Optimal layout of edge-weighted forests

The layout problem for trees with weighted edges is motivated by the design of very-large-scale integrated circuits. Some of the nodes are fixed and the object is to position the remainder so that the total weighted edge cost is minimized. The cost of each edge is the product of its weight and its length under some appropriate norm. Optimization for planar layouts is shown to be NP-hard. If crossings are permitted, then optimal layouts under the L₁ norm can be efficiently computed. Suitable algorithms and data structures are presented, and explicit exact cost functions are given for two classes of weighted complete binary trees.