Abstract
This paper presents an optimal \(\varTheta (n \log n)\) algorithm for determining time-minimal rectilinear paths among n transient rectilinear obstacles. An obstacle is transient if it exists in the scene only for a specific time interval, i.e., it appears and then disappears at specific times. Given a point robot moving with bounded speed among transient rectilinear obstacles and a pair of points s, d, we determine a time-minimal, obstacle-avoiding path from s to d. The main challenge in solving this problem arises as the robot may be required to wait for an obstacle to disappear, before it can continue moving toward the destination. Our algorithm builds on the continuous Dijkstra paradigm, which simulates propagating a wavefront from the source point. We also solve a query version of this problem. For this, we build a planar subdivision with respect to a fixed source point, so that minimum arrival time to any query point can be reported in \(O(\log n)\) time, using point location for the query point in this subdivision.
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Agarwal, P.K., Arge, L., Yi, K.: An optimal dynamic interval stabbing-max data structure? In: SODA 2005, pp. 803–812 (2005)
Agarwal, P.K., Kumar, N., Sintos, S., Suri, S.: Computing shortest paths in the plane with removable obstacles. In SWAT 2018, pp. 5:1–5:15 (2018)
Chazelle, B.: Functional approach to data structures and its use in multidimensional searching. SIAM J. Comput. 17(3), 427–462 (1988)
de Rezende, P.J., Lee, D.T., Wu, Y.F.: Rectilinear shortest paths with rectangular barriers. In: SCG 1985, pp. 204–213. ACM, New York (1985)
Fujimura, K.: On motion planning amidst transient obstacles. In: ICRA 1992, vol. 2, pp. 1488–1493, May 1992
Fujimura, K.: Motion planning using transient pixel representations. In: ICRA 1993, vol. 2, pp. 34–39, May 1993
Fujimura, K.: Motion planning amid transient obstacles. Int. J. Robot. Res. 13(5), 395–407 (1994)
Hershberger, J., Kumar, N., Suri, S.: Shortest paths in the plane with obstacle violations. In: ESA 2017, pp. 49:1–49:14 (2017)
Hershberger, J., Suri, S.: An optimal algorithm for euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)
Kirkpatrick, D.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983)
LaValle, S.M.: Planning Algorithms, pp. 495–558. Cambridge University Press, Cambridge (2006)
LaValle, S.M., Sharma, R.: Robot motion planning in a changing, partially predictable environment. In: ISIC 1994, pp. 261–266 (1994)
Lee, D.T., Yang, C.D., Wong, C.K.: Rectilinear paths among rectilinear obstacles. Discret. Appl. Math. 70(3), 185–215 (1996)
Maheshwari, A., Nouri, A., Sack, J.R.: Rectilinear shortest paths among transient obstacles. CoRR arXiv: abs/1809.08898 (2018)
Mitchell, J.S.: L1 shortest paths among polygonal obstacles in the plane. Algorithmica 8(1–6), 55–88 (1992)
Mitchell, J.S.B.: Shortest paths among obstacles in the plane. In: SCG 1993, pp. 308–317. ACM, New York (1993)
Vaishnavi, V.K., Wood, D.: Rectilinear line segment intersection, layered segment trees, and dynamization. J. Algorithms 3(2), 160–176 (1982)
Yang, C., Lee, D., Wong, C.: Rectilinear path problems among rectilinear obstacles revisited. SIAM J. Comput. 24(3), 457–472 (1995)
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Maheshwari, A., Nouri, A., Sack, JR. (2018). Rectilinear Shortest Paths Among Transient Obstacles. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_2
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