On the approximability of two tree drawing conventions - Penna, Paolo (Information Processing Letters, 2002)
We consider two aesthetic criteria for the visualization of rooted trees: inclusion and tip-over. Finding the minimum area layout according to either of these two standards is an NP-hard task, even when we restrict ourselves to binary trees. We provide a fully polynomial time approximation scheme for this problem. This result applies to any tree for tip-over layouts and to bounded degree trees in the case of the inclusion convention. We also prove that such restriction is necessary since, for unbounded degree trees, the inclusion problem is strongly NP-hard. Hence, neither a fully polynomial time approximation scheme nor a pseudopolynomial time algorithm exists, unless P=NP. Our technique, combined with the parallel algorithm by Metaxas et al., also yields an NC fully parallel approximation scheme. This latter result holds for inclusion of binary trees and for the slicing floorplanning problem. Although this problem is in P, it is unknown whether it belongs to NC or not. All the above results also apply to other size functions of the drawing (e.g., the perimeter).