In-Place Radix Sort
This is a long text. Please bear with me. Boiled down, the question is: Is there a workable in-place radix sort algorithm?
Preliminary
I've got a huge number of small fixed-length strings that only use the letters “A”, “C”, “G” and “T” (yes, you've guessed it: DNA) that I want to sort.
At the moment, I use std::sort which uses introsort in all common implementations of the STL. This works quite well. However, I'm convinced that radix sort fits my problem set perfectly and should work much better in practice.
Details
I've tested this assumption with a very naive implementation and for relatively small inputs (on the order of 10,000) this was true (well, at least more than twice as fast). However, runtime degrades abysmally when the problem size becomes larger (N > 5,000,000).
The reason is obvious: radix sort requires copying the whole data (more than once in my naive implementation, actually). This means that I've put ~ 4 GiB into my main memory which obviously kills performance. Even if it didn't, I can't afford to use this much memory since the problem sizes actually become even larger.
Use Cases
Ideally, this algorithm should work with any string length between 2 and 100, for DNA as well as DNA5 (which allows an additional wildcard character “N”), or even DNA with IUPAC ambiguity codes (resulting in 16 distinct values). However, I realize that all these cases cannot be covered, so I'm happy with any speed improvement I get. The code can decide dynamically which algorithm to dispatch to.
Research
Unfortunately, the Wikipedia article on radix sort is useless. The section about an in-place variant is complete rubbish. The NIST-DADS section on radix sort is next to nonexistent. There's a promising-sounding paper called Efficient Adaptive In-Place Radix Sorting which describes the algorithm “MSL”. Unfortunately, this paper, too, is disappointing.
In particular, there are the following things.
First, the algorithm contains several mistakes and leaves a lot unexplained. In particular, it doesn’t detail the recursion call (I simply assume that it increments or reduces some pointer to calculate the current shift and mask values). Also, it uses the functions dest_group and dest_address without giving definitions. I fail to see how to implement these efficiently (that is, in O(1); at least dest_address isn’t trivial).
Last but not least, the algorithm achieves in-place-ness by swapping array indices with elements inside the input array. This obviously only works on numerical arrays. I need to use it on strings. Of course, I could just screw strong typing and go ahead assuming that the memory will tolerate my storing an index where it doesn’t belong. But this only works as long as I can squeeze my strings into 32 bits of memory (assuming 32 bit integers). That's only 16 characters (let's ignore for the moment that 16 > log(5,000,000)).
Another paper by one of the authors gives no accurate description at all, but it gives MSL’s runtime as sub-linear which is flat out wrong.
To recap: Is there any hope of finding a working reference implementation or at least a good pseudocode/description of a working in-place radix sort that works on DNA strings?
Well, here's a simple implementation of an MSD radix sort for DNA. It's written in D because that's the language that I use most and therefore am least likely to make silly mistakes in, but it could easily be translated to some other language. It's in-place but requires 2 * seq.length passes through the array.
void radixSort(string[] seqs, size_t base = 0) {
if(seqs.length == 0)
return;
size_t TPos = seqs.length, APos = 0;
size_t i = 0;
while(i < TPos) {
if(seqs[i][base] == 'A') {
swap(seqs[i], seqs[APos++]);
i++;
}
else if(seqs[i][base] == 'T') {
swap(seqs[i], seqs[--TPos]);
} else i++;
}
i = APos;
size_t CPos = APos;
while(i < TPos) {
if(seqs[i][base] == 'C') {
swap(seqs[i], seqs[CPos++]);
}
i++;
}
if(base < seqs[0].length - 1) {
radixSort(seqs[0..APos], base + 1);
radixSort(seqs[APos..CPos], base + 1);
radixSort(seqs[CPos..TPos], base + 1);
radixSort(seqs[TPos..seqs.length], base + 1);
}
}
Obviously, this is kind of specific to DNA, as opposed to being general, but it should be fast.
Edit:
I got curious whether this code actually works, so I tested/debugged it while waiting for my own bioinformatics code to run. The version above now is actually tested and works. For 10 million sequences of 5 bases each, it's about 3x faster than an optimized introsort.
I've never seen an in-place radix sort, and from the nature of the radix-sort I doubt that it is much faster than a out of place sort as long as the temporary array fits into memory.
Reason:
The sorting does a linear read on the input array, but all writes will be nearly random. From a certain N upwards this boils down to a cache miss per write. This cache miss is what slows down your algorithm. If it's in place or not will not change this effect.
I know that this will not answer your question directly, but if sorting is a bottleneck you may want to have a look at near sorting algorithms as a preprocessing step (the wiki-page on the soft-heap may get you started).
That could give a very nice cache locality boost. A text-book out-of-place radix sort will then perform better. The writes will still be nearly random but at least they will cluster around the same chunks of memory and as such increase the cache hit ratio.
I have no idea if it works out in practice though.
Btw: If you're dealing with DNA strings only: You can compress a char into two bits and pack your data quite a lot. This will cut down the memory requirement by factor four over a naiive representation. Addressing becomes more complex, but the ALU of your CPU has lots of time to spend during all the cache-misses anyway.
You can certainly drop the memory requirements by encoding the sequence in bits. You are looking at permutations so, for length 2, with "ACGT" that's 16 states, or 4 bits. For length 3, that's 64 states, which can be encoded in 6 bits. So it looks like 2 bits for each letter in the sequence, or about 32 bits for 16 characters like you said.
If there is a way to reduce the number of valid 'words', further compression may be possible.
So for sequences of length 3, one could create 64 buckets, maybe sized uint32, or uint64.
Initialize them to zero.
Iterate through your very very large list of 3 char sequences, and encode them as above.
Use this as a subscript, and increment that bucket.
Repeat this until all of your sequences have been processed.
Next, regenerate your list.
Iterate through the 64 buckets in order, for the count found in that bucket, generate that many instances of the sequence represented by that bucket.
when all of the buckets have been iterated, you have your sorted array.
A sequence of 4, adds 2 bits, so there would be 256 buckets. A sequence of 5, adds 2 bits, so there would be 1024 buckets.
At some point the number of buckets will approach your limits. If you read the sequences from a file, instead of keeping them in memory, more memory would be available for buckets.
I think this would be faster than doing the sort in situ as the buckets are likely to fit within your working set.
Here is a hack that shows the technique
#include <iostream>
#include <iomanip>
#include <math.h>
using namespace std;
const int width = 3;
const int bucketCount = exp(width * log(4)) + 1;
int *bucket = NULL;
const char charMap[4] = {'A', 'C', 'G', 'T'};
void setup
(
void
)
{
bucket = new int[bucketCount];
memset(bucket, '\0', bucketCount * sizeof(bucket[0]));
}
void teardown
(
void
)
{
delete[] bucket;
}
void show
(
int encoded
)
{
int z;
int y;
int j;
for (z = width - 1; z >= 0; z--)
{
int n = 1;
for (y = 0; y < z; y++)
n *= 4;
j = encoded % n;
encoded -= j;
encoded /= n;
cout << charMap[encoded];
encoded = j;
}
cout << endl;
}
int main(void)
{
// Sort this sequence
const char *testSequence = "CAGCCCAAAGGGTTTAGACTTGGTGCGCAGCAGTTAAGATTGTTT";
size_t testSequenceLength = strlen(testSequence);
setup();
// load the sequences into the buckets
size_t z;
for (z = 0; z < testSequenceLength; z += width)
{
int encoding = 0;
size_t y;
for (y = 0; y < width; y++)
{
encoding *= 4;
switch (*(testSequence + z + y))
{
case 'A' : encoding += 0; break;
case 'C' : encoding += 1; break;
case 'G' : encoding += 2; break;
case 'T' : encoding += 3; break;
default : abort();
};
}
bucket[encoding]++;
}
/* show the sorted sequences */
for (z = 0; z < bucketCount; z++)
{
while (bucket[z] > 0)
{
show(z);
bucket[z]--;
}
}
teardown();
return 0;
}
If your data set is so big, then I would think that a disk-based buffer approach would be best:
sort(List<string> elements, int prefix)
if (elements.Count < THRESHOLD)
return InMemoryRadixSort(elements, prefix)
else
return DiskBackedRadixSort(elements, prefix)
DiskBackedRadixSort(elements, prefix)
DiskBackedBuffer<string>[] buckets
foreach (element in elements)
buckets[element.MSB(prefix)].Add(element);
List<string> ret
foreach (bucket in buckets)
ret.Add(sort(bucket, prefix + 1))
return ret
I would also experiment grouping into a larger number of buckets, for instance, if your string was:
GATTACA
the first MSB call would return the bucket for GATT (256 total buckets), that way you make fewer branches of the disk based buffer. This may or may not improve performance, so experiment with it.
I'm going to go out on a limb and suggest you switch to a heap/heapsort implementation. This suggestion comes with some assumptions:
- You control the reading of the data
- You can do something meaningful with the sorted data as soon as you 'start' getting it sorted.
The beauty of the heap/heap-sort is that you can build the heap while you read the data, and you can start getting results the moment you have built the heap.
Let's step back. If you are so fortunate that you can read the data asynchronously (that is, you can post some kind of read request and be notified when some data is ready), and then you can build a chunk of the heap while you are waiting for the next chunk of data to come in - even from disk. Often, this approach can bury most of the cost of half of your sorting behind the time spent getting the data.
Once you have the data read, the first element is already available. Depending on where you are sending the data, this can be great. If you are sending it to another asynchronous reader, or some parallel 'event' model, or UI, you can send chunks and chunks as you go.
That said - if you have no control over how the data is read, and it is read synchronously, and you have no use for the sorted data until it is entirely written out - ignore all this. :(
See the Wikipedia articles:
- Heapsort
- Binary heap