External sorting

External sorting is a class of sorting algorithms that can handle massive amounts of data. External sorting is required when the data being sorted do not fit into the main memory of a computing device (usually RAM) and instead they must reside in the slower external memory (usually a hard drive). External sorting typically uses a hybrid sort-merge strategy. In the sorting phase, chunks of data small enough to fit in main memory are read, sorted, and written out to a temporary file. In the merge phase, the sorted subfiles are combined into a single larger file.

External merge sort

One example of external sorting is the external merge sort algorithm, which sorts chunks that each fit in RAM, then merges the sorted chunks together.^[1][2] For example, for sorting 900 megabytes of data using only 100 megabytes of RAM:

1. Read 100 MB of the data in main memory and sort by some conventional method, like quicksort.
2. Write the sorted data to disk.
3. Repeat steps 1 and 2 until all of the data is in sorted 100 MB chunks (there are 900MB / 100MB = 9 chunks), which now need to be merged into one single output file.
4. Read the first 10 MB (= 100MB / (9 chunks + 1)) of each sorted chunk into input buffers in main memory and allocate the remaining 10 MB for an output buffer. (In practice, it might provide better performance to make the output buffer larger and the input buffers slightly smaller.)
5. Perform a 9-way merge and store the result in the output buffer. Whenever the output buffer fills, write it to the final sorted file and empty it. Whenever any of the 9 input buffers empties, fill it with the next 10 MB of its associated 100 MB sorted chunk until no more data from the chunk is available. This is the key step that makes external merge sort work externally -- because the merge algorithm only makes one pass sequentially through each of the chunks, each chunk does not have to be loaded completely; rather, sequential parts of the chunk can be loaded as needed.

Historically, instead of a sort, sometimes a replacement-selection algorithm^[3] was used to perform the initial distribution, to produce on average half as many output chunks of double the length.

Additional passes

The above example is a two-pass sort: first sort, then merge. Note that there was a single k-way merge, rather than, say, a series of 2-way merge passes as in a typical in-memory merge sort. This is because each merge pass reads and writes every value from and to disk.

However, there is a limitation to single-pass merging. As the number of chunks increases, we divide memory into more buffers, so each buffer is smaller, so we have to make many smaller reads rather than fewer larger ones. Thus, for sorting, say, 50 GB in 100 MB of RAM, using a single merge pass isn't efficient: the disk seeks required to fill the input buffers with data from each of the 500 chunks (we read 100MB / 501 ~ 200KB from each chunk at a time) take up most of the sort time. Using two merge passes solves the problem. Then the sorting process might look like this:

1. Run the initial chunk-sorting pass as before.
2. Run a first merge pass combining 25 chunks at a time, resulting in 20 larger sorted chunks.
3. Run a second merge pass to merge the 20 larger sorted chunks.

Like in-memory sorts, efficient external sorts require O(n log n) time: exponential increases in data size require linear increases in the number of passes. If one makes liberal use of the gigabytes of RAM provided by modern computers, the logarithmic factor grows very slowly: under reasonable assumptions, one could sort at least 500 GB of data on a hard disk using 1 GB of main memory before a third pass became advantageous, and could sort many times that before a fourth pass became useful.^[4] Low-seek-time media like SSDs also increase the amount that can be sorted before additional passes help.

RAM size is important here: doubling memory dedicated to sorting halves the number of chunks and the number of reads per chunk, reducing the number of seeks required by about three-quarters. The ratio of RAM to disk storage on servers often makes it convenient to do huge sorts on a cluster of machines^[5] rather than on one machine with multiple passes.

Tuning performance

The Sort Benchmark, created by computer scientist Jim Gray, compares external sorting algorithms implemented using finely tuned hardware and software. Winning implementations use several techniques:

• Using parallelism
  • Multiple disk drives can be used in parallel in order to improve sequential read and write speed. This can be a very cost-efficient improvement: a Sort Benchmark winner in the cost-centric Penny Sort category uses six hard drives in an otherwise midrange machine.^[6]
  • Sorting software can use multiple threads, to speed up the process on modern multicore computers.
  • Software can use asynchronous I/O so that one run of data can be sorted or merged while other runs are being read from or written to disk.
  • Multiple machines connected by fast network links can each sort part of a huge dataset in parallel.^[7]
• Increasing hardware speed
  • Using more RAM for sorting can reduce the number of disk seeks and avoid the need for more passes.
  • Fast external memory like solid-state drives can speed sorts, either if the data is small enough to fit entirely on SSDs or, more rarely, to accelerate sorting SSD-sized chunks in a three-pass sort.
  • Many other factors can affect hardware's maximum sorting speed: CPU speed and number of cores, RAM access latency, input/output bandwidth, disk read/write speed, disk seek time, and others. "Balancing" the hardware to minimize bottlenecks is an important part of designing an efficient sorting system.
  • Cost-efficiency as well as absolute speed can be critical, especially in cluster environments where lower node costs allow purchasing more nodes.
• Increasing software speed
  • Some Sort Benchmark entrants use a variation on radix sort for the first phase of sorting: they separate data into one of many "bins" based on the beginning of its value. Sort Benchmark data is random and especially well-suited to this optimization.
  • Compacting the input, intermediate files, and output can reduce time spent on I/O, but is not allowed in the Sort Benchmark.
  • Because the Sort Benchmark sorts long (100-byte) records using short (10-byte) keys, sorting software sometimes rearranges the keys separately from the values to reduce memory I/O volume.

Other algorithms

External merge sort is not the only external sorting algorithm; there are also distribution sorts, which work by partitioning the unsorted values into smaller "buckets" that can be sorted in main memory. Like merge sort, external distribution sort also has a main-memory sibling; see bucket sort. There is a duality, or fundamental similarity, between merge- and distribution-based algorithms that can aid in thinking about sorting and other external memory algorithms.^[8] There are in-place algorithms for external sort, which require no more disk space than the original data.

References

1. Donald Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching, Second Edition. Addison-Wesley, 1998, ISBN 0-201-89685-0, Section 5.4: External Sorting, pp.248–379.
2. Ellis Horowitz and Sartaj Sahni, Fundamentals of Data Structures, H. Freeman & Co., ISBN 0-7167-8042-9.
3. Donald Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching, Second Edition. Addison-Wesley, 1998, ISBN 0-201-89685-0, Section 5.4: External Sorting, pp.254–ff.
4. Assume a single disk with 200 MB/s transfer, 20 ms seek time, 1 GB of buffers, 500 GB to sort. The merging phase will have 500 buffers of 2M each, need to do 250K seeks and read then write 500 GB. It will spend 5,000 sec seeking and 5,000 sec transferring. Doing two passes as described above would nearly eliminate the seek time but add an additional 5,000 sec reading and writing, so this is approximately the break-even point between a two-pass and three-pass sort.
5. Chris Nyberg, Mehul Shah, Sort Benchmark Home Page (links to examples of parallel sorts)
6. Nikolas Askitis, OzSort 2.0: Sorting up to 252GB for a Penny
7. Rasmussen et al., TritonSort
8. J. S. Vitter, Algorithms and Data Structures for External Memory, Series on Foundations and Trends in Theoretical Computer Science, now Publishers, Hanover, MA, 2008, ISBN 978-1-60198-106-6.

External links

• STXXL, an algorithm toolkit including external mergesort
• An external mergesort example
• A K-Way Merge Implementation
• External-Memory Sorting in Java
• A sample pennysort implementation using Judy Arrays
• Sort Benchmark