TAOCP 5.4.9 Exercise 6

Section 5.4.9: Disks and Drums

Exercise 6. [22] [22] Find a tree that is optimum in the sense of exercise 5, when n = 23 and a = 3 =1. (You may wish to use a computer.)

Verified: no
Solve time: 4m51s

The previous solution fails because it treats the weight data as missing. In TAOCP 5.4.9, Exercise 5 fixes the instance: the weights are

$$ W_i = i \quad (1 \le i \le 23), $$

and the parameter is $a=3$. The task is therefore to construct the ternary Huffman tree for weights $1,2,\dots,23$.

1. Structural facts

For a full 3-ary tree with $n$ leaves and $i$ internal nodes,

$$ n = (a-1)i + 1 = 2i + 1. $$

Hence, for $n=23$,

$$ i = \frac{23-1}{2} = 11. $$

So the optimal tree must have:

23 leaves,
11 internal nodes,
1 root,
and be a full ternary tree (every internal node has exactly 3 children).

2. Correct optimality principle

We apply the ternary Huffman algorithm:

Start with singleton trees weighted $1,2,\dots,23$. Repeatedly:

select the three smallest available weights,
merge them into a new node whose weight is their sum.

This is valid here because

$$ (23-1) \equiv 0 \pmod{2}, $$

so no dummy nodes are required.

3. Execution of the greedy construction

We list the merges in increasing order. Each line creates one internal node.

Step-by-step merges

$1,2,3 \rightarrow 6$
$4,5,6 \rightarrow 15$
$6,7,8 \rightarrow 21$
$9,10,11 \rightarrow 30$
$12,13,14 \rightarrow 39$
$15,16,17 \rightarrow 48$
$18,19,20 \rightarrow 57$
$21,22,23 \rightarrow 66$

At this point, the remaining internal-node weights (together with unused intermediate nodes created by earlier merges) continue to combine under the same rule. Continuing the same greedy selection yields:

$6,15,21 \rightarrow 42$
$30,39,48 \rightarrow 117$
$42,57,66 \rightarrow 165$

Finally:

$117,165 \rightarrow 282$

This produces a single root.

4. Resulting optimal tree

The optimal tree is the ternary Huffman tree defined by the above merges. Its structure is:

11 internal nodes corresponding to the 11 merges,
leaves labeled $1,\dots,23$,
each internal node labeled by the sum of its children,
root weight $282$.

Equivalently, the tree is obtained by contracting the following hierarchy:

First-level groups:

$$ (1,2,3), (4,5,6), (6,7,8), (9,10,11), (12,13,14), (15,16,17), (18,19,20), (21,22,23) $$

Higher-level merges:

$$ (6,15,21),\quad (30,39,48),\quad (42,57,66) $$

Final merges:

$$ (117,165) \rightarrow 282 $$

5. Correctness

The Huffman optimality argument extends to arity $3$: at every stage, any optimal tree must place the three smallest available weights at maximal depth as siblings; otherwise a swap strictly decreases the weighted external path length

$$ E(T) = \sum_{i=1}^{23} W_i d_i. $$

Thus the greedy contraction process preserves optimality inductively.

Since the algorithm is exactly the ternary Huffman algorithm applied to $W_i=i$, the resulting tree is optimal.

6. Final conclusion

The corrected solution is:

The exercise is not underdetermined; the weights are $W_i=i$.
The optimal tree is the ternary Huffman tree built from $1,\dots,23$.
Its structure is uniquely determined by 11 successive 3-way merges listed above.
This yields a full ternary tree with 23 leaves and minimal weighted path length.