1. 2-Dimensional Pattern Matching
Amihood Amir, Dina Sokol, Shoshana Neuburger
UWSL 2006

2. 2-Dimensional Pattern Matching
Perform pattern matching on images
MRI
FAX

3. Searching Aerial Photographs

4. Historic Two Dimensional Model:

5. 2D Pattern Matching - Example
Input: = {A,B} Pattern: Text Output: { (1,4),(2,2),(4, 3)} A B A B A B A B A B

6. Bird-Baker Algorithm (1976)
Time: for bounded fixed
alphabets.
for infinite
Technique: linearization.

7. Bird / Baker First linear-time 2D pattern matching algorithm.
View each pattern row as a metacharacter to linearize problem.
Convert 2D pattern matching to 1D.

8. Linearization Concatenate rows of Text and use string matching tools.
In this case – The Aho and Corasick algorithm for a dictionary of patterns.
The dictionary consists of all pattern rows.

9. Find all pattern rows… then align them.

10. Bird / Baker Preprocess pattern:
Name rows of pattern using AC automaton.
Using names, pattern has 1D representation.
Construct KMP automaton of pattern.
Identical rows receive identical names.

11. Bird / Baker - Example Preprocess pattern:
Name rows of pattern using AC automaton.
Using names, pattern has 1D representation.
Construct KMP automaton of pattern. A B 1 2

12. Bird / Baker
Scan text:
Name positions of text that match a row of pattern, using AC automaton within each row.
Run KMP on named columns of text.
Since the 1D names are unique, only one name can be given to a text location.

13. Bird / Baker - Example Scan text:
Name positions of text that match a row of pattern, using AC automaton within each row.
Run KMP on named columns of text. A B 2 1 2 1

14. Another linearization- pad with “don’t cares”m
n-m
Time: Fischer-Paterson (1972)

15. Witnesses Popular paradigm in pattern matching:
find consistent candidates
verify candidates
consistent candidates → verification is linear

16. Dueling Algorithm

17. Data Structure List of potential candidates
R = rightmost element of that list
N = new element R N

18. Case 1: N dies X R N N

19. Case 2: R dies X R N

20. Case 3: noone dies add N to list of consistent candidates
Since N is consistent with R, and R is consistent with the rest of the list,
by transitivity, N is consistent with the list

21. Witnesses
Vishkin introduced the duel to choose between two candidates by checking the value of a witness.
Alphabet-independent method.

22. Dueling Paradigm [Vishkin 1985]T=
witness
i j ? b a
A duel chooses between two possible candidates by checking the value of a ‘witness.’

23. Witness Table P T Witness table
A witness table is a table of size |P|, which stores a location of a conflict for each location of P (w/r to left cand). P
Witness table T
i j

24. Dueling Method in 2D How do we arrange for candidates to agree on
overlap? – duel!
When there is conflict between two candidates, a single text check eliminates at least one candidate.
The text location can be pre-computed because of transitivity.
The dueling phase is thus linear time.
A A A A A A A
A A A A V A A
A A A A A A A
A A A A V A A

25. A duel in 2-dimensions
Witness[3,3]=(4,3) a b 1 2 3 4 a b b

26. 2-D Witness Table P Witness Table a b *
A 2-D Witness table is a table of size m2, storing a witness for each location of P. P
Witness Table a b *
4,3
4,2
4,1

27. 2D Witnesses
Amir et. al. – 2D witness table can be used for linear time and space alphabet-independent 2D matching.
The order of duels is significant, it is done in 2 waves:
1: duel within each column, bottom to top.
2: duel between columns from right to left.

28. First Truly 2d Algorithm – The Dueling Method
(A-Benson-
Farach 1991)
Once duels are over, the situation is:
All potential pattern
“starts” agree on
overlap.
A i.e. all want to see
the same symbol
in every text location.

29. Verification
Do a forward wave down the columns to label starts of pattern rows.
Do a forward wave on each row, beginning anew for each new row. Label positions of mismatch.
Kill all candidates that contain a mismatch (using 2 similar backwards waves)

30. Dueling Method …
Time for checking every text element’s correctness: linear.
Every candidate with incorrect element in its range is eliminated.
Method: The “wave”.
Total Time:

31. 2D Dictionary Matching
Suppose we are given a set of 2d patterns, called a dictionary.
Goal: search for all Patterns in Text simultaneously, in linear time.
Bird/Baker can be extended, if all patterns have uniform width.
(How?)