New string search algorithm(volnitsky.com) |
New string search algorithm(volnitsky.com) |
Note that the worst case of complexity for this algorithm is much, much worse than the worst case complexity for Boyer Moore. Do not use this algorithm carelessly. For example, if you use it in a thoughtless way in your web server, you may open yourself to a DoS attack.
Note that the author nicely characterizes it as of potential use for small alphabets and possibly multiple substrings (in a single search). That immediately made me think he might have devised it for genomics research. In most applications I would think you'd also want regexp features. Interestingly, DNA research and use in a regexp engine is something he goes on to suggest. (If you are searching for a very large number of regexps in a big genome database, I would not use this algorithm. I found that some simple variants on classic NFA techniques work very well for a wide class of typical regexps (e.g., regexps modeling SNPs, small read position errors, small numbers of read errors, etc. There probably isn't any one obviously right answer, though, and a lot depends on your particular hardware situation, data set sizes, etc.).
The HN headline is very bogus hype. "X2 times faster than Boyer-Moore" is far from true in the general case. "breakthrough" is a gross exaggeration: this is a technique that anyone with some good algorithms course or two under the belt should be able to think of an, for most applications, decide to not use because of the limitations of the thing. I can definitely see it being nice for some applications tolerant of its limitations but... breakthrough it ain't.
See http://en.wikipedia.org/wiki/Burrows–Wheeler_transform or http://bioinformatics.oxfordjournals.org/content/early/2009/...
Can you explain your thought process for why worst case complexity is worse than other? Did you do any measurements? I believe you are incorrect. I am author of this page. I've just did quick test for SS = "aaa ...aaaBaaa...aaa", with SS_size=240. My algorithm is faster than 3 BMs out of 4 tested.
> Do not use this algorithm carelessly. For example, if you use it in a thoughtless way in your web server, you may open yourself to a DoS attack.
Same can be said about all BM, naive and BSD's memmem/strstr. Possibility of DOS for substring search algorithm was known for a long time - but it never materialized. The cure is trivial - limit substring size.
> That immediately made me think he might have devised it for genomics research.
No, it wasn't. It was devised when I was preparing for Google interview (which I failed). And of cause algorithms which pre-index haystack will always be faster. I myself do not consider haystack pre-indexing algorithms a "text search" algorithms (maybe incorrectly).
> gross exaggeration
If you count pre-indexing algorithms and edge cases - then you are correct. For most common case - can you show me something faster (from a student or even yourself)?
You write on the page that your algorithm has "O(NM) worst case complexity." Compare Boyer-Moore's time complexity. The grandfather was interested in asymptotics. Not in some examples.
> Same can be said about all BM, naive and BSD's memmem/strstr.
Red herring.
> Possibility of DOS for substring search algorithm was known for a long time - but it never materialized.
What are you talking about?
> If you count pre-indexing algorithms and edge cases - then you are correct. For most common case - can you show me something faster (from a student or even yourself)?
What do you mean by egde cases? All the cases that make your program run slow?
Using your terminology, KMP has complexity O(N + M). That's better than O(MN).
EDIT: Am I missing something???
Complexity analysis according to the author: m = search term, n = text
O(m) preprocessing -- that's right, O(search term). And O(n times m) worst-case query string search, so the worst case traverses the whole text.
Now compare that to suffix trees:
O(n) preprocessing O(m) string search
where worst case complexity is linear in search term.
The algorithm itself looks very similar to the one used in agrep proposed by Wu and Manber [1].
I also found the book "Flexible Pattern Matching in Strings" to be a very good reference on all things related to pattern matching [2].
[1] S. Wu and U. Manber. A fast algorithm for multi-pattern searching. Report TR-94-17, Department of Computer Science, University of Arizona, 1994.
[2] http://www.amazon.com/Flexible-Pattern-Matching-Strings-Line...
In the experiment he used patterns of different length on the same text collection. As you can see in the graph, different algorithms perform best for a certain alphabet size.
He describes the text collection as "text corpus taken from wikipedia text dump" so I'm guessing the alphabet size is around 90?
It's also probably not a good thing that all the strings he is searching for are prefixes of the same pattern.
References:
Shift-Or: http://www-igm.univ-mlv.fr/~lecroq/string/node6.html#SECTION...
BNDM: http://www-igm.univ-mlv.fr/~lecroq/string/bndm.html#SECTION0...
BOM: http://www-igm.univ-mlv.fr/~lecroq/string/bom.html#SECTION00...
A quick review of the code makes me think that its worst case is actually on the order of O((s * (s + 1)) / 2), where s = m / 2.
The Achilles heel is the hash function. It's trivial to create collisions and have the insertion time for word w turn from O(1) to O(w).
But - and I don't want to sound pedantic - how is m^2 not exponential growth?
Edit: mea culpa guys, I carelessly translated from Dutch. You're all right: quadratic, not exponential growth.
>Preprocessing phase in O(M) space and time complexity. Searching phase average O(N) time complexity and O(N*M) worst case complexity.
I don't trust the analysis of someone referring to "average O(N) time"; Big O notation refers to boundary times.
Edit: Okay, based on arguments here and on [1], I'm going to accept that maybe he's just bastardizing the notation.
[1] http://stackoverflow.com/questions/3905355/meaning-of-averag...
No it doesn't. You can have an O(N) amortized time. Big-O is a bounding function up to a constant factor, not necessarily a boundary (as in worst-case) time.
I concur with the other commenters that it's silly for you to complain about his not comparing his online string-search algorithm against offline string-search algorithms that search an index of the text, such as suffix-tree algorithms.
As for the online algorithm issue, see my reply below.
Dual-Pivot Quicksort is a demonstration that someone didn't read the existing literature; nothing more.
If, on the other hand, you have a fixed "corpus" and a dynamic query, O(n) search time (this algo, purportedly) is terrible.
"This algorithm especially well suited for long S, multi-substrings, small alphabet, regex/parsing and search for common substrings."
Long source text: the O(n times m) worst-case time per search kills it. Even for a single search, O(n times m) worst case here versus O(n + m) for suffix trees.
multi-substrings: suffix trees do each substring of length m in O(m), but are not compared.
search for common substrings: again, suffix trees would be more appropriate.
(You could also use the O(n) time median algorithm to construct a truly O(n lg n) Quicksort, but that's just a fun theoretical side-note here.)
I think I may have misunderstood what you meant by this. If I have, I apologise, if I haven't...
Talking about the average-case complexity of algorithms with Big-Oh notation is in no way an "abuse" of that notation. If the average time complexity of an algorithm is `O(f(n))`, then its running time averaged over all inputs of size `n` is bounded above (in some sense) by `f(n)`.
I guess that statement doesn't say that the number of comparisons or swaps or "steps" the algorithms must make to complete is `O(n lg n)`. Perhaps that's what you meant.
Using terminology like "exponential" and "quadratic" correctly in a discussion of algorithms is not pedantic...
m^2 is exponential -- in the value 2. But the value 2 doesn't change very rapidly, so we tend to focus on the m component of the expression. :-)
Exponential growth would be 2^m.
Have a look at http://en.wikipedia.org/wiki/Time_complexity , it's pretty comprehensive.
In most academic papers I have read that deal with online pattern matching suffix arrays/trees are generally not compared.
If you want to tell me that BWT is more useful for something like searching an FBI or CIA DNA database, and that intel types want to encourage commercial development of BWT (even if irrational for the customers) to subsidize its covert use in intel -- that I can believe. I can see where it would be helpful not for resequencing (a way to read off interesting details of an individuals genotype based on a lot of reads) but rather for fingerprinting of suspects and surveillance targets against a large database of reference genomes.
As for the FBI's DNA database, this does not consist of sequence data, but, I believe, just microsatellite data. Even if they had full sequence data, it wouldn't make sense to search for new samples against the whole database, but to align the sample with a reference genome and then match the variations across a database of variations.
BWT =~ suffix tree with far far greater memory efficiency
It's true that dual-pivot quicksort is a step in the direction of Samplesort. (I do understand Samplesort well enough to say that.) That doesn't mean it's not a worthwhile contribution in its own right. Jon Bentley (advisor of Brian Reid, Ousterhout, Josh Bloch, and Gosling while at CMU; later at Bell Labs in its glory days) was quoted as having a substantially different opinion from yours:
http://permalink.gmane.org/gmane.comp.java.openjdk.core-libs...
> I think that Vladimir's contributions to Quicksort go way beyond
> anything that I've ever done, and rank up there with Hoare's original
> design and Sedgewick's analysis. I feel so privileged to play a very,
> very minor role in helping Vladimir with the most excellent work!
I am not sure Bentley will be persuaded by your claim that he didn't read the existing literature.
Here's an excerpt from the email I wrote to Prof. Sedgewick back then; he didn't reply, so I've no idea what he thinks about this:
"I believe that your PhD thesis studies this under the name of "double partition modification". I could find no real difference between the two algorithms. Yaroslavsky claims to achieve a factor of 0.8 improvements over the standard quicksort on the number of exchanges (neutral on comparisons), while your analysis led you to reject the technique after concluding it required many more comparisons than the standard quicksort. I've only attempted very cursorily to reconcile the math; it seems that both your thesis and Yaroslavsky arrive at 0.8n(log n) as the average number of swaps, but you have (1/3) n(log n) as the average number of swaps in the classic algorithm while Yaroslavsky has 1.0 n(log n) for the same thing, so what he sees as an improvement you viewed as a disaster. My interpretation of this may be wrong, and I haven't attempted to verify either analysis yet."
The "classic algorithm" in the above is the classic quicksort as presented by Sedgewick in the thesis, which is slightly different than e.g. the standard Java implementation. I lost interest soon thereafter and stopped investigating this, but it shouldn't be too difficult to find out who was right in that algorithm comparison, if someone's interested enough.
It's entirely possible that a modification which was a significant loss in 1978 is a significant gain in 2008.
Bentley's comments from that email don't make any sense to me. I can't understand what he was thinking, unless he's just unusually prone to flattering other people's work (I don't know Bentley personally, but I've met other people who do this, much to my irritation).
Why isn't it used everywhere that people currently use Quicksort? I've never actually heard of anyone using Samplesort except in a parallel context.
Essentially, any alignment technique that can come close enough to I/O saturating reading both the dominant factor (the reads) and the minor factor (the reference) is optimal.
When (for a slightly different kind of gapped, error-prone read) I wrote alignment software, I also beat MAQ by 10-20X, like the BWA algorithm but without using anything like the BWT hair. I contemplated how the gapping rules and error rules looked in a brute force NFA translation. I found simple ways to optimize that NFA very cheaply. I found that it paid to maximize read-data throughput by devoting all of memory to the NFA built from as many reads as possible. It was not hard, this way, to get in the ballpark (small enough constant factor) of I/O bounding read processing (where "not hard" means something like 3-6 months of staring at walls and scribbling stuff on paper until I found some solutions that were simple enough it is only bad luck I didn't find them right off the bat). Building a prefix or suffix tree or any other kind of index to the reference was the approach my boss suggested at the start and that I explicitly set out to beat (and did!). Streaming the reference is not the bottleneck if you can do it in an I/O bound way because (for resequencing, at least), the size of the read data is much larger and you can do alignment while coming within a small constant of I/O saturating the input of that read data using a simpler NFA approach.
Looking at the description of MAQ, besting its performance by any number of means is not so surprising. Reaching for the hair of using BWT does surprise me.
Its different for an imagined database where you are trying to align reads against a lot of personal genomes, e.g., to identify someone given a pretty complete genomic database of the usual suspects plus a few reads from the crime scene -- more around-the-corner "GATTACA" scenario than today's FBI, perhaps. There, perhaps, its economic to dedicate one server to ever N "usual suspects", loading the server with BWT of the suspect database, then stream reads out to the servers rather than loading servers with set of reads and streaming a single-human reference over that. (Of course, maybe you could just align the crime scene reads against a single human reference and then .... So even in GATTACA land use of BWT seems like dubious hair against a lightweight opponent like MAQ)
Last I heard, python's built-in sort was a type of samplesort, though, so it isn't completely unused.
Python's built-in sort is timsort, last I heard, which is a variant of mergesort that looks for existing sorted or backwards runs in order to run faster in the common cases of mostly-sorted or mostly-backwards data. It's true that Python's built-in sort was a samplesort variant until 2002, though, which I didn't know.
(It does seem like Samplesort's virtue of minimizing comparisons would be particularly desirable in an environment like Python, where every comparison potentially involves a call into slow interpreted code.)
- From an input sequence of length n, choose a random subsequence of length k.
- Sort the subsequence.
- Partition the remaining (n-k) elements of the original sequence into the (k+1) partitions given by the subsequence.
- Sort the (k+1) partitions.
In the paper, the partitioning and sorting is always done using quicksort. Any n*log(n) sorting algorithm should work though, which includes using samplesort recursively.
If you use an optimal value for k (and this might be a significant fraction of n), you can prove that lim_{n->infty} E(C_n)/log(n!) = 1, where E(C_n) is the expected number of comparisons for a sequence of length n.
Now, using samplesort recursively, quicksort is samplesort with k = 1. Double-Pivot quicksort is samplesort with k = 2.
