12,545 research outputs found

    Orderly Spanning Trees with Applications

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    We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph GG, consisting of a plane graph HH of GG, and an orderly spanning tree of HH. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of GG, and (3) the best known encodings of GG with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

    Isomorphism of graph classes related to the circular-ones property

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    We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs.Comment: 25 pages, 9 figure

    Computational Complexity for Physicists

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    These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

    Schnyder woods for higher genus triangulated surfaces, with applications to encoding

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    Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus gg and compute a so-called gg-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus gg and nn vertices in 4n+O(glog(n))4n+O(g \log(n)) bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time O((n+g)g)O((n+g)g), hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr
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