Search methods datastructures - search

I have a problem with a question in a paper that was handed out to me which I am hoping that someone will be able to give me some help with. The question is:
a) The little book of birds with 100 headwords and the big book of birds with 1000 headwords is stored in a computer searchable form. It takes approximately 50% as long time to search through the big bird book as the small bird book. Which search method is used?
b) Now the storing- and search method is exchanged. Now it takes the same amount of time to search through both books. Which is the new search method?
So the first question is of complexity O(log(n)) and I don't know any search method with that time complexity. The second one should be of order O(1), as they take the same amount of time??

Binary search has asymptotic complexity O(log n). If the length of headwords is assumed to be constant, a hash-based algorithm can retrieve elements in constant time, see e.g hash tables.

Related

Find the most similar text in a list of strings

First of all, I want to acknowledge that this is perhaps a very sophisticated problem; however, I have not been able to find a definitive answer for it online so I'm looking for suggestions.
Suppose I want to collect a list containing over hundred thousand strings, values of these strings are sentences that a user has typed. The values are added to the list as soon as a user types a new message. For example:
["Hello world!", "Good morning, my name is John", "Good morning, everyone"]
But I also want to have a timeout for each string so if they are not repeated within 5 min, they should be removed, so I change it to following format:
[{message:"Hello world!", timeout: NodeJS.Timeout, count: 1}, {message:"Good morning, my name is John", timeout: NodeJS.Timeout, count: 1}, {message:"Good morning, everyone", timeout: NodeJS.Timeout, count: 1}]
Now suppose a user types the following message:
Good morning, everyBODY
I want to compare this string to all the messages in list and if one is 70% or more similar, update the count of that message, otherwise insert it as a new message. For this message for example, the application should update the count for Good morning, everyone to be equal to 2.
Since users can type a lot of messages in a short amount of time, the algorithm must also support fast insertion, searching, and deleting after the timeout.
What is the best way to implement this? or are there any libraries to help me with this?
NOTE: The strings do not need to be in an array, any data structure would work.
The main purpose of this algorithm is to detect similar messages when the count reaches a predefined value. For example warning: Over 5 users typed messages similar to "Hello everybody" within 5 minutes
I have looked at B-Trees, Nearest Neighbor, etc but I can't figure out what would be the best solution.
Update:
I plan on using Levenshtein distance for string similarity, however the main problem is how to apply that to a list of strings in most time efficient way, without having to check every single string every time a new message is added.
Levenshtein distance
Unlike the other answer I think Levenshtein distance is perfectly capable of dealing with spelling mistakes. Indeed, Levenshtein and LevXenshtein only have Levenshtein distance 1, and thus can be concluded to likely be the same message.
However, if you want to use this distance, you will have to compute the distance between the new message and every message stored, every time a new message comes in. There is likely no way around this.
Unfortunately there is no real useful pre-processing you can do for this.
Other possibilities
If you can find a way to map every message to a fixed-size vector, you can use essentially any nearest neighbor search technique. I suggest doing so.
This leaves us with two problems to solve. Generating the fixed-length vector, and doing the search.
Fixed-size vector representation
There are multiple ways of doing this, all with their own set of drawbacks. I'll specifically mention two, but it will depend on your architecture and data which method is best for you.
First, you could go the machine-learning way. You could map every word to a pre-trained vector with fastText, average the words in the message, and use that as your vector. The drawbacks of this method are that it will ignore word order, and it will work less well if the words used tend to be very informal. If your messages have their own culture to them (such as for example Twitch chat) you would have to retrain these vectors instead of using pre-trained ones.
Alternatively, you could use the structure of the text directly, and make an occurrence vector of bigrams. That is, jot down how often every 2-character combination occurs in a message. This is fairly robust, but has the drawback that the vectors will become relatively large.
Regardless, these are just two options, and it's impossible to tell what method is ideal for you. Unless of course someone has a brilliant idea.
Nearest neighbor search
Given that we have fixed length vectors, we can now do nearest neighbor search. As you've probably found, there are once again many different methods for this, all with their own drawbacks. Exhausting, I know.
I'll choose to discuss three categories.
Approximate search: This method may seem a little silly, but it could be what you want. Specifically, Locality-sensitive hashing is essentially just making some hashing function where "similar" vectors are likely to end up in the same bucket. You could then do anything you want, such as Levenshtein, with all of the other members of the bucket, because there should not be too many of them. The advantage of such an approximate algorithm is that it can be fast, and with some smart hashing you don't even need fixed-length vectors. A downside, of course, is that it is not guaranteed to work.
Exact search: We can also choose to instead solve the problem of Fixed-radius near neighbors. That is, find the points within some distance of the target point. You could do this by mapping vectors to integers (if they aren't already) and simply checking every lattice point within the distance you want to search. The primary drawback here is that the search time grows very fast not with the number of points, but with the number of dimensions of the vector. This method would necessitate small vectors.
Fancy datastructures: This seems to me most likely to be the right solution. Unfortunately you have a lot of letter-trees. You mention B-trees, but there's also R-trees, R+-Trees, R*-Trees, X-Trees, and that's just the direct descendants of the R-tree. With the risk of missing the trees for the forest, I'd suggest taking a look at the k-d tree. It can do nearest neighbor search in logarithmic time, as well as insertion and deletion.
You want to covert all of the words to their Soundex value.
Then you need a database for the soundex values that ranks the importance of the word in the sentence, e.g. the should probably get 0. The more information the word carries the higher its value.
Then sort the words in the sentence into a list of integers.
Use the list of integers as the key to find similar sentences.
Since the key is a list of integers a Rose tree should work as data structure.
While some may suggest measuring using something like Levenshtein distance that presupposes that the sentences have no spelling mistakes or such. You need something that is flexible enough to deal with human error.
I would suggest you to use Algolia. Which has their own ranking algorithm rates each matching record on several criteria (such as the number of typos or the geo-distance), to which they individually assign a integer value score.
I would totally take a loook on it, since they have Search-as-you-type and different Ranking algorithm criterias.
https://blog.algolia.com/search-ranking-algorithm-unveiled/
I think Search Engine like SOLR or Elastic Search are best fit for your problem.
You have to create single collection in which you can store data as you have mention in the question after that you just have to add data to solr and search it in the solr search with your time limit.

Efficient sampling of discrete random variable [duplicate]

I have a list of US names and their respective names from the US census website. I would like to generate a random name from this list using the given probability. The data is here: US Census data
I have seen algorithms like the roulette wheel selection algorithm that are easy to implement, but I wanted to know if there was any way to generate random names in O(1). For histogram data this is easier, as you could create a hash of integers to birthdays, but I would like to do this for a continuous distribution.
If this is not possible, are there any python modules that take in probability distributions and generate random values based on those distributions?
There is an O(1)-time method See this detailed description of Vose's "alias" method. Unfortunately, it suffers from high initialization cost. For comparative timings of simpler methods, see Eli Bendersky's blog post. More timings can be found in this from the Python issue tracker.
These days it's practical to enumerate the entire US population (~317 million) if you really need O(1) lookup. Just pick a number up to 317 million and get the name from there. (317000000*4 bytes = 1.268GB)
I think there are lots of O(log n) ways. Is there a particular reason you need O(1) (They will use a lot less memory)

Most efficient algorithm for comparing a string with a group of strings

I have a scenario where a user can post a number of responses or phrases via a form field. I would like to be able to take the response and determine what they are asking for. For instance if the user types in car, train, bike, jet .... I can assume they are talking about a vehicle, and respond accordingly. I understand that I could use a switch statement or perhaps a regexp as well, however the larger the number of possible responses, the less efficient that computation will be. I'm wondering if there is an efficient algorithm for comparing a string with a group of strings. Any info would be great.
You may want to look into the Aho-Corasick algorithm. If you have a collection of strings that you want to search for, you can spend linear time doing preprocessing on those strings and from that point forward can, in O(n) time, check for all possible matches of those strings in a text corpus of length n. In other words, with a small preprocessing time to set up the algorithm once, you can extremely efficiently scan over numerous inputs again and again searching for those keywords.
Interestingly enough, the algorithm was specifically invented to build a fast index (that is, to look for a lot of different keywords in a huge body of text), and allegedly outperformed other methods by a factor of ten. I think it would work great in your application.
Hope this helps!
If you have a large number of "magic" words, I would suggest splitting the query into words, and using a hash-based lookup to check whether the words are recognized.
You can check Trie structure. I think one of best solution for your problem.

How do search engines conduct 'AND' operation?

Consider the following search results:
Google for 'David' - 591 millions hits in 0.28 sec
Google for 'John' - 785 millions hits in 0.18 sec
OK. Pages are indexed, it only needs to look up the count and the first few items in the index table, so speed is understandable.
Now consider the following search with AND operation:
Google for 'David John' ('David' AND 'John') - 173 millions hits in 0.25 sec
This makes me ticked ;) How on earth can search engines get the result of AND operations on gigantic datasets so fast? I see the following two ways to conduct the task and both are terrible:
You conduct the search of 'David'. Take the gigantic temp table and conduct a search of 'John' on it. HOWEVER, the temp table is not indexed by 'John', so brute force search is needed. That just won't compute within 0.25 sec no matter what HW you have.
Indexing by all possible word
combinations like 'David John'. Then
we face a combinatorial explosion on the number of keys and
not even Google has the storage
capacity to handle that.
And you can AND together as many search phrases as you want and you still get answers under a 0.5 sec! How?
What Markus wrote about Google processing the query on many machines in parallel is correct.
In addition, there are information retrieval algorithms that make this job a little bit easier. The classic way to do it is to build an inverted index which consists of postings lists - a list for each term of all the documents that contain that term, in order.
When a query with two terms is searched, conceptually, you would take the postings lists for each of the two terms ('david' and 'john'), and walk along them, looking for documents that are in both lists. If both lists are ordered the same way, this can be done in O(N). Granted, N is still huge, which is why this will be done on hundreds of machines in parallel.
Also, there may be additional tricks. For example, if the highest-ranked documents were placed higher on the lists, then maybe the algorithm could decide that it found the 10 best results without walking the entire lists. It would then guess at the remaining number of results (based on the size of the two lists).
I think you're approaching the problem from the wrong angle.
Google doesn't have a tables/indices on a single machine. Instead they partition their dataset heavily across their servers. Reports indicate that as many as 1000 physical machines are involved in every single query!
With that amount of computing power it's "simply" (used highly ironically) a matter of ensuring that every machine completes their work in fractions of a second.
Reading about Google technology and infrastructure is very inspiring and highly educational. I'd recommend reading up on BigTable, MapReduce and the Google File System.
Google have an archive of their publications available with lots of juicy information about their techologies. This thread on metafilter also provides some insight to the enourmous amount of hardware needed to run a search engine.
I don't know how google does it, but I can tell you how I did it when a client needed something similar:
It starts with an inverted index, as described by Avi. That's just a table listing, for every word in every document, the document id, the word, and a score for the word's relevance in that document. (Another approach is to index each appearance of the word individually along with its position, but that wasn't required in this case.)
From there, it's even simpler than Avi's description - there's no need to do a separate search for each term. Standard database summary operations can easily do that in a single pass:
SELECT document_id, sum(score) total_score, count(score) matches FROM rev_index
WHERE word IN ('david', 'john') GROUP BY document_id HAVING matches = 2
ORDER BY total_score DESC
This will return the IDs of all documents which have scores for both 'David' and 'John' (i.e., both words appear), ordered by some approximation of relevance and will take about the same time to execute regardless of how many or how few terms you're looking for, since IN performance is not affected much by the size of the target set and it's using a simple count to determine whether all terms were matched or not.
Note that this simplistic method just adds the 'David' score and the 'John' score together to determine overall relevance; it doesn't take the order/proximity/etc. of the names into account. Once again, I'm sure that google does factor that into their scores, but my client didn't need it.
I did something similar to this years ago on a 16 bit machine. The dataset had an upper limit of around 110,000 records (it was a cemetery, so finite limit on burials) so I setup a series of bitmaps each containing 128K bits.
The search for "david" resulting in me setting the relevant bit in one of the bitmaps to signify that the record had the word "david" in it. Did the same for 'john' in a second bitmap.
Then all you need to do is a binary 'and' of the two bitmaps, and the resulting bitmap tells you which record numbers had both 'david' and 'john' in them. Quick scan of the resulting bitmap gives you back the list of records that match both terms.
This technique wouldn't work for google though, so consider this my $0.02 worth.

What tried and true algorithms for suggesting related articles are out there?

Pretty common situation, I'd wager. You have a blog or news site and you have plenty of articles or blags or whatever you call them, and you want to, at the bottom of each, suggest others that seem to be related.
Let's assume very little metadata about each item. That is, no tags, categories. Treat as one big blob of text, including the title and author name.
How do you go about finding the possibly related documents?
I'm rather interested in the actual algorithm, not ready solutions, although I'd be ok with taking a look at something implemented in ruby or python, or relying on mysql or pgsql.
edit: the current answer is pretty good but I'd like to see more. Maybe some really bare example code for a thing or two.
This is a pretty big topic -- in addition to the answers people come up with here, I recommend tracking down the syllabi for a couple of information retrieval classes and checking out the textbooks and papers assigned for them. That said, here's a brief overview from my own grad-school days:
The simplest approach is called a bag of words. Each document is reduced to a sparse vector of {word: wordcount} pairs, and you can throw a NaiveBayes (or some other) classifier at the set of vectors that represents your set of documents, or compute similarity scores between each bag and every other bag (this is called k-nearest-neighbour classification). KNN is fast for lookup, but requires O(n^2) storage for the score matrix; however, for a blog, n isn't very large. For something the size of a large newspaper, KNN rapidly becomes impractical, so an on-the-fly classification algorithm is sometimes better. In that case, you might consider a ranking support vector machine. SVMs are neat because they don't constrain you to linear similarity measures, and are still quite fast.
Stemming is a common preprocessing step for bag-of-words techniques; this involves reducing morphologically related words, such as "cat" and "cats", "Bob" and "Bob's", or "similar" and "similarly", down to their roots before computing the bag of words. There are a bunch of different stemming algorithms out there; the Wikipedia page has links to several implementations.
If bag-of-words similarity isn't good enough, you can abstract it up a layer to bag-of-N-grams similarity, where you create the vector that represents a document based on pairs or triples of words. (You can use 4-tuples or even larger tuples, but in practice this doesn't help much.) This has the disadvantage of producing much larger vectors, and classification will accordingly take more work, but the matches you get will be much closer syntactically. OTOH, you probably don't need this for semantic similarity; it's better for stuff like plagiarism detection. Chunking, or reducing a document down to lightweight parse trees, can also be used (there are classification algorithms for trees), but this is more useful for things like the authorship problem ("given a document of unknown origin, who wrote it?").
Perhaps more useful for your use case is concept mining, which involves mapping words to concepts (using a thesaurus such as WordNet), then classifying documents based on similarity between concepts used. This often ends up being more efficient than word-based similarity classification, since the mapping from words to concepts is reductive, but the preprocessing step can be rather time-consuming.
Finally, there's discourse parsing, which involves parsing documents for their semantic structure; you can run similarity classifiers on discourse trees the same way you can on chunked documents.
These pretty much all involve generating metadata from unstructured text; doing direct comparisons between raw blocks of text is intractable, so people preprocess documents into metadata first.
You should read the book "Programming Collective Intelligence: Building Smart Web 2.0 Applications" (ISBN 0596529325)!
For some method and code: First ask yourself, whether you want to find direct similarities based on word matches, or whether you want to show similar articles that may not directly relate to the current one, but belong to the same cluster of articles.
See Cluster analysis / Partitional clustering.
A very simple (but theoretical and slow) method for finding direct similarities would be:
Preprocess:
Store flat word list per article (do not remove duplicate words).
"Cross join" the articles: count number of words in article A that match same words in article B. You now have a matrix int word_matches[narticles][narticles] (you should not store it like that, similarity of A->B is same as B->A, so a sparse matrix saves almost half the space).
Normalize the word_matches counts to range 0..1! (find max count, then divide any count by this) - you should store floats there, not ints ;)
Find similar articles:
select the X articles with highest matches from word_matches
This is a typical case of Document Classification which is studied in every class of Machine Learning. If you like statistics, mathematics and computer science, I recommend that you have a look at the unsupervised methods like kmeans++, Bayesian methods and LDA. In particular, Bayesian methods are pretty good at what are you looking for, their only problem is being slow (but unless you run a very large site, that shouldn't bother you much).
On a more practical and less theoretical approach, I recommend that you have a look a this and this other great code examples.
A small vector-space-model search engine in Ruby. The basic idea is that two documents are related if they contain the same words. So we count the occurrence of words in each document and then compute the cosine between these vectors (each terms has a fixed index, if it appears there is a 1 at that index, if not a zero). Cosine will be 1.0 if two documents have all terms common, and 0.0 if they have no common terms. You can directly translate that to % values.
terms = Hash.new{|h,k|h[k]=h.size}
docs = DATA.collect { |line|
name = line.match(/^\d+/)
words = line.downcase.scan(/[a-z]+/)
vector = []
words.each { |word| vector[terms[word]] = 1 }
{:name=>name,:vector=>vector}
}
current = docs.first # or any other
docs.sort_by { |doc|
# assume we have defined cosine on arrays
doc[:vector].cosine(current[:vector])
}
related = docs[1..5].collect{|doc|doc[:name]}
puts related
__END__
0 Human machine interface for Lab ABC computer applications
1 A survey of user opinion of computer system response time
2 The EPS user interface management system
3 System and human system engineering testing of EPS
4 Relation of user-perceived response time to error measurement
5 The generation of random, binary, unordered trees
6 The intersection graph of paths in trees
7 Graph minors IV: Widths of trees and well-quasi-ordering
8 Graph minors: A survey
the definition of Array#cosine is left as an exercise to the reader (should deal with nil values and different lengths, but well for that we got Array#zip right?)
BTW, the example documents are taken from the SVD paper by Deerwester etal :)
Some time ago I implemented something similiar. Maybe this idea is now outdated, but I hope it can help.
I ran a ASP 3.0 website for programming common tasks and started from this principle: user have a doubt and will stay on website as long he/she can find interesting content on that subject.
When an user arrived, I started an ASP 3.0 Session object and recorded all user navigation, just like a linked list. At Session.OnEnd event, I take first link, look for next link and incremented a counter column like:
<Article Title="Cookie problem A">
<NextPage Title="Cookie problem B" Count="5" />
<NextPage Title="Cookie problem C" Count="2" />
</Article>
So, to check related articles I just had to list top n NextPage entities, ordered by counter column descending.

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