External Sorting

One method for sorting a file is to load the file into memory, sort the data in memory, then write the results. When the file cannot be loaded into memory due to resource limitations, an external sort applicable. We will implement an external sort using replacement selection to establish initial runs, followed by a polyphase merge sort to merge the runs into one sorted file. I highly recommend you consult Knuth [1998], as many details have been omitted.

Theory

For clarity, I'll assume that data is on one or more reels of magnetic tape. Figure 4-1 illustrates a 3-way polyphase merge. Initially, in phase A, all data is on tapes T1 and T2. Assume that the beginning of each tape is at the bottom of the frame. There are two sequential runs of data on T1: 4-8, and 6-7. Tape T2 has one run: 5-9. At phase B, we've merged the first run from tapes T1 (4-8) and T2 (5-9) into a longer run on tape T3 (4-5-8-9). Phase C is simply renames the tapes, so we may repeat the merge again. In phase D we repeat the merge, with the final output on tape T3.

Figure 4-1: Merge Sort
PhaseT1T2T3
A 7
6
8
4		 9
5
B 7
6
9
8
5
4
C 9
8
5
4		 7
6
D

9
8
7
6
5
4

Several interesting details have been omitted from the previous illustration. For example, how were the initial runs created? And, did you notice that they merged perfectly, with no extra runs on any tapes? Before I explain the method used for constructing initial runs, let me digress for a bit.

In 1202, Leonardo Fibonacci presented the following exercise in his Liber Abbaci (Book of the Abacus): "How many pairs of rabbits can be produced from a single pair in a year's time?" We may assume that each pair produces a new pair of offspring every month, each pair becomes fertile at the age of one month, and that rabbits never die. After one month, there will be 2 pairs of rabbits; after two months there will be 3; the following month the original pair and the pair born during the first month will both usher in a new pair, and there will be 5 in all; and so on. This series, where each number is the sum of the two preceeding numbers, is known as the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... .
Curiously, the Fibonacci series has found wide-spread application to everything from the arrangement of flowers on plants to studying the efficiency of Euclid's algorithm. There's even a Fibonacci Quarterly journal. And, as you might suspect, the Fibonacci series has something to do with establishing initial runs for external sorts.

Recall that we initially had one run on tape T2, and 2 runs on tape T1. Note that the numbers {1,2} are two sequential numbers in the Fibonacci series. After our first merge, we had one run on T1 and one run on T2. Note that the numbers {1,1} are two sequential numbers in the Fibonacci series, only one notch down. We could predict, in fact, that if we had 13 runs on T2, and 21 runs on T1 {13,21}, we would be left with 8 runs on T1 and 13 runs on T3 {8,13} after one pass. Successive passes would result in run counts of {5,8}, {3,5}, {2,3}, {1,1}, and {0,1}, for a total of 7 passes. This arrangement is ideal, and will result in the minimum number of passes. Should data actually be on tape, this is a big savings, as tapes must be mounted and rewound for each pass. For more than 2 tapes, higher-order Fibonacci numbers are used.

Initially, all the data is on one tape. The tape is read, and runs are distributed to other tapes in the system. After the initial runs are created, they are merged as described above. One method we could use to create initial runs is to read a batch of records into memory, sort the records, and write them out. This process would continue until we had exhausted the input tape. An alternative algorithm, replacement selection, allows for longer runs. A buffer is allocated in memory to act as a holding place for several records. Initially, the buffer is filled. Then, the following steps are repeated until the input is exhausted:

Figure 4-2 illustrates replacement selection for a small file. The beginning of the file is to the right of each frame. To keep things simple, I've allocated a 2-record buffer. Typically, such a buffer would hold thousands of records. We load the buffer in step B, and write the record with the smallest key (6) in step C. This is replaced with the next record (key 8). We select the smallest key >= 6 in step D. This is key 7. After writing key 7, we replace it with key 4. This process repeats until step F, where our last key written was 8, and all keys are less than 8. At this point, we terminate the run, and start another.

Figure 4-2: Replacement Selection
StepInputBufferOutput
A 5-3-4-8-6-7

B 5-3-4-8 6-7
C 5-3-4 8-7 6
D 5-3 8-4 7-6
E 5 3-4 8-7-6
F
5-4 3 | 8-7-6
G
5 4-3 | 8-7-6
H

5-4-3 | 8-7-6

This strategy simply utilizes an intermediate buffer to hold values until the appropriate time for output. Using random numbers as input, the average length of a run is twice the length of the buffer. However, if the data is somewhat ordered, runs can be extremely long. Thus, this method is more effective than doing partial sorts.

When selecting the next output record, we need to find the smallest key >= the last key written. One way to do this is to scan the entire list, searching for the appropriate key. However, when the buffer holds thousands of records, execution time becomes prohibitive. An alternative method is to use a binary tree structure, so that we only compare lg n items.

Implementation

An ANSI-C implementation of an external sort is included. Function makeRuns calls readRec to read the next record. Function readRec employs the replacement selection algorithm (utilizing a binary tree) to fetch the next record, and makeRuns distributes the records in a Fibonacci distribution. If the number of runs is not a perfect Fibonacci number, dummy runs are simulated at the beginning of each file. Function mergeSort is then called to do a polyphase merge sort on the runs.