1. Less Than Matching
Orgad Keller

2. Orgad Keller - Algorithms 2 - Recitation 12
Less Than Matching
Input: A text , a pattern over alphabet with order relation .
Output: All locations where
Can we use the regular methods?
Orgad Keller - Algorithms 2 - Recitation 12

3. Orgad Keller - Algorithms 2 - Recitation 12
Transitivity
Less Than Matching is in fact transitive, but that is not enough for us:
does not imply anything about the relation between and .
Orgad Keller - Algorithms 2 - Recitation 12

4. Orgad Keller - Algorithms 2 - Recitation 12
Approach
A good approach for solving Pattern Matching problems is sometimes solving:
The problem for a binary alphabet
The problem for a bounded alphabet
The problem for an ubounded alphabet .
In that order.
Orgad Keller - Algorithms 2 - Recitation 12

5. Orgad Keller - Algorithms 2 - Recitation 12
Binary Alphabet
The only case that prevents a match at location is the case where:
This is equivalent to:
So how can we solve this case?
Orgad Keller - Algorithms 2 - Recitation 12

6. Orgad Keller - Algorithms 2 - Recitation 12
Binary Alphabet
So if , there is no match at .
We can calculate
Then we’ll calculate using FFT.
We’ll return all locations where
Time:
Orgad Keller - Algorithms 2 - Recitation 12

7. Orgad Keller - Algorithms 2 - Recitation 12
Bounded Alphabet
We need reductions to binary alphabet.
For each we’ll define:
We notice are binary.
Orgad Keller - Algorithms 2 - Recitation 12

8. Orgad Keller - Algorithms 2 - Recitation 12
Bounded Alphabet
Theorem: (less than) matches at location if and only if , (less than) matches at location .
Proof: does not match at iff
that is true iff , meaning that does not (less than) match at location .
Orgad Keller - Algorithms 2 - Recitation 12

9. Orgad Keller - Algorithms 2 - Recitation 12
Bounded Alphabet
So for each , we’ll run the binary alphabet algorithm on
We’ll return only the locations that matched in all iterations.
Time:
Orgad Keller - Algorithms 2 - Recitation 12

10. Orgad Keller - Algorithms 2 - Recitation 12
Unbounded Alphabet
Running the bounded alphabet algorithm could result in a time algorithms (We’ll run it only for alphabet symbols which are actually in pattern).
Can be worse than the naïve algorithm.
We present an improvement on the next slides.
Orgad Keller - Algorithms 2 - Recitation 12

11. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
First, use the segment splitting trick. Therefore we can assume
For each location in text, we’ll produce a triplet: , where
For each location in pattern, we’ll produce a triplet: , where
We now have triplets all together.
Orgad Keller - Algorithms 2 - Recitation 12

12. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
We’ll hold all triplets together.
Sort all triplets according to symbol.
We’ll define a symbol that has more than triplets as a “frequent symbol”.
There are frequent symbols.
Put all frequent symbols’ triplets aside.
Orgad Keller - Algorithms 2 - Recitation 12

13. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
Split non-frequent symbols’ triplets to groups of size in the following manner:
Orgad Keller - Algorithms 2 - Recitation 12

14. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
The rule is that there can’t be two triplets of the same symbol in different groups.
Orgad Keller - Algorithms 2 - Recitation 12

15. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
For each such group, choose the symbol of the first triplet in group as the group’s representative.
For instance, on previous example, group 1’s representative is and group 2’s representative is .
There are representatives all together.
Orgad Keller - Algorithms 2 - Recitation 12

16. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
To sum up:
frequent symbols.
representatives of non-frequent symbols.
We’ll swap each non-frequent symbol in pattern and text with its representative.
Now our text and pattern are over sized alphabet.
Orgad Keller - Algorithms 2 - Recitation 12

17. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
We want to run our algorithm over the new text and pattern to count the mismatches between symbols of different groups.
But we have a problem:
Let’s say is a frequent symbol, but:
Orgad Keller - Algorithms 2 - Recitation 12

18. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
The representative of group 2 is , which is smaller than , but the group also contains which is greater than .
Orgad Keller - Algorithms 2 - Recitation 12

19. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
In that case we’ll split group 2 to two groups with their own representatives.
Since we performed at most such splits, we still have representatives.
Orgad Keller - Algorithms 2 - Recitation 12

20. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
We can now run our algorithm over the new text and pattern in
But we still haven’t handled comparisons between two non-frequent symbols that are in the same group.
Orgad Keller - Algorithms 2 - Recitation 12

21. Orgad Keller - Algorithms 2 - Recitation 12
Abrahamson-Kosaraju Method
We’ll do so naively in each group:
For each triplet in the group
For each triplet of the form in the group, if , then add an error at location .
Time:
Orgad Keller - Algorithms 2 - Recitation 12

22. Orgad Keller - Algorithms 2 - Recitation 12
Running Time
For one segment:
Sorting the triplets and representatives:
Running the algorithm:
Correcting results (Adding in-group errors):
Overall for one segment:
Overall for all segments:
Orgad Keller - Algorithms 2 - Recitation 12

23. Orgad Keller - Algorithms 2 - Recitation 12
Running Time
We can improve to
Left as an exercise.
Orgad Keller - Algorithms 2 - Recitation 12