1. S C A L E D Pattern Matching Amihood Amir Ayelet Butman Bar-Ilan University Moshe Lewenstein and Johns Hopkins University Bar-Ilan University

2. Motivation Searching for Templates in Aerial Photographs Input: Aerial photo Template Task: Search for all locations where the template appears in the image.

4. Model Low level (pixel level) avoid costly processing Asymptotically efficient solutions. Serial, exact algorithms.

5. Types of Approximations Local errors: Level of detail Occlusion Noise results: O(n² log m) mismatches O(n²k²( edit distance, k errors, rectangular patterns. O(n²k√(m log m) √(k log k) edit distance, k errors, half rectangular patterns AL-88 AF-95

6. Types of Approximation Orientation. results: O(n²m ) FU-98 O(n²m³) ACL-98 Scaling: Natural scales: results: O(n) 1-d EV-88 O(n² log |Σ|) 2-d ALV-92 O(n²) dictionary AC-96 Real scales: this result: O(n) 1-d, truncation 5

7. It seems daunting, but…

8. CPM 2003: Morelia, Mexico

9. Problem inherently inexact What if occurrence is 1½ times bigger? What is the meaning of “½ a pixel”? Solutions until now: Natural Scales - Consider only discrete scales: 1, 2, 3, 4, 5,...

10. Definition: Text: Pattern: Find all occurrences of the pattern in the text in all discrete sizes. m m n n

11. Discrete exact Scaled Matching T P A A A A A A A A A A A A A A A A A A C C A A C A A A A C C A A A A A C C A A A A A A A C C A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A C C A A A A A C C C A A A A A A A A A A C C C A A A A C A A A A A A A A A A A A A A A A A A A A A A A A C C A C A A A A A A A A A A A A A

12. Discrete exact Scaled Matching P Z U Y K V S X E T P³ Z Z Z U U U Y Y Y K K K V V V S S S X X X E E E T T T

13. Idea: Fix a scale s Constant amount of work for each square (s-block) s s n n/s

14. Algorithm time Time for scale s: Total time: converges to a constant Making the total time O(n²)

15. Problem: Real scales Was open even for strings … How do we define? aabcccbb Scaled to 2: aaaabbccccccbbbb Scaled to 1½: aaab cccc bbb truncate truncate ½b ½c

16. Formally: Denote: a aaa... a Problem Definition 1: Input: Pattern Text: Output: All text locations where appears for some r times r

17. Remark α ≥ 1 means we only scale “up” Reasons: Avoid conceptual problem of loss of resolution. From “far enough” away everything looks the same. By our definition, for k<1/m there is a match at every text location.

18. Simplify definition Definition 2: Look for in the text. Example: P=aabcccbbbb Match by definition 2: daaabccccbbbbbbe Match by definition 1 but not by def 2: daaaabccccbbbbbbbe

19. Why are definitions equivalent? Split text and pattern to symbol part T s, P s and length part T L, P L. Example: P= aabcccbbbb P s =abcb P L =2134 T=daaabccccbbbbbbe T s =dabcbe T L =131461

20. Time Time for split: O(n+m) Finding P s in T s : O(n+m) (e.g. KMP) HARD PART: Finding P L in T L.

21. Definitions are Equivalent Claim: Solving def 2 in time O(f(n)) Solving def 1 in time O(f(n)). Why? - Find in time O(f(n)) - For each match verify 1 st and last symbol in constant time in T s and T L. Total time: O(f(n)+n)=O(f(n)).

22. Na ï ve algorithm for matching P L in T L For each text location, position pattern starting at that location and calculate interval [t/p, (t+1)/p) for each resulting pair. This is the interval of possible scales since t/p·p = t for every α < t/p, |αp| < t (t+1)/p ·p = t+1 for every α ≥ t/p, |αp| > t

23. Check intersection If intersection of all intervals is not empty then there is a match. Time: O(nm) Example: P L : 2 1 2 3 2 T L : 2 4 2 4 7 4 5 3 [1,3/2) [4,5) The intersection is empty thus no scaled match in location 1. But…

24. Check intersection If intersection of all intervals is not empty then there is a match. Time: O(nm) Example: P L : 2 1 2 3 2 T L : 2 4 2 4 7 4 5 3 [2,5/2) [2,3) [2,5/2)[7/3,8/3)[2,5/2) The intersection is [7/3,5/2) thus there is a scaled match in location 2.

25. Improvement – Parameterized Matching Introduced: Baker 1994. Motivation: “copying” code.

26. Parameterized Matching Input: two strings s and t |s|=|t|, over alphabets ∑ s and ∑ t. s parameterize matches t: if bijection : ∑ s ∑ t, such that (s) = t. (a)=x (b)=y aa b bb xx yyy Example:

27. Parameterized Matching Claim (AFM-94): For Σ that can be sorted in linear time (e.g. Σ={1,..., n}) Parameterized matching can be done in time O(n).

28. The reduction Lemma: for which P L matches T L at location i scaled to α only if P L p-matches T L at i. Proof: Assume P L does not p-match T L at location i. The possible situations are:

29. Possibility 1 w.l.o.g. c ≥ a+1 For c = a+1 (smallest possible): TLTL PLPL a bb c≠a

30. Possibility 2 w.l.o.g. c ≥ b+1 Intersection not empty only if: (a+1)/(b+1) > a/b i.e. ab+b > ab+a b>a But this can never happen if α ≥ 1. TLTL PLPL a bc≠b a

31. Algorithm for Real Scaled String Matching Let { P i 1, P i 2,..., P i j } be the different numbers in P L. 1.P-match P L in T L. 2.For each match, chack intersection of intervals between P i 1,..., P i j and corresponding symbols in T L. End Algorithm

32. P L = 2 3 2 3 2 P i 1 =2 P i 2 =3 p-matches T L = 5 6 5 6 5 6 10 6 10 6 10 7 scaled match Example:

33. Important Fact: So there are at most O(√m) different P i k ’s. Time: O(n) for parameterized matching (Σ={1,2,…,n}). O(√m) verification for each location. Total: O(n√m).

34. Tighter analysis Upper bound number of possible p-matches. Lemma: Let |P|=m, |T|=n, { P i 1, P i 2,..., P i j } be the different numbers in P L. Then there are at most n/2j p-matches of P L in T L. Meaning: Since verification time is O(j) per p-match, the lemma implies that total verification time is: O((n/2j) · j) = O(n)

35. Proof of Lemma: 1 st appearance of P i 1,..., P i j P L P i 1 P i 2 P i j T L a 1 a 2 a j m-match

36. Lemma’s proof (cont.) Let x be the total number of p-matches in the text. The sum of all text elements that match 1 st occurrences of P i k ‘s in the pattern ≥ (xj²)/2 But: There are overlaps! How many?

37. Lemma’s proof (cont.) For each text location, at most j matches will count it. Therefore… Total count without overlaps ≥ Clearly: x·j/2 ≤ n thus x ≤ (2n)/j

38. Open Problem: Give 1-d algorithm linear in run-length compressed text and pattern.