Ternary Directed Acyclic Word Graphs (TDAWG)

1. Ternary Directed Acyclic Word Graphs (TDAWG) Satoru Miyamoto, Shunsuke Inenaga, Masayuki Takeda and Ayumi Shinohara Present by Peera Liewlom (The Last Algorithm Group)

2. CIAA 2003 • Eighth International Conference on Implementation and Application of Automata • July 16-18, 2003, Santa Barbara, CA, USA • Topic / Committee / Community

3. Why did I select this paper ? • DAWG start 1985… not so far • Continueing development • cDAWG, ASDAWG, morphic DAWG, WDAWG, SDAWG, two-tree DAWG, DASG, CSDAWG etc. • TST : 1997 – 98, TDAWG : 2003 • DAWG : Widely Apply by Bioinformatics, NLP, Graph Theory, String Matching, Automata etc. • Speed & Space Trends in Huge Data Management • Topic for Algorithm Group • Matching the interesting topics in this seminar group

4. Content • DFA (use in string matching’s problem) • DAWG • Ternary Search Tree • Paper : TDAWG, Experiment & Result • Paper : Conclusion • Paper : Discussion

5. DFADeterministic Finite Automata

6. Formalities • Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states

7. Set of States

8. Input Aplhabet

9. Initial State

10. Set of Final States

11. Transition Function

15. Transition Function

16. Another Example accept accept accept

17. = { all substrings with prefix } accept

18. = { all strings without substring }

19. DAWGDirected Acyclic Word Graph

20. DAWG

21. DAWG

22. DAWG

23. cDAWG

25. TSTTernary Search Tree

26. TST History • Jon L. Bentley and Robert Sedgewick • Algorithms for Sorting and Searching Strings, Proceeding. 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), January 1997. • Ternary Search Trees, Dr. Dobb's Journal, April 1998. • Dictionary of Algorithms and Data Structures, National Institute of Standard and Technology, http://www.nist.gov/

27. DST BST TST

29. TDAWGTernary Directed Acyclic Word Graph

30. Introduction • DFA  how to implement the transitions of each state ? (Time & Space efficiency) • TST “implant” BST for transitions • Good Time • DAWG smallest DFA for all suffixes • Good Space • TDAWG • Proof : TDAWG VS. DAWG

31. Hypothesis / Theorem (1/2) • Time = Construct + Search (useable for online) • DFA function •  = Alphabet (Chinese & Japan ~ 1000 chars) • State • Table  O(|p|) p = length of pattern • Table use very large memory • Link List  O(| | x |p|) search time • If  is large … problem for search time

32. Hypothesis / Theorem (2/2) • For TDAWG • Use O(|S|) space • Use O(log|| x |p|) for search time • Use O(|| x |S|2) construct time (Bentley & Sedwick) • Use O(|| x |S|) construct time (this paper … apply from Blummer’s online DAWG construction) • Comparison : TDAWG VS. DAWG(table & link list) • Space , Search Time , Construction Time

33. TST  TDAWG

34. Online DAWG Construction

35. Online TDAWG Construction

36. Experiment Result

37. Conclusion • New data structure … TDAWG • Construction time (English text 256) • TDAWG < linklistDAWG < tableDAWG • Space Requirment • linklistDAWG < TDAWG ~ 20 % • tableDAWG not compare in same scale • Search Time • Short pattern: tableDAWG best , TDAWG < linklistDAWG • Log curve VS. Linear Curve (long pattern?)

38. Discussion & Future Work • In Asian Language (characters~1000s) should have better search time than English (character 256) because log(||x|p|) • Apply to other DAWG… cDAWG, minimumDAWG …etc. • More efficiency by AVL tree (AVL-balance) • Bioinformatic have 4 character . But, Sliding window with 12 characters = 412