Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 174
We are now ready to construct the optimization algorithm in detail. We recall the following definitions,
which are based on optimal trees T ij consisting of nodes with keys k i+1 ... k j :
1. a i : the frequency of a search for k i .
2. b j : the frequency of a search argument x between k j and k j+1 .
3. w ij : the weight of T ij .
4. p ij : the weighted path length of T ij .
5. r ij : the index of the root of T ij .
We declare the following arrays:
a: ARRAY n+1 OF INTEGER; (*a[0] not used*)
b: ARRAY n+1 OF INTEGER;
p,w,r: ARRAY n+1, n+1 OF INTEGER;
Assume that the weights w ij have been computed from a and b in a straightforward way. Now consider w
as the argument of the procedure OptTree to be developed and consider r as its result, because r
describes the tree structure completely. p may be considered an intermediate result. Starting out by
considering the smallest possible subtrees, namely those consisting of no nodes at all, we proceed to larger
and larger trees. Let us denote the width j-i of the subtree T ij by h. Then we can trivially determine the
values p ii for all trees with h = 0 according to the definition of p ij :
FOR i := 0 TO n DO p[i,i] := b[i] END
In the case h = 1 we deal with trees consisting of a single node, which plainly is also the root (see Fig.
4.38).
FOR i := 0 TO n-1 DO
j := i+1; p[i,j] := w[i,j] + p[i,i] + p[j,j]; r[i,j] := j
END
k j|a j
b j-1
b j
w j-1, j-1
w j-1, j
Fig. 4.38. Optimal search tree with single node
Note that i denotes the left index limit and j the right index limit in the considered tree T ij . For the cases
h > 1 we use a repetitive statement with h ranging from 2 to n, the case h = n spanning the entire tree
T 0,n . In each case the minimal path length p ij and the associated root index r ij are determined by a simple
repetitive statement with an index k ranging over the interval given for r ij :
FOR h := 2 TO n DO
FOR i := 0 TO n-h DO
j := i+h;
find k and min = MIN k: i < k < j : (p i,k-1 + p kj ) such that r i,j-1 < k < r i+1,j ;
p[i,j] := min + w[i,j]; r[i,j] := k
END
END