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Path schematization for route sketches. (English)
Kaplan, Haim (ed.), Algorithm theory ‒ SWAT 2010. 12th Scandinavian symposium and workshops on algorithm theory, Bergen, Norway, June 21‒23, 2010. Proceedings. Berlin: Springer (ISBN 978-3-642-13730-3/pbk). Lecture Notes in Computer Science 6139, 285-296 (2010).
Summary: Motivated from drawing route sketches, we consider the following path schematization problem. We are given a simple embedded polygonal path $P = (v _{1}, \dots , v _{n })$ and a set $\mathcal{C}$ of admissible edge orientations including the coordinate axes. The problem is to redraw $P$ schematically such that all edges are drawn as line segments that are parallel to one of the specified orientations. We also require that the path preserves the orthogonal order and that it remains intersection-free. Finally, we want the drawing to maximize the number of edges having their preferred edge direction and to minimize the path length. In this paper we first present an efficient two-step approach for schematizing monotone paths. It consists of an $O(n ^{2})$-time algorithm to assign edge directions optimally and a subsequent linear program to minimize the path length. In order to schematize non-monotone paths we propose a heuristic that first splits the input into $k$ monotone subpaths and then combines the optimal embeddings of the monotone subpaths into a single, intersection-free embedding of the initial path in $O(k ^{2} + n)$ time.