Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 175
The details of the refinement of the statement in italics can be found in the program presented below. The
average path length of T 0,n is now given by the quotient p 0,n /w 0,n , and its root is the node with index r 0,n .
Let us now describe the structure of the program to be designed. Its two main components are the
procedures to find the optimal search tree, given a weight distribution w, and to display the tree given the
indices r. First, the counts a and b and the keys are read from an input source. The keys are actually not
involved in the computation of the tree structure; they are merely used in the subsequent display of the tree.
After printing the frequency statistics, the program proceeds to compute the path length of the perfectly
balanced tree, in passing also determining the roots of its subtrees. Thereafter, the average weighted path
length is printed and the tree is displayed.
In the third part, procedure OptTree is activated in order to compute the optimal search tree; thereafter,
the tree is displayed. And finally, the same procedures are used to compute and display the optimal tree
considering the key frequencies only, ignoring the frequencies of non-keys. To summarize, the following are
the global constants and variables:
CONST N = 100; (*max no. of keywords*) (* ADenS46_OptTree *)
WordLen = 16; (*max keyword length*)
VAR key: ARRAY N+1, WordLen OF CHAR;
a, b: ARRAY N+1 OF INTEGER;
p, w, r: ARRAY N+1, N+1 OF INTEGER;
PROCEDURE BalTree (i, j: INTEGER): INTEGER;
VAR k, res: INTEGER;
BEGIN
k := (i+j+1) DIV 2; r[i, j] := k;
IF i >= j THEN res := 0
ELSE res := BalTree(i, k-1) + BalTree(k, j) + w[i, j]
END;
RETURN res
END BalTree;
PROCEDURE ComputeOptTree (n: INTEGER);
VAR x, min, tmp: INTEGER;
i, j, k, h, m: INTEGER;
BEGIN
(*argument: w, results: p, r*)
FOR i := 0 TO n DO p[i, i] := 0 END;
FOR i := 0 TO n-1 DO
j := i+1; p[i, j] := w[i, j]; r[i, j] := j
END;
FOR h := 2 TO n DO
FOR i := 0 TO n-h DO
j := i+h; m := r[i, j-1]; min := p[i, m-1] + p[m, j];
FOR k := m+1 TO r[i+1, j] DO
tmp := p[i, k-1]; x := p[k, j] + tmp;
IF x < min THEN m := k; min := x END
END;
p[i, j] := min + w[i, j]; r[i, j] := m
END
END
END ComputeOptTree;