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Compatible Paths on Labelled Point Sets
Elena Arseneva,Yeganeh Bahoo,Ahmad Biniaz,Pilar Cano,Farah Chanchary,John Iacono,Kshitij Jain,Anna Lubiw,Debajyoti Mondal,Khadijeh Sheikhan,Csaba D. Tóth +10 more
TLDR
In this article, the authors give polynomial-time algorithms to find compatible geometric paths or report that none exist in three scenarios: O(n)$ time for points in convex position, O( n 2 ) time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and O (n 2 √ log n) time for the points in general position.Abstract:
Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with the same clockwise ordering of the vertices on its boundary as in $f$. In particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same connected $n$-vertex graph.
Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: $O(n)$ time for points in convex position; $O(n^2)$ time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for points in general position if the paths are restricted to be monotone.read more
Citations
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On compatible triangulations with a minimum number of Steiner points
Anna Lubiw,Debajyoti Mondal +1 more
TL;DR: This paper proves the problem to be NP-hard for polygons with holes, and proves that two compatible polygons have compatible triangulations with at most Steiner points.
References
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