Minimum-Layer Upward Drawings of Trees

An upward drawing of a rooted tree T is a planar straight-line drawing of T where the vertices of T are placed on a set of horizontal lines, called layers, such that for each vertex u of T , no child of u is placed on a layer vertically above the layer on which u has been placed. In this paper we give a linear-time algorithm to obtain an upward drawing of a given rooted tree T on the minimum number of layers. Moreover, if the given tree T is not rooted, we can select a vertex r of T in linear time such that an upward drawing of T rooted at r would require the minimum number of layers among all the upward drawings of T with any of its vertices as the root. We also extend our results on a rooted tree to give an algorithm for an upward drawing of a rooted ordered tree. To the best of our knowledge, there is no previous algorithm for obtaining an upward drawing of a tree on the minimum number of layers. Submitted: August 2009 Reviewed: January 2010 Revised: February 2010 Accepted: March 2010 Final: April 2010 Published: June 2010 Article type: Regular paper Communicated by: G. Liotta E-mail addresses: [email protected] (Muhammad Jawaherul Alam) [email protected] (Md. Abul Hassan Samee) [email protected] (Mashfiqui Rabbi) [email protected] (Md. Saidur Rahman) 246 M. J. Alam et al. Minimum-Layer Upward Drawings of Trees