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<!DOCTYPE html>
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<html>
<head>
<title>Packing Non-Self-crossing Edge-disjoint Spanning Paths</title>
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><strong></strong> Portfolio - Packing Non-Self-crossing
Edge-disjoint Spanning Paths</a
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<h1>
Packing Non-Self-crossing Edge-disjoint Spanning Paths into a
Point Set
</h1>
</header>
<span class="image main"
><img src="images/thesis.PNG" alt=""
/></span>
<p>
The term packing refers to the arrangement of multiple geometrical
structures or shapes such as circles, squares, triangles, or
polygons into a fixed and finite set of points.
</p>
<p>
Packing strategies help to construct and arrange multiple
geometric structures in a predetermined bounded space; hence, it
can be classified as an optimization problem, as we are trying to
allocate the optimal space for resources in a finite bounded
space. The better the efficiency of the algorithm, the greater
number of items that can be packed.
</p>
<p>
In our research, we have studied the packing of non-self-crossing,
edge-disjoint plane spanning paths and have obtained some
promising results.
</p>
<hr class="major" />
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<div class="content">
<h2>Problem Statement</h2>
<p>
In our research, the topic of interest is packing spanning
paths into any given point set P. The spanning paths to be
packed should be non-self-crossing and edge-disjoint.
</p>
<p>
A path is called non-self-crossing if no two edges of the path
cross. Two paths are edge-disjoint if they do not share an
edge. Our goal is to pack two non-self-crossing and
edge-disjoint spanning paths into P, such that all the edges
of each path cross some line in the plane.
</p>
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<img src="images/thesis1.PNG" alt="" />
</span>
</section>
<hr class="major" />
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<img src="images/thesis2.PNG" alt="" />
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<h2>New Idea</h2>
<p>
Packing multiple geometric structures can raise some design
considerations such as a concept called a
<i>blocked configuration</i>. This blocked state prevents us
from appyling any algorithm to acheive the desired output.
</p>
<p>
Hence, we introduce the new idea of <i>Tangent Switch</i>. We
introduce an approach to exchange vertices incident on
separate convex hulls of two different point sets, which in
turn resolves the blocked configuration.
</p>
<p>
We implement the tangent switch rotating a
<i
><a
href="https://en.wikipedia.org/wiki/Tangent_lines_to_circles"
>
common separating tangent
</a></i
>
until a pair of vertices between 2 point sets are exchanged.
</p>
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<h2>Final Result</h2>
<p>
Once we resolve the blocked configuration with the help of the
tangent-switch algorithm, the rest of the procedure is simply
repeating the tangent-switch unless all blocked vertices are
exchanged.
</p>
<p>
Thus, with my solution we demonstrated an alternate approach
to pack 2 non-self-crossing and edge-disjoint spanning paths
into a point set.
</p>
<p>
With more research it might be even possible to embed another
path into the point set.
</p>
</div>
<span class="image object">
<img src="images/thesis3.PNG" alt="" />
</span>
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<h2>Retrospective</h2>
<p>
An open question still exists: Is it possible to extend the number
of non-self-crossing edge-disjoint spanning paths that can be
packed in a point set consisting of points in general position in
the plane to three or more? Furthermore, instead of having
multiple cases for each variation of a blocked configuration, a
much more generic approach to the problem might also exist.
</p>
<p>
See my research
<a
href="https://www.proquest.com/openview/ed91ec0653078eadec068438dd9c310a/1.pdf?pq-origsite=gscholar&cbl=18750&diss=y"
>
here</a
> for more details.
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