Damn Cool Algorithms: Levenshtein Automata

In a previous Damn Cool Algorithms post, I talked about BK-trees, a clever indexing structure that makes it possible to search for fuzzy matches on a text string based on Levenshtein distance - or any other metric that obeys the triangle inequality. Today, I'm going to describe an alternative approach, which makes it possible to do fuzzy text search in a regular index: Levenshtein automata.

Introduction

The basic insight behind Levenshtein automata is that it's possible to construct a Finite state automaton that recognizes exactly the set of strings within a given Levenshtein distance of a target word. We can then feed in any word, and the automaton will accept or reject it based on whether the Levenshtein distance to the target word is at most the distance specified when we constructed the automaton. Further, due to the nature of FSAs, it will do so in O(n) time with the length of the string being tested. Compare this to the standard Dynamic Programming Levenshtein algorithm, which takes O(mn) time, where m and n are the lengths of the two input words! It's thus immediately apparrent that Levenshtein automaton provide, at a minimum, a faster way for ...

Damn Cool Algorithms: Log structured storage

Typically, if you're designing a storage system - such as a filesystem, or a database - one of your major concerns is how to store the data on disk. You have to take care of allocating space for the objects to be stored, as well as storing the indexing data; you have to worry about what happens when you want to extend an existing object (eg, appending to a file), and you have to take care of fragmentation, which happens when old objects are deleted, and new ones take their place. All of this adds up to a lot of complexity, and the solutions are often buggy or inefficient.

Log structured storage is a technique that takes care of all of these issues. It originated as Log Structured File Systems in the 1980s, but more recently it's seeing increasing use as a way to structure storage in database engines. In its original filesystem application, it suffers from some shortcomings that have precluded widespread adoption, but as we'll see, these are less of an issue for database engines, and Log Structured storage brings additional advantages for a database engine over and above easier storage management.

The basic organization of a ...

Damn Cool Algorithms: Spatial indexing with Quadtrees and Hilbert Curves

Last Thursday night at Oredev, after the sessions, was "Birds of a Feather" - a sort of mini-unconference. Anyone could write up a topic on the whiteboard; interested individuals added their names, and each group got allocated a room to chat about the topic. I joined the "Spatial Indexing" group, and we spent a fascinating hour and a half talking about spatial indexing methods, reminding me of several interesting algorithms and techniques.

Spatial indexing is increasingly important as more and more data and applications are geospatially-enabled. Efficiently querying geospatial data, however, is a considerable challenge: because the data is two-dimensional (or sometimes, more), you can't use standard indexing techniques to query on position. Spatial indexes solve this through a variety of techniques. In this post, we'll cover several - quadtrees, geohashes (not to be confused with geohashing), and space-filling curves - and reveal how they're all interrelated.

Quadtrees are a very straightforward spatial indexing technique. In a Quadtree, each node represents a bounding box covering some part of the space being indexed, with the root node covering the entire area. Each node is either a leaf node - in which case it contains ...

Update on Anagram Trees

Original Post

One nice thing about working at Google is that you are surrounded by very smart people. I told one of my coworkers about the anagram tree idea, and he immediately pointed out that reordering the alphabet so that the least frequently used letters come first would reduce the branching factor early in the tree, which has the effect of reducing the overall size of the tree substantially. While this seems obvious in retrospect, it's kind of unintuitive - usually we try to _increase_ the branching factor of n-ary trees to make them shallower and require fewer operations, rather than trying to reduce it.

Trying it out with an ordering determined by looking at the branching factor for each letter produces results that bear this out: Memory is reduced by about a third, and the number of internal nodes is reduced to 858,858 from 1,874,748, a reduction of more than 50! Though I haven't benchmarked it, difficult lookups are substantially faster, too.

The next logical development to try is to re-evaluate the order of the alphabet on a branch-by-branch basis. While I doubt this will have a substantial impact, it seems worth a try, so ...

Damn Cool Algorithms, Part 3: Anagram Trees

I hesitate to call this algorithm "damn cool", since it's something I invented* it myself, but I think it _is_ rather cool, and it fits the theme of my algorithms posts, so here it is anyway.

When it comes to finding anagrams of words, a frequent approach is to use an anagram dictionary - simply put, sort the letters in your word to provide a unique key that all anagrams of a word have in common. Another approach is to generate a letter-frequency histogram for each letter in your word. (Both these approaches are more or less equivalent, in fact.) These approaches make the problem of finding exact single-word anagrams for strings very efficient - O(1) if you use a hashtable.

However, the problem of finding subset anagrams - a word that contains a subset of the letters in a string - is still rather inefficient, requiring either a brute force O(n) search through the dictionary, or looking up every substring of the sorted input string, which is O(2^l) with the number of letters in the input string. Finding subset anagrams is significantly more interesting, too, as it has applications in finding multi-word anagrams, as well as being applicable ...

Damn Cool Algorithms, Part 2: Secure permutations with block ciphers

It's been too long since I blogged about anything much, and way too long since I posted the first Damn Cool Algorithms post, which I promised would be a series. So here's part 2.

To start, I'm assuming you know what a permutation is - basically a shuffling of a sequence of items in a particular order. A permutation of the range 1-10, for example, is {5,2,1,6,8,4,3,9,7,10}. A secure permutation is one in which an attacker, given any subset of the permutation, cannot determine the order of any other elements. A simple example of this would be to take a cryptographically secure pseudo-random number generator, seed it with a secret key, and use it to shuffle your sequence.

What if you want to generate a really, really big permutation - one so big precomputing and storing it isn't practical or desirable? Furthermore, what if you want it to be a secure permutation? There's a really neat trick we can pull with block ciphers that allows us to generate a secure permutation over any range of numbers without first having to precompute it.

A block cipher, for anyone that ...

Damn Cool Algorithms, Part 1: BK-Trees

This is the first post in (hopefully) a series of posts on Damn Cool Algorithms - essentially, any algorithm I think is really Damn Cool, particularly if it's simple but non-obvious.

BK-Trees, or Burkhard-Keller Trees are a tree-based data structure engineered for quickly finding near-matches to a string, for example, as used by a spelling checker, or when doing a 'fuzzy' search for a term. The aim is to return, for example, "seek" and "peek" if I search for "aeek". What makes BK-Trees so cool is that they take a problem which has no obvious solution besides brute-force search, and present a simple and elegant method for speeding up searches substantially.

BK-Trees were first proposed by Burkhard and Keller in 1973, in their paper "Some approaches to best match file searching". The only copy of this online seems to be in the ACM archive, which is subscription only. Further details, however, are provided in the excellent paper "Fast Approximate String Matching in a Dictionary".

Before we can define BK-Trees, we need to define a couple of preliminaries. In order to index and search our dictionary, we need a way to compare strings. The canonical method for this is the Levenshtein ...