TAOCP 5.4.1 Exercise 9
Let the comparison used in Algorithm R for the selection tree be denoted by $\prec$, where in the original algorithm $a \prec b$ means that key $a$ is smaller than key $b$.
Section 5.4.1: Multiway Merging and Replacement Selection
Exercise 9. [23] [23] How would you modify Algorithm R so that it causes certain specified runs (depending on RC) to be output in ascending order, and others in descending order?
Verified: yes
Solve time: 1m17s
Solution
Let the comparison used in Algorithm R for the selection tree be denoted by $\prec$, where in the original algorithm $a \prec b$ means that key $a$ is smaller than key $b$. The algorithm constructs runs by repeatedly extracting the minimal key from the root of the loser tree and replacing it by the next input record, provided that the new record does not violate the monotonicity condition of the current run.
The modification introduces a run-direction parameter determined by $\mathrm{RC}$. For each run, $\mathrm{RC}$ specifies whether the output sequence must be ascending or descending. This requires replacing the fixed comparison relation $\prec$ by a run-dependent comparison relation $\prec_{\mathrm{RC}}$ that is used consistently both in the selection tree and in the placement of replacement records.
For runs designated as ascending, $\prec_{\mathrm{RC}}$ coincides with the original relation $\prec$. For runs designated as descending, $\prec_{\mathrm{RC}}$ is the reverse order, defined by $a \prec_{\mathrm{RC}} b$ if and only if $b \prec a$. This ensures that the tournament structure continues to select the “winner” according to the required direction of monotonicity.
Algorithm R maintains a current run number, say $\mathrm{RN}$, associated with each record stored in the external nodes. The modification does not change the mechanism by which run boundaries are created; instead, it changes the ordering constraint imposed when deciding whether a newly read input record can be inserted into the current run or must be deferred to the next run.
When a new record $x$ is read from the input, it is compared with the last output record $y$ of the current run using the run-dependent order. If the current run is ascending, the condition for continuation remains $x \ge y$. If the current run is descending, the condition becomes $x \le y$. If the condition is satisfied, $x$ is inserted into the selection tree in the usual manner with its run number equal to the current $\mathrm{RN}$. If the condition fails, the current run is terminated and $\mathrm{RC}$ is toggled or advanced to the specification for the next run, and $x$ is inserted into the structure as the first element of the next run with its updated run number.
The internal operation of the loser tree is modified only through the comparison operator. In each internal node where Algorithm R compares two keys to determine the loser, the comparison is performed using $\prec_{\mathrm{RC}}$ instead of $\prec$. Thus, for descending runs, every comparison that previously selected the smaller key now selects the larger key, and the root of the tree yields the maximum element instead of the minimum.
The sentinel key $+\infty$ used in the ascending case must be replaced in the descending case by a sentinel that behaves as a global minimum under the reversed order. This can be achieved by reusing the same representation of $\infty$ but interpreting comparisons under $\prec_{\mathrm{RC}}$, since under reversed ordering the role of $\infty$ becomes that of a minimal element and therefore never interferes with extraction of valid keys.
The replacement selection property is preserved because every structural update in the tree depends only on pairwise comparisons, and the modified relation $\prec_{\mathrm{RC}}$ remains a total order on the key set for each fixed run orientation. Consequently, the tournament invariant that each internal node stores the loser of its comparison remains valid, and the root continues to represent the extreme element with respect to the chosen direction.
Thus Algorithm R is modified by introducing a run-dependent comparison relation $\prec_{\mathrm{RC}}$, replacing all occurrences of the fixed ordering in both selection and insertion logic, and adjusting the continuation test for runs to $x \ge y$ in ascending mode and $x \le y$ in descending mode. This completes the required modification. ∎