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Witness Tables

Witness Tables. Tsvi Kopelowitz Modified by Orgad Keller. Patten Matching. Input: A (large) text and a (smaller) pattern . Output: All locations where pattern matches text, i.e. all locations where

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Witness Tables

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  1. Witness Tables Tsvi Kopelowitz Modified by Orgad Keller

  2. Patten Matching • Input: A (large) text and a (smaller) pattern . • Output: All locations where pattern matches text, i.e. all locations where • In the lecture you will see an algorithm using a Witness Table.

  3. Witness Tables • A witness table is an array W of size m • W[i]= • * if p1p2…pm-i+1= pipi+1…pm • k if p1p2…pk-i= pipi+1…pk-1 and pk pk-i+1 • Example:

  4. Example: P: * 7 8 3 8 6 5 2 W:

  5. Another Example

  6. Construction • Naïve: O(m2) • Idea: when calculating W[i], use W[1]…W[i-1] • Let W[k] be the maximum of {W[2], W[3],…,W[i-1]} • What is W[k] for i=5? i=10? i=12? i=18? • What is k for i=5? i=10? i=12? i=18?

  7. A Few Notes • W[1] isn’t interesting. • If i>1, and W[i]=*, then we can treat W[i] as being m+1.

  8. Construction • So we compared alignment 1 and alignment k, but we will also want to compare alignments k and i. We already compared these. We want to compare these. W[k] i i-k Also already compared these. k-1 i-k i-1

  9. Three (+1) Cases • Case 0 (?): i>W[k] • Compare from i. W[k] i k-1 i-1

  10. Three Cases • Case 1: W[i-k+1]+k-1<W[k] • We found the first mismatch. • W[i]=W[i-k+1]+k-1 iff W[i-k+1]+k-1<W[k] W[i] W[i-k+1] W[k] i W[i-k+1]+k-1 i-k k-1 i-k i-1

  11. Three Cases • Case 2: W[i-k+1]+k-1>W[k] • W[i]=W[k]! W[i-k+1] W[k] i i-k k-1 i-1

  12. ? ? = Three Cases We have already “scanned” the pattern till W[k] • Case 3: W[i-k+1]+k-1=W[k] • If W[i-k+1]+k-1=W[k] compare from W[k] • Total of O(m) construction time. W[k] W[i] W[i-k+1] i i-k ? k-1 i-1 ?

  13. Example: k W[i-k+1]+k-1=W[5-4+1]+4-1=W[2]+3=5<6 W[i-k+1]+k-1=W[6-4+1]+4-1=W[3]+3=6=6=W[k] W:

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