1. Pattern Matching With Don’t Cares Clifford & Clifford’s Algorithm
Orgad Keller
Modified by Ariel Rosenfeld

2. Orgad Keller - Algorithms 2 - Further Reading
PM with Don’t Cares
Input: A text , a pattern over alphabet ( is the don’t care symbol).
Output: All locations where
Orgad Keller - Algorithms 2 - Further Reading

3. Orgad Keller - Algorithms 2 - Further Reading
Idea
Let be a symbol from the pattern and be a symbol from the text.
As mentioned, In our matching relation, iff:
Orgad Keller - Algorithms 2 - Further Reading

4. Orgad Keller - Algorithms 2 - Further Reading
Idea
we encode as 0, which is like saying:
Orgad Keller - Algorithms 2 - Further Reading

5. Orgad Keller - Algorithms 2 - Further Reading
Idea
As mentioned, we seek All locations where
As we’ve seen, that’s true iff:
Orgad Keller - Algorithms 2 - Further Reading

6. Orgad Keller - Algorithms 2 - Further Reading
Problem
We have a problem because, generally speaking:
but the opposite is not true!
On the other hand, if we know that
then
Orgad Keller - Algorithms 2 - Further Reading

7. Orgad Keller - Algorithms 2 - Further Reading
Problem
So we can easily fix the problem:
Now:
Orgad Keller - Algorithms 2 - Further Reading

8. Orgad Keller - Algorithms 2 - Further Reading
Idea (cont’d)
To sum up, we got that:
Problem left: How can we compute
for all efficiently?
Orgad Keller - Algorithms 2 - Further Reading

9. Orgad Keller - Algorithms 2 - Further Reading
Computing Efficiently
Orgad Keller - Algorithms 2 - Further Reading

10. Orgad Keller - Algorithms 2 - Further Reading
Convolutions
We can easily compute ,
and for all ‘s using FFT-based
multiplication in time
Then we’ll return all locations where
Orgad Keller - Algorithms 2 - Further Reading

11. Orgad Keller - Algorithms 2 - Further Reading

12. Orgad Keller - Algorithms 2 - Further Reading
Running Time
Most costly operations are 3 polynomial multiplications, so overall time is
Orgad Keller - Algorithms 2 - Further Reading