On Building Trees with Minimum Height, Relationally

Relationally Shin-Cheng Mu Computing Laboratory, University of Oxford s m omlab.ox.a .uk Abstra t The algebrai style of reasoning about programs has been proposed and studied by omputing s ientists. We rephrase the old problem of building trees of minimum height as an optimisation problem and apply the greedy theorem to derive a linear time algorithm. To put the problem in the right form, we nd it ne essary to generalise from fun tions to relations and make use of the onverse of a fun tion theorem to write the inverse of a fun tion as a fold. 1 Introdu tion Given a list of trees. The task is to ombine them into one, retaining the leftto-right order of the trees. How an we ombine them so that the height of the resulting tree is as small as possible? The a tual ontents of the subtrees are not relevant. Therefore we an think of the input as a list of numbers representing the heights of the subtrees. Fig. 1 illustrates one of the best arrangement of subtrees with heights [2; 9; 8; 3; 6; 9℄, whose height is 11.