Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 173
Instead of using the probabilities p i and q j , we will subsequently use such frequency counts and denote
them by
a i = number of times the search argument x equals k i
b j = number of times the search argument x lies between k j and k j+1
By convention, b 0 is the number of times that x is less than k 1 , and b n is the frequency of x being greater
than k n (see Fig. 4.37). We will subsequently use P to denote the accumulated weighted path length instead
of the average path length:
P = (Si: 1 ≤ i ≤ n : a i *h i ) + (Si: 1 ≤ i ≤ m : b i *h' i )
Thus, apart from avoiding the computation of the probabilities from measured frequency counts, we gain
the further advantage of being able to use integers instead of fractions in our search for the optimal tree.
Considering the fact that the number of possible configurations of n nodes grows exponentially with n,
the task of finding the optimum seems rather hopeless for large n. Optimal trees, however, have one
significant property that helps to find them: all their subtrees are optimal too. For instance, if the tree in Fig.
4.37 is optimal, then the subtree with keys k 3 and k 4 is also optimal as shown. This property suggests an
algorithm that systematically finds larger and larger trees, starting with individual nodes as smallest possible
subtrees. The tree thus grows from the leaves to the root, which is, since we are used to drawing trees
upside-down, the bottom-up direction [4-6].
The equation that is the key to this algorithm is derived as follows: Let P be the weighted path length of a
tree, and let P L and P R be those of the left and right subtrees of its root. Clearly, P is the sum of P L and P R ,
and the number of times a search travels on the leg to the root, which is simply the total number W of
search trials. We call W the weight of the tree. Its average path length is then P/W:
P = P L + W + P R
W = (Si: 1 ≤ i ≤ n : a i ) + (Si: 1 ≤ i ≤ m : b i )
These considerations show the need for a denotation of the weights and the path lengths of any subtree
consisting of a number of adjacent keys. Let T ij be the optimal subtree consisting of nodes with keys k i+1 ,
k i+2 , ... , k j . Then let w ij denote the weight and let p ij denote the path length of T ij . Clearly P = p 0,n and
W = w 0,n . These quantities are defined by the following recurrence relations:
w ii = b i (0 ≤ i ≤ n)
w ij = w i,j-1 + a j + b j (0 ≤ i < j ≤ n)
p ii = w ii (0 ≤ i ≤ n)
p ij = w ij + MIN k: i < k ≤ j : (p i,k-1 + p kj ) (0 ≤ i < j ≤ n)
The last equation follows immediately from the definitions of P and of optimality. Since there are
approximately n 2 /2 values p ij , and because its definition calls for a choice among all cases such that 0 < j-
i ≤ n, the minimization operation will involve approximately n 3 /6 operations. Knuth pointed out that a
factor n can be saved by the following consideration, which alone makes this algorithm usable for practical
purposes.
Let r ij be a value of k which achieves the minimum for It is possible to limit the search for r ij to a much
smaller interval, i.e., to reduce the number of the j-i evaluation steps. The key is the observation that if we
have found the root r ij of the optimal subtree T ij , then neither extending the tree by adding a node at the
right, nor shrinking the tree by removing its leftmost node ever can cause the optimal root to move to the
left. This is expressed by the relation
r i,j-1 ≤ r ij ≤ r i+1,j
which limits the search for possible solutions for r ij to the range r i,j-1 ... r i+1,j . This results in a total
number of elementary steps in the order of n 2 .