Minimum-area h-v drawings of complete binary trees

Extended abstract
Packing, Compressing, and Touching
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)

Abstract

We study the area requirement of h-v drawings of complete binary trees. An h-v drawing of a binary tree t is a drawing of t such that (a) nodes are points with integer coordinates, (b) each edge is either a rightward-horizontal or a downward-vertical straight-line segment from a node to one of its children, (c) edges do not intersect, and (d) if t1 and t2 are immediate subtrees of a node u, the enclosing rectangles of the drawings of t1 and t2 are disjoint. We prove that, for any complete binary tree t of height h ≥ 3 and with n nodes, the area of the optimum h-v drawing of t is equal to (a) 2.5n − 4.5 \(\sqrt {(n + 1)/2} \)+ 3.5 if h is odd, (b) 2.5n − 3.25 \(\sqrt {n + 1} \)+ 3.5 otherwise. As far as we know, this is one of the few examples in which a closed formula for the minimum-area drawing of a graph has been explicitly found. Furthermore this minimum-area h-v drawing can be constructed in linear time. As a consequence of this result and the result of Trevisan (1996), we have that h-v drawings are provably less area-efficient than strictly upward drawings when we restrict ourselves to complete binary trees. We also give analogous results for the minimum-perimeter and the minimum-enclosing square area h-v drawings.

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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  1. 1.Dipartimento di Scienze dell'InformazioneRoma
  2. 2.Dipartimento di Sistemi ed InformaticaUniversità di FirenzeFirenzeItaly

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