# Deterministic acyclic finite state automaton

In computer science, a **deterministic acyclic finite state automaton** (**DAFSA**),^{[1]} also called a **directed acyclic word graph** (**DAWG**; though that name also refers to a related data structure that functions as a suffix index^{[2]}) is a data structure that represents a set of strings, and allows for a query operation that tests whether a given string belongs to the set in time proportional to its length. Algorithms exist to construct and maintain such automata, while keeping them minimal.

A DAFSA is a special case of a finite state recognizer that takes the form of a directed acyclic graph with a single source vertex (a vertex with no incoming edges), in which each edge of the graph is labeled by a letter or symbol, and in which each vertex has at most one outgoing edge for each possible letter or symbol. The strings represented by the DAFSA are formed by the symbols on paths in the graph from the source vertex to any sink vertex (a vertex with no outgoing edges). In fact, a deterministic finite state automaton is acyclic if and only if it recognizes a finite set of strings.^{[1]}

## Comparison to tries

By allowing the same vertices to be reached by multiple paths, a DAFSA may use significantly fewer vertices than the strongly related trie data structure. Consider, for example, the four English words "tap", "taps", "top", and "tops". A trie for those four words would have 11 vertices, one for each of the strings formed as a prefix of one of these words, or for one of the words followed by the end-of-string marker. However, a DAFSA can represent these same four words using only six vertices *v _{i}* for 0 ≤

*i*≤ 5, and the following edges: an edge from

*v*

_{0}to

*v*

_{1}labeled "t", two edges from

*v*

_{1}to

*v*

_{2}labeled "a" and "o", an edge from

*v*

_{2}to

*v*

_{3}labeled "p", an edge

*v*

_{3}to

*v*

_{4}labeled "s", and edges from

*v*

_{3}and

*v*

_{4}to

*v*

_{5}labeled with the end-of-string marker. There is a tradeoff between memory and functionality, because a standard DAFSA can tell you if a word exists within it, but it cannot point you to auxiliary information about that word, whereas a trie can.

The primary difference between DAFSA and trie is the elimination of suffix and infix redundancy in storing strings. The trie eliminates prefix redundancy since all common prefixes are shared between strings, such as between *doctors* and *doctorate* the *doctor* prefix is shared. In a DAFSA common suffixes are also shared, for words that have the same set of possible suffixes as each other. For dictionary sets of common English words, this translates into major memory usage reduction.

Because the terminal nodes of a DAFSA can be reached by multiple paths, a DAFSA cannot directly store auxiliary information relating to each path, e.g. a word's frequency in the English language. However, if for each node we store the number of unique paths through that point in the structure, we can use it to retrieve the index of a word, or a word given its index.^{[3]} The auxiliary information can then be stored in an array.

## References

- ↑
^{1.0}^{1.1}Jan Daciuk, Stoyan Mihov, Bruce Watson and Richard Watson (2000). Incremental construction of minimal acyclic finite state automata. Computational Linguistics**26**(1):3-16. - ↑ Black, Paul E. "directed acyclic word graph".
*Dictionary of Algorithms and Data Structures*. NIST.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Kowaltowski, T.; CL Lucchesi (1993). "Applications of finite automata representing large vocabularies".
*Software-Practice and Experience*.**1993**: 15–30. CiteSeerX: 10.1.1.56.5272.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

- Blumer, A.; Blumer, J.; Haussler, D.; Ehrenfeucht, A.; Chen, M.T.; Seiferas, J. (1985), "The smallest automation recognizing the subwords of a text",
*Theoretical computer science*,**40**: 31–55, doi:10.1016/0304-3975(85)90157-4<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Appel, Andrew; Jacobsen, Guy (1988), "The World's Fastest Scrabble Program" (PDF),
*Communications of the ACM*, doi:10.1145/42411.42420<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. One of the early mentions of the data structure. - Jansen, Cees J. A.; Boekee, Dick E. (1990), "On the significance of the directed acyclic word graph in cryptology",
*Advances in Cryptology — AUSCRYPT '90*, Lecture Notes in Computer Science,**453**, Springer-Verlag, pp. 318–326, doi:10.1007/BFb0030372, ISBN 3-540-53000-2<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. - Epifanio, Chiara; Mignosi, Filippo; Shallit, Jeffrey; Venturini, Ilaria (2004), "Sturmian graphs and a conjecture of Moser", in Calude, Cristian S.; Calude, Elena; Dineen, Michael J. (eds.),
*Developments in language theory. Proceedings, 8th international conference (DLT 2004), Auckland, New Zealand, December 2004*, Lecture Notes in Computer Science,**3340**, Springer-Verlag, pp. 175–187, ISBN 3-540-24014-4, Zbl 1117.68454<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## External links

- http://pages.pathcom.com/~vadco/dawg.html - JohnPaul Adamovsky teaches how to construct a DAFSA using an array of integers.
- http://pages.pathcom.com/~vadco/cwg.html - JohnPaul Adamovsky teaches how to construct a DAFSA hash function using a novel encoding with multiple integer arrays. This encoding is called the Caroline Word Graph (CWG).