Parameterized Pattern Matching

1. Parameterized Pattern Matching Amihood Amir Martin Farach V. Muthukrishnan

2. 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. Example: a a b b b (a)=x x x y y y (b)=y

3. Parameterized Matching Input: Two strings T, P; |T|=n, |P|=m. Output: All text locations i, such that (P)=Ti …Ti+m-1.

4. Parameterized Matching History • Introduced by Brenda Baker [Baker93]. • Others: [AFM94], [Bak95], [Bak97]. • Two Dimensions: [AACLP03]. • Used in scaled matching [ABL99]. • Periodicity of parameterized matching [ApostolicoGiancarlo]. • Approximate parameterized matching [HLS04].

5. Alternate Definition: Notice: Alphabet bijection between S and T means: S[i] ≈ T[i] for all I Where: S[i] ≈ T[i] if S[i] ≠ S[k], k=1,…,i-1 and T[i] ≠ T[k], k=1,…,i-1 or for all k=1,…,i-1 S[i]=S[k] iff T[i]=T[k]

6. Parameterized Matching Algorithm: Run KMP with the following modifications: • Construct table: A[1],…,A[m] where largest k, 1≤k<i, s.t. P[i]=P[k] A[i]= i , if no such k exists

7. 2. Replace equality checks as follows: Instead of P[i]=T[j]? do: Compare (P[i],T[j]) If A[i]=i and T[j]≠T[k], k=j-i+1,…,j then return equal If A[i]≠i and T[j]=T[j-i+A[i]] then return equal return not equal End

8. Instead of P[i]=P[j]? do: Compare (P[i],P[j]) If (A[i]=i or i-A[i]≥j) and P[j]≠P[k], k=1,…,j then return equal If i-A[i]<j and P[j]=P[j-i+A[i]] then return equal return not equal End

9. Correctness: Automaton construction guarantees that failure arrow points to largest prefix that parameter matches the suffix.

10. TIME: KMP is linear time, but we have a new Compare subroutine. Take text size to be ≤ 2m, and Compare takes time O(log σ), where σ=min(|Σ|,m). This is the time to search if T[j] or P[j] appears in a balanced tree.

11. TIME: Automaton Construction: O(m log σ) . Text Scanning: O(n log σ) . Can we do better?

12. Alphabet Σ={1,…,n} Can be done in linear time. How? Construct array: 1: list of indices of symbol 1 2: list of indices of symbol 2 . . m: list of indices of symbol m.

13. To check if T[j]≠T[k], k=j-i+1,…,j Assume the symbol in T[j] is a. Check if previous index to j in a’s list < j-i+1

14. LOWER BOUNDS What about general alphabets? Element distincness Problem (EDP) Input: Array A[1],…,A[n] of natural numbers. Decide: If all elements of A are distinct (i.e. no i≠j where A[i]=a[j])

15. TIME FOR EDP: In comparison model: General alphabets:Ω(n log n) Alphabet Σ={1,…,n}:linear time. (construct array of indices)

16. Linear Reduction Claim: EDP is linearly reducible to Parameterized Matching. Proof: Let A[1],…,A[n] be an array of numbers. In linear time, check if A[1] is unique. If so, construct S=A[2],A[3],…,A[n],A[1]

17. Linear Reduction (cont.) A ≈ S iff all elements of A are distinct. trivial By induction on the prefixes of A. A[1] is unique – we checked. Assume A[1],…,A[k] are distinct. In particular, A[k] is unique.

18. Linear Reduction (cont.) But A[k] was parameter-matched to S[k], so S[k] only appears once in S. But S[k]=A[k+1]. This means that A[k+1] is unique.