mathematical Sciences, Montana State University, Bozeman, Montana, USA

    Brittany Terese Fasy

  3. Department of Mathematics, Statistics, & Computer Science, Macalester College, Saint Paul, Minnesota, USA

    Lori Ziegelmeier

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Adams, B., Adams, H., Roberts, C. (2018). Sweeping Costs of Planar Domains. In: Chambers, E., Fasy, B., Ziegelmeier, L. (eds) Research in Computational Topology. Association for Women in Mathematics Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-89593-2_5

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