TAOCP 5.4.5 Exercise 1

Section 5.4.5: The Oscillating Sort

Exercise 1. [22] [22] The text illustrates Sobel’s original oscillating sort for T = 5 and S = 16. Give a precise specification of an algorithm that generalizes the procedure, sorting S = P®* initial runs on T = P +1 > 3 tapes. Strive for simplicity.

Verified: no
Solve time: 1m10s

Solution

Let the tapes be $0,1,\dots,P$, where tape $q$ is the designated output tape and the remaining $P$ tapes are work tapes. Let $\sigma$ denote the cyclic permutation on ${0,1,\dots,P}$ defined by $\sigma(j) = (j+1) \bmod (P+1)$. At each stage, $\sigma$ is applied to all tape indices so that the role of each work tape is rotated while $q$ remains the output tape of the current merge phase.

Assume $S = P^m$ initial runs are present on the input, each run of unit weight.

For each integer $s$ with $1 \le s \le m$, define a level consisting of one distribution phase followed by one merge phase.

At the beginning of level $s$, the input consists of $P^{m-s+1}$ runs distributed one per work tape in blocks determined by previous levels, each run having length $P^{s-1}$.

The distribution phase at level $s$ proceeds by writing runs onto the $P$ work tapes in cyclic order $\sigma^{s-1}(1), \sigma^{s-1}(2), \dots, \sigma^{s-1}(P)$, skipping tape $q$. Each write operation places one run on the next available tape in this order until all $P^{m-s+1}$ runs have been exhausted or each tape has received one block of consecutive runs. The resulting configuration places runs of length $P^{s-1}$ on each work tape, preserving their internal ordering.

The merge phase at level $s$ performs repeated $P$-way merges into tape $q$. At each merge operation, the algorithm selects one run from each of the $P$ work tapes, reading each tape in the direction consistent with the construction of the runs (forward or backward as required by their orientation in the current level), and merges these $P$ runs into a single run on tape $q$. Each such operation replaces $P$ runs of length $P^{s-1}$ by one run of length $P^s$. This process continues until each work tape has contributed all its runs of the current level, producing exactly $P^{m-s}$ runs on tape $q$.

After completion of the merge phase, the roles of tapes are rotated by $\sigma$, and level $s+1$ begins.

After level $m$, the process produces a single run of length $P^m$ on tape $q$, which is sorted in ascending order.

The construction ensures that after completion of level $s$, every produced run has length $P^s$, and the total number of runs across the $P$ work tapes is $P^{m-s}$ before the merge and $P^{m-s-1}$ after it. This invariant follows from the fact that each merge replaces exactly $P$ runs by one run and that each level processes all available runs exactly once.

This completes the specification of the generalized oscillating sort. ∎