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

(a) Rejected path (in
blue)
(b) Rejected path (in
blue)
BT1
ζ2
ζ1
(c) ζ1 contains BT1
Figure 6: Rejected and merged paths (in blue)
(d) Merged paths
section 4.2. To do this, a dummy tangent line starting at S in the direction φ as clockwise
is added to the BT processing algorithm. Similar procedure is followed for counter-clockwise
direction and for the end point E.
4.3 Further elimination of paths
Sections 4.1 and 4.2 detailed how the PCTs/BTs can be eliminated based on the region
formulation using Lemma 4. However, there are further elimination of PCTs/BTs is possible.
This is done through the application of Lemma 3 which stated that a path cannot intersect
certain PCTs/BTs in specific ways. Thus a check is made to eliminate paths that are not
valid. Figures 6(a) and 6(b) show the path in blue that can be eliminated using Lemma 3.
4.4 Merging of paths
A tree structure is maintained to store the computed paths thus far. The tree starts with a
single node S, and the children for each node is the list of valid tangents obtained using the
node. The procedure processes every child recursively till E is reached via. the termination
list (which contains the PCTs processed from E) or no further valid tangent is available.
Along with every node, the length from S up to that node is also stored by computing the
Euclidean distance for tangents and line integral for curved portions. Thus the length of leaf
node when it reaches E will be the length of that particular path.
While the above procedures are followed recursively, a merging step is performed to avoid
redundant computations. Simply, paths are merged if a bitanget to be processed has been
already traversed. In this case all subtree of paths already computed is added to the current
path. The length corresponding to each node is updated accordingly. An example of merging
is shown in Figures 6(c) and 6(d). In Figure 6(c), the bitangent BT1 is already traversed by
the path ζ1 and hence the path ζ2 is merged when BT1 is reached. Algorithm 2 (in appendix)
gives the pseudocode for processing the bitangents.
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