Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 177
a 2 = 2 key 2 = Ernst b 2 = 1
a 3 = 4 key 3 = Peter b 3 = 1
The results of procedure Find are shown in Fig. 4.39 and demonstrate that the structures obtained for the
three cases may differ significantly. The total weight is 40, the path length of the balanced tree is 78, and
that of the optimal tree is 66.
balanced tree
optimal tree
not considering key misses
Ernst
Albert
Peter
Albert
Ernst
Peter
Peter
Ernst
Albert
Fig. 4.39. The 3 trees generated by the Optimal Tree procedure
It is evident from this algorithm that the effort to determine the optimal structure is of the order of n 2 ;
also, the amount of required storage is of the order n 2 . This is unacceptable if n is very large. Algorithms
with greater efficiency are therefore highly desirable. One of them is the algorithm developed by Hu and
Tucker [4-5] which requires only O(n) storage and O(n*log(n)) computations. However, it considers only
the case in which the key frequencies are zero, i.e., where only the unsuccessful search trials are registered.
Another algorithm, also requiring O(n) storage elements and O(n*log(n)) computations was described by
Walker and Gotlieb [4-7]. Instead of trying to find the optimum, this algorithm merely promises to yield a
nearly optimal tree. It can therefore be based on heuristic principles. The basic idea is the following.
Consider the nodes (genuine and special nodes) being distributed on a linear scale, weighted by their
frequencies (or probabilities) of access. Then find the node which is closest to the center of gravity. This
node is called the centroid, and its index is
(Si: 1 ≤ i ≤ n : i*a i ) + (Si: 1 ≤ i ≤ m : i*b i ) / W
rounded to the nearest integer. If all nodes have equal weight, then the root of the desired optimal tree
evidently coincides with the centroid Otherwise - so the reasoning goes - it will in most cases be in the
close neighborhood of the centroid. A limited search is then used to find the local optimum, whereafter this
procedure is applied to the resulting two subtrees. The likelihood of the root lying very close to the centroid
grows with the size n of the tree. As soon as the subtrees have reached a manageable size, their optimum
can be determined by the above exact algorithm.
4.7 B-Trees
So far, we have restricted our discussion to trees in which every node has at most two descendants, i.e.,
to binary trees. This is entirely satisfactory if, for instance, we wish to represent family relationships with a
preference to the pedigree view, in which every person is associated with his parents. After all, no one has
more than two parents. But what about someone who prefers the posterity view? He has to cope with the
fact that some people have more than two children, and his trees will contain nodes with many branches.
For lack of a better term, we shall call them multiway trees.
Of course, there is nothing special about such structures, and we have already encountered all the
programming and data definition facilities to cope with such situations. If, for instance, an absolute upper
limit on the number of children is given (which is admittedly a somewhat futuristic assumption), then one
may represent the children as an array component of the record representing a person. If the number of
children varies strongly among different persons, however, this may result in a poor utilization of available
storage. In this case it will be much more appropriate to arrange the offspring as a linear list, with a pointer