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Compatible Paths on Labelled Point Sets

par Arseneva, Elena;Bahoo, Yeganeh;Biniaz, Ahmad;Cano Vila, Maria Del Pilar ;Chanchary, Farah;Iacono, John ;Jain, Kshitij;Lubiw, Anna;Mondal, Debajyoti;Sheikhan, Khadijeh;Tóth, Csaba C.D.
Référence arXiv.org
Publication Publié, 2020-06-01

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Résumé :

Let P and Q be finite point sets of the same cardinality in R2, each labelled from 1 to n. Two noncrossing geometric graphs GP and GQ spanning P and Q, respectively, are called compatible if for every face f in GP, there exists a corresponding face in GQ with the same clockwise ordering of the vertices on its boundary as in f. In particular, GP and GQ 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(n2) time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and O(n2logn) time for points in general position if the paths are restricted to be monotone.