TAOCP 5.4.1 Exercise 15

Section 5.4.1: Multiway Merging and Replacement Selection

Exercise 15. [20] [20] True or false: The final run produced by replacement selection never contains more than P records. Discuss your answer.

Verified: yes
Solve time: 1m47s

Solution

Let the algorithm be replacement selection with a selection tree containing $P$ external nodes as defined in Section 5.4.1. At every moment during execution, each external node contains either a record currently eligible for output or a marker indicating absence of a record, and the tree contains at most one record per external node.

The final run begins at the moment the input stream is exhausted. From that point onward, no further input records are inserted into the selection tree, so the structure evolves only by repeatedly outputting the current smallest key and promoting existing internal structure updates without introducing new records.

At the start of the final run, every record that remains in internal memory is already stored in some external node of the selection tree. Since there are $P$ external nodes, the number of such records is at most $P$.

Each step of the final run removes exactly one record from the tree and outputs it. Because no new records are inserted after input exhaustion, no step can increase the number of records in the tree. Hence the number of output steps in the final run is exactly the number of records initially present in the tree at the start of the final run.

Since that initial number is at most $P$, the total number of records output in the final run is at most $P$. Therefore the final run never contains more than $P$ records.

This completes the proof. ∎