Optimal Algorithms to Embed Trees in a Point Set
Bose, Prosenjit and McAllister, Michael and Snoeyink, Jack (1996) Optimal Algorithms to Embed Trees in a Point Set. In: Symposium on Graph Drawing, GD 1995, September 20-22, 1995, Passau, Germany , pp. 64-75 (Official URL: http://dx.doi.org/10.1007/BFb0021791).
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We present optimal \Theta(n log n) time algorithms to solve two tree embedding problems whose solution previously took quadratic time or more: rooted-tree embeddings and degree-constrained embeddings. In the rooted-tree embedding problem we are given a rooted-tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the root at point p. In the degree-constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2n-2 and are asked to embed a tree in P using straight lines that respects the degrees assigned to each point of P. In both problems, the points of P must be in general position and the embeddings have no crossing edges.
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