A Linear Time Algorithm for Optimum Tree Placement

LetN be a point placed directed graph andG be a sub-graph of N . We study the point placement problem of optimally re-placing the nodes in G assuming that the remaining nodes in N are fixed. The goal is to minimize the total “wiring” cost i.e. the cost of the internal nets of G plus the external nets connecting G to other other nodes of N . The cost of a net is the semi-perimeter of the smallest bounding box of the nodes of the net. The main result of this paper is a linear time algorithm (DP) for this problem that is optimum when G is a tree; nodes of G can have multiple fanouts, but for each net in G there can be no more than one fanout to other nodes of G. Experiments verify the optimality of the algorithm by comparisons with a linear programming (LP) formulation of the problem. The algorithm is much faster (linear) than the LP (quadratic) in practice, and for non-trees was only marginally sub-optimal in the application tested. Several application areas are suggested, one being placement-aware logic synthesis.