Deterministic acyclic finite state automaton

In computer science, a deterministic acyclic finite state automaton (DAFSA),^[1] 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,^[1] while keeping them minimal. DAFSA is the rediscovery of a data structure called Directed Acyclic Word Graph (DAWG),^[2] although the same name had already been given to a different data structure which is related to suffix automaton.^[3]

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]

Blumer et al^[3] first defined terminology Directed Acyclic Word Graph (DAWG) in 1983. Appel and Jacobsen^[2] used the same naming for a different data structure in 1988. Independent of earlier work, Daciuk et al^[1] rediscovered the latter data structure in 2000 but called it DAFSA.

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 12 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 v₀ to v₁ labeled "t", two edges from v₁ to v₂ labeled "a" and "o", an edge from v₂ to v₃ labeled "p", an edge v₃ to v₄ labeled "s", and edges from v₃ and v₄ to v₅ 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.^[4] The auxiliary information can then be stored in an array.

• Blumer, A.; Blumer, J.; Haussler, D.; Ehrenfeucht, A.; Chen, M.T.; Seiferas, J. (1985), "The smallest automaton recognizing the subwords of a text", Theoretical Computer Science, 40: 31–55, doi:10.1016/0304-3975(85)90157-4
• Appel, Andrew; Jacobsen, Guy (1988), "The World's Fastest Scrabble Program" (PDF), Communications of the ACM, 31 (5): 572–578, doi:10.1145/42411.42420. 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, vol. 453, Springer-Verlag, pp. 318–326, doi:10.1007/BFb0030372, ISBN 3-540-53000-2.
• 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, vol. 3340, Springer-Verlag, pp. 175–187, ISBN 3-540-24014-4, Zbl 1117.68454
• Tresoldi, Tiago (2020), "DAFSA: a Python library for Deterministic Acyclic Finite State Automata", Journal of Open Source Software, 5 (46): 1986, doi:10.21105/joss.01986, hdl:21.11116/0000-0005-AD0D-B An open source Python implementation.

Wikimedia Commons has media related to Deterministic acyclic finite state automaton.

• "Directed Acyclic Word Graph or DAWG" – JohnPaul Adamovsky teaches how to construct a DAFSA using an array of integers (Archived 22 July 2022 at the Wayback Machine)
• "Caroline Word Graph or CWG" – JohnPaul Adamovsky teaches how to construct a DAFSA hash function using a novel encoding with multiple integer arrays (Archived 27 July 2022 at the Wayback Machine)