| Summary: | In the island adoption problem from geographical information system we are asked
to identify which islands are located in which lakes. This problem translates directly to
polygon nesting in computational geometry: given a set of polygons, find their nesting
structure. We present our research into a broader nesting problem, namely connected
component nesting, beginning with the underlying concept of topology-building and a
related issue of random polygon generation.
Topology building is a process of structuring data. We develop a plane sweep al
gorithm for building a quad-edge data structure that captures the topological structure
of connected components of a set of line segments. The algorithm starts with a data
structure representing a single edge then adds edges into the data structure at each
step while sweeping across the connected components The algorithm’s time complexity
is determined by the time to sort the vertices of the line segments.
We develop two approaches for obtaining the nesting structure of polygons. The
first adopts a basic idea of Bajaj and Dey [1], but introduces a new notch definition to
simplify their algorithm. The second generalizes the nesting problem to a broader class
including the nesting of connected components. We present a sweep algorithm, based on
a union-find data structure, that computes the nesting of the connected components.
In order to test and verify the time complexity of our polygon nesting algorithm, we
present an algorithm that generates x-monotone polygons uniformly at random over a
vertex set of n points. This algorithm scans the point set to calculate the total number
of monotone polygons that can be created, then reverses the scan to generate a random
monotone polygon. This process generates a random polygon over the n vertices in 0(K)
time, where n K n2 is the number edges of the visibility graph of the x-monotone
chain whose vertices are the given n points. The space complexity of our algorithm is
0(n).
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