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

More documents

Recommendations

Info

(a) All potential SIPs (b) SIP (c) All potential SIPs (d) SIP (e) All potential SIPs (f) SIP (g) All potential SIPs (h) SIP Figure 8: All potential shortest paths (in blue) and SIP (in black) for outer boundary as a bounding box. 5.2 Discussion To the best of the knowledge of the authors, not much work has been done on MCD. Ref. 2 in [3], which appear to be related to MCD is not accessible. [3, 10] seem to have handled only SCD. However, both are theoretical in nature and no implementation result have been provided. As the SIP is not unique in MCD, the algorithm has to be tuned to handle holes. The MCD can be considered some what similar to the calculation of path from start to end point through obstacles (i.e. a domain without the outer boundary in MCD). Most of the algorithms related to SIP of obstacles handle polygonal boundaries, including a distance function approach [20]. For curved obstacles, [16] proposed the computation of visibility graph, assuming the obstacles as convex curves. For disk obstacles, shortest path based on visibility graph and Dijkstra was proposed in [4]. Approach in [6] seems to be based on Voronoi diagram and it is well known that computation of the diagram is very difficult and not robust for arbitrary shaped curves. All the algorithms for curved obstacles appear to be theoretical in nature and no implementation results have been provided (in [17], an approach using grid-based approximation has been described used fast-sweeping method in [22]). To use an algorithm such as Dijsktra for curved obstacles, visibility graph of the curved obstacles can be used. However, to compute the visibility graph, the footpoints of the PCTs/BTs are required. In our work, while traversal of the algorithm, all the potential paths are identified and hence the application of visibility graph and Dijkstra need not be computed. 13
(a) All potential SIPs (b) SIP (c) All potential SIPs (d) SIP (e) SIP (f) SIP Figure 9: All potential shortest paths (in blue) and SIP (in black) for curves having C 1 discontinuities. 5.2.1 Complexity of the algorithm In this paper, as the computation of SIP involves PCTs/BTs, the complexity is analyzed based on the number of PCTs/BTs that may be involved (the analysis does not take into C 1 discontinuity points). Let n be the number of concave portions from the outer boundary and h be the number of inner loops. Between two concave regions there can be at most four completely contained BTs (this can be shown by employing the convex hull between two curves). Thus maximum number of BTs for the outer boundary will be 4(nC2) (nC2 = n(n − 1)/2). For h inner loops, number of BTs is 4(hC2) and between inner and outer boundary, it will be 4nh. From the starting point S, there can be at most two completely contained PCTs for a concave portion. This implies that at most 2n PCTs from S and 2n PCTs from E will be available when n concave portions are considered. Similarly 2h from S and 2h from E will be the number of PCTs for the inner loops. Let T = 4(nC2 + nh + n + h) be the total number of tangents (PCTs + BTs). Tangent lists are searched for the closest tangents (to E or S) as in discussed in section 4.1.1 and for mutually farthest tangents as in section 4.1.2. No tangent is processed twice because of merging criteria followed in section 4.4. Thus the complexity of the algorithm is O(T ) in the worst case. In general, not all BTs will be used for our computation (as inner loops may be eliminated) 14
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: (a) All potential SIPs (b) SIP (c)
Page 17 and 18: compute the tangency points to use
Page 19 and 20: 7 Appendix: Pseudocode of the algor
Page 21 and 22: S. Bharath Ram is currently a resea

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?