TAOCP 5.4.6 Exercise 6

Section 5.4.6: Practical Considerations for Tape Merging

Exercise 6. [22] [22] What changes should be made to Algorithm 5.4.3C in order to convert it into an algorithm for cascade merge with rewind overlap, on T + 1 tapes?

Verified: no
Solve time: 1m05s

Solution

Algorithm 5.4.3C performs a cascade merge on $T$ tapes by repeated distribution of runs followed by successive merging passes in which the number of runs per file is doubled. Between passes, all tapes that will participate in the next pass are rewound, and this rewinding is treated as a separate phase.

Let the tapes be $A_1,\dots,A_T,A_{T+1}$, where $A_{T+1}$ is the auxiliary tape used in the cascade pattern. In Algorithm 5.4.3C, each merge pass uses a subset of $T$ tapes as alternating input and output, while the remaining tape is idle or being prepared for the next pass. The inefficiency arises because the rewind of all tapes is postponed until a global synchronization point, so no useful tape activity occurs during rewinding.

The modification is to eliminate global rewind phases and replace them by per-tape asynchronous rewinding that is overlapped with merging activity in the next pass. The key structural change is that a tape is rewound immediately when it ceases to be an output tape in a pass, and this rewind proceeds concurrently with all reading, writing, and computation in subsequent operations.

Formally, suppose pass $k$ uses a set of tapes $S_k \subseteq {A_1,\dots,A_T,A_{T+1}}$ as input/output tapes according to the cascade schedule of Algorithm 5.4.3C. When a tape $A_i$ completes its role as an output tape in pass $k$, it is marked idle and a rewind operation on $A_i$ is initiated immediately. While this rewind is in progress, $A_i$ is not assigned any input or output role in pass $k+1$.

The structural constraint that makes this safe is that in each pass of a cascade merge on $T$ tapes, at most $T$ tapes are required for actual merging activity, while $T+1$ tapes are available. Therefore, even if one tape is simultaneously engaged in rewinding, the remaining $T$ tapes suffice to execute the next merge pass without waiting for global synchronization.

To see that no conflict occurs, consider any pass boundary. Each tape is either (i) actively participating in merging in the current pass, or (ii) idle. Any tape of type (ii) begins rewinding immediately upon becoming idle. The cascade schedule ensures that a tape that has just finished being an output tape is not required again as an input tape until the next pass, and before it is needed again it has been idle for a full pass duration of other tapes’ activity. During this interval, its rewind completes, since rewind time is bounded above by a constant factor of the time required to process a full pass over the data (by the assumption in Section 5.4.6 that rewind time is roughly a fixed fraction of read/write time).

Thus the modification consists of replacing the statement in Algorithm 5.4.3C that performs a collective rewind step between passes by the rule that each tape begins rewinding at the instant it becomes idle, and the next pass selects only tapes whose rewind has completed. Because $T+1$ tapes are available while only $T$ are needed per pass, at least one tape is always available to absorb the overlap, so merging is never blocked by rewind latency.

This completes the transformation of Algorithm 5.4.3C into a cascade merge algorithm with rewind overlap on $T+1$ tapes. ∎