Shortest path in a multiply-connected domain having curved ...

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(a) All BTs in the given shape (568 BTs) (b) Completely contained BTs (183 BTs) Figure 10: Number of bitangents though it cannot be quantified exactly how many tangents will be removed as it depends on the shape of the curves. The number of BTs that will be used will be a substantial reduction of the total number of BTs, when we compute on the fly over any algorithm that might use all the completely contained tangents. Consider the input curves in Figure 7(a). The curves have 568 BTs (Figure10(a)) and 183 completely contained BTs (Figure10(b)). By including the PCTs, approximately there are 200 completely contained tangents. These are the input for graph based Dijkstra’s algorithm. However, our algorithm has been found to use far lesser number of tangents as shown in the Table 1. The table shows the number of tangents selected/number contained tangents at each stage along the recursion. Blank entry in the table means all paths have terminated. Table 1: Number of tangents selected/Number contained tangents at each stage Figure PCTs from E PCTs from S BTs BTs BTs BTs BTs BTs BTs 7(a) 4/11 5/12 2/8 1/10 1/6 1/12 7(c) 3/3 4/10 1/12 2/4 1/10 1/16 2/3 2/12 1/6 7(e) 3/14 6/10 3/11 2/12 1/15 2/16 1/12 1/12 It should be noted that application of Lemma 4 leads to inner loops and regions get rejected at each stage, BTs are computed and used only in a closed band around the shortest path as could be seen from figures 7(a) and 7(e). Thus the number of computations are also greatly reduced. 5.2.2 Using discretization of the curve to compute SIP Discretizing the curve into set of points or polylines and then employing Dijkstra’s algorithm could be another approach to compute SIP. However, to compute a path, it is imperative that the footpoints of the tangent (PCTs/BTs) were part of sample points or vertices of polylines. However, there is no effective sampling strategy that can precisely capture all footpoints of tangents and bitangents and hence more inaccuracies are introduced. As one has to anyway 15
compute the tangency points to use an algorithm like Dijkstra, which will be of O(T 2 ), the complexity of our algorithm is O(T ). Also, it is worth noting that the algorithm directly computes all the potential path on the fly. 6 Conclusion In this paper, an algorithm to compute SIP of curved boundaries, forming a multiplyconnected domain, extending the algorithm in [18] has been presented. It is shown that all potential paths can be computed, eliminating many PCTs/BTs/inner loops using a generalized region lemma and then specifically finding exterior and interior regions using PCTs/BTs. Merging of paths that will avoid redundant computation is also done. It is also shown that the algorithm can be applied to any number of holes and also to curves having C 1 discontinuity points. Test results on a few cases indicate that the algorithm is very amenable to implementation. SIP can be computed without employing visibility graph and Dijkstra. Further work involves focussing on applications based on [18] and this paper. References [1] Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. Data Structures and Algorithms. Addison-Wesley, 1983. [2] H. Alt and E. Welzl. Visibility graphs and obstacle-avoiding shortest paths. Mathematical Methods of Operations Research, 32:145–164, 1988. 10.1007/BF01928918. [3] Richard D. Bourgin and Sally E. Howe. Shortest curves in planar regions with curved boundary. Theoretical Computer Science, 112(2):215 – 253, 1993. [4] Ee-Chien Chang, Sung Woo Choi, DoYong Kwon, Hyungju Park, and Chee K. Yap. Shortest path amidst disc obstacles is computable. In Proceedings of the twenty-first annual symposium on Computational geometry, SCG ’05, pages 116–125, New York, NY, USA, 2005. ACM. [5] Danny Z. Chen and Haitao Wang. Computing shortest paths amid pseudodisks. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’11, pages 309–326. SIAM, 2011. [6] Danny Z. Chen and Haitao Wang. A nearly optimal algorithm for finding L1-shortest paths among polygonal obstacles in the plane. In Proceedings of the 19th European conference on Algorithms, ESA’11, pages 481–492, Berlin, Heidelberg, 2011. Springer- Verlag. [7] David P. Dobkin and Diane L. Souvaine. Computational geometry in a curved world. Algorithmica, 5(1-4):421–457, 1988. [8] Gershon Elber. IRIT 10.0 User’s Manual. The Technion—IIT, Haifa, Israel, 2009. 16
Page 1 and 2: Accepted Manuscript Shortest path i
Page 3 and 4: non-intersecting polygonal obstacle
Page 5 and 6: It is to be noted that when both S
Page 7 and 8: E S (a) Sample set of curves (b) PC
Page 9 and 10: C1 I1 I2 CSM (a) An example of inte
Page 11 and 12: (a) Rejected path (in blue) (b) Rej
Page 13 and 14: (a) All potential SIPs (b) SIP (c)
Page 15: (a) All potential SIPs (b) SIP (c)
Page 19 and 20: 7 Appendix: Pseudocode of the algor
Page 21 and 22: S. Bharath Ram is currently a resea

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