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

(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.
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