The manipulation of trees and linear graphs within a computer and some applications

Abstract

A digraph of z points and br arcs can be represented by its adjacency matrix. Within a computer this means a storage of z elements. By suppress1ng obv1ous information, a reduction can be made 1n the storage reqUired. The branches list representation stores the non-zero elements of the adjacency matr1x and requ1res only (br + z) elements. Any trees reqUired for computer manipulation are rooted and ordered. They can be represented in the two arrays below[j] and posnbr[j]1 where below[j] stores the below of a point j and posnbr[j] its pos1tive neighbour. However, th1s representation is very inconvenient for going up the tree. Thus another representation called the rd, lu representation is defined such that it is nearly as easy to go up the tree as to go down it. A few procedures were written which enabled an ordered-rooted tree to be divided into two parts and rejoined together at different points. This technique forms a basis for Top tree and Transportree. A succesfUl investigation was also carried out to find a relationship between labelled ordered-rooted trees and labelled binary pendant trees. Top tree is a heuristic method of obtaining a good solution in a relatively short time to the Travelling Salesman Problem. It is based on the observation that the majority of lines of a minimal solution (to the problem) appear in the minimal spanning tree (for that same graph). The technique is to reduce multi-membered stars of the minimal spanning tree so as to have all points incident to at most two lines. This seems to give very good results on both random data and published examples. The problem of minimising the bandwidth of a matrix was also examined. The problem was re-stated as that of having to label the points of a large graph so that the maximum difference between the labels of adjacent pOints is a minimum. The problem of doing this quickly was not solved but here again, techniques based on the spanning tree for that graph were evolved which reduced the initial bandwidth considerably. An algorithm was written which did find the minimum bandwidth labelling by going through the permutation list. But due to the size of the list this was slow and impractical for graphs with z greater than 20. The nature or this work was such that it was suitable to tackle the Shortest Paths ( through a digraph) Problem. The tree spanning technique was developed so that for large, highly sparse digraphs ( or networks), it was found to be more erficient than the Cascade method, one of the better matrix type methods. Finally H.I.Scoins method of solving the Transportation Problem was refined (and called Transportree ) so that the tree was not kept in the below array (i.e. as a rooted tree) but in the rd, lu representation. This results in the time spent list processing in order to go up the tree being drasticaly reduced. This last section was merely an exercise in showing how ordered-rooted trees and their manipulation are of use in a wide array of problems.