TAOCP 5.4.8 Exercise 10

Exercise 10. [20] [20] How would you sort a multireel file with only two tapes?

*5.4.9. Disks and Drums

So far we have considered tapes as the vehicles for external sorting, but more flexible types of mass storage devices are generally available. Although such “bulk memory” or “direct-access storage” units come in many different forms, they may be roughly characterized by the following properties:

Verified: yes
Solve time: 1m13s

Solution

Let the multireel file consist of a sequence of records distributed over several reels, with no restriction on where a run begins or ends relative to reel boundaries. The storage device behaves as a single sequential tape that happens to require occasional physical reel changes; this does not alter the logical access pattern of forward reading and backward rewinding assumed in Section 5.4.8.

The sorting procedure is therefore implemented as a standard two-tape external merge sort, with the only modification that tape motion continues across reel boundaries without changing direction or logical position in the file.

The initial phase constructs sorted runs using replacement selection or any internal method that produces runs of average length substantially exceeding the internal memory size $M$. Each run is written sequentially onto tape $T_0$, advancing across reels as needed. When the end of a reel is reached, writing continues on the next reel of $T_0$ without interruption of the run structure. This produces a sequence of runs stored consecutively on $T_0$.

The merging phase proceeds in passes. On a given pass, the runs currently on $T_0$ are distributed alternately onto $T_0$ and $T_1$, preserving run boundaries. Then a balanced two-way merge is performed, reading simultaneously from both tapes and writing merged runs back onto the opposite tape. The merging uses the standard two-buffer discipline from Section 5.4.8, and whenever the end of a reel is reached on either input or output tape, the head moves to the next reel while preserving continuity of the run being processed.

After each pass, the roles of $T_0$ and $T_1$ are interchanged. The process repeats until a single run remains, which is the sorted file. The final output may be left distributed across multiple reels of either tape, or consolidated by a final copy pass if required.

Since the algorithm performs the same sequence of run generation and balanced merging as the ordinary two-tape external merge sort, the number of passes and asymptotic running time remain unchanged from the analysis in Section 5.4.8. Reel boundaries introduce only constant-factor overhead due to physical repositioning, without affecting the logical number of comparisons or data transfers per pass.

This completes the construction of a sorting method for multireel files using only two tapes. ∎