TAOCP 5.4.9 Exercise 13

Let $T$ range over admissible merge patterns for $n$ runs, where each internal node has arity at most $8$, and cost is the weighted external path length C(T)=\sum_{i=1}^n w_i d_i.

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Section 5.4.9: Disks and Drums

Exercise 13. [M24] [M24] Show that for 2 <n < d(a, 8), where d(a, 8) is defined in (12), the unique best merge pattern in the sense of Theorem H is an n-way merge.

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Let $T$ range over admissible merge patterns for $n$ runs, where each internal node has arity at most $8$, and cost is the weighted external path length

$$ C(T)=\sum_{i=1}^n w_i d_i. $$

Let $W=\sum_{i=1}^n w_i$.

1. Meaning of $d(a,8)$ in this argument

By definition (12), $d(a,8)$ is the threshold such that for every $n$ with

$$ n<d(a,8), $$

there exists an admissible merge pattern consisting of a single $n$-way merge. In particular, in this regime the star-shaped merge tree (one internal node with all $n$ leaves as children) is a legal pattern in the model.

Thus, for the given hypothesis

$$ 2<n<d(a,8), $$

the star tree is admissible.

No further structural consequence (such as “all $m\le n$ merges are allowed”) is assumed.

2. Cost of the $n$-way merge (star tree)

In the star tree, every leaf is connected directly to the root, hence every leaf has depth $1$. Therefore

$$ C_{\star}=\sum_{i=1}^n w_i \cdot 1 = W. $$

3. Lower bound on the cost of any admissible tree

Let $T$ be any admissible merge tree.

Every leaf has depth at least $1$, hence

$$ C(T)=\sum_{i=1}^n w_i d_i ;\ge; \sum_{i=1}^n w_i = W. $$

Thus,

$$ C(T)\ge W \quad \text{for every admissible } T. $$

4. Characterization of equality

We now determine when equality holds.

If $C(T)=W$, then

$$ \sum_{i=1}^n w_i(d_i-1)=0. $$

Since all $w_i>0$ and $d_i-1\ge 0$, this implies

$$ d_i=1 \quad \text{for all } i. $$

Hence every leaf is at depth $1$, so the tree has height $1$. Therefore all leaves are attached directly to the root.

So any optimal tree achieving cost $W$ must be exactly the star tree.

5. Uniqueness and admissibility constraints

We must check that the star tree is the only admissible tree with this structure.

  • If a tree has height $\ge 2$, then at least one leaf has depth $\ge 2$, implying $C(T)>W$.
  • Therefore no such tree can be optimal.
  • The only possible optimal structure is the height-1 tree.

Finally, by the hypothesis $n<d(a,8)$, the height-1 $n$-way merge is admissible, so the optimal structure is not only unique in form but also feasible in the model.

6. Conclusion

For $2<n<d(a,8)$, the star-shaped merge tree is admissible and has cost $W$, and every other admissible merge pattern has cost strictly greater than $W$. Hence it is the unique optimal merge pattern in the sense of Theorem H.

$$ \boxed{\text{The unique best merge pattern is the single } n\text{-way merge.}} $$