Orgad Keller

Less Than Matching

Orgad Keller - Algorithms 2 - Recitation 12 2

Less Than Matching

Input: A text , a pattern

over alphabet with order relation . Output: All locations where

Can we use the regular methods?

i

0 1, j i jj m p t

0 1... nT t t 0 1... mP p p

i jt

jp

iT

P

Orgad Keller - Algorithms 2 - Recitation 12 3

Transitivity

Less Than Matching is in fact transitive, but that is not enough for us:

does not imply anything about the relation between and .

,a c b c a b

Orgad Keller - Algorithms 2 - Recitation 12 4

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.

0,1

Orgad Keller - Algorithms 2 - Recitation 12 5

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?

0 1, 1 0j i jj m p t

i

0 1, 1 1j i jj m p t

Orgad Keller - Algorithms 2 - Recitation 12 6

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: .

1

0

0m

j i jj

p t

i

0 1... nT t t RT P

( )[ 1] 0RT P i m

( log )O n m

i

Orgad Keller - Algorithms 2 - Recitation 12 7

Bounded Alphabet

We need reductions to binary alphabet. For each we’ll define:

We notice are binary.

0 1

1

0

...

ii

i

n

tt

t

T t t

0 1

1

0

...

ii

i

m

pp

p

P p p

,T P

Orgad Keller - Algorithms 2 - Recitation 12 8

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 .

PP T

iT i

P T i, j i jj p t

1 0j i jp t

P

iT

Orgad Keller - Algorithms 2 - Recitation 12 9

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: .

,T P

( log )O n m

Orgad Keller - Algorithms 2 - Recitation 12 10

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.

(min , log )O m n m

Orgad Keller - Algorithms 2 - Recitation 12 11

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.

Abrahamson-Kosaraju Method

2T m

( , ' ', )a T ii

ip bi

( , ' ', )b P i

3m

it a

Orgad Keller - Algorithms 2 - Recitation 12 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.

m

( )O m

Orgad Keller - Algorithms 2 - Recitation 12 13

Abrahamson-Kosaraju Method

Split non-frequent symbols’ triplets to groups of size in the following manner:

2m S m

2 1

3 2

Group 1

1 3

2 4

( , ' ', 4), ( , ' ',7),..., ( , ' ',300) , ( , ' ',3),..., ( , ' ', 200) ,

( , ' ',5),..., ( , ' ',1000) , ( , ' ',5),..., ( , ' ',150)

m m

m m

a T a T a P b T b T

d P d T g P g T

Group 2

,...

Orgad Keller - Algorithms 2 - Recitation 12 14

Abrahamson-Kosaraju Method

The rule is that there can’t be two triplets of the same symbol in different groups.

2 1

3 2

Group 1

1 3

2 4

( , ' ', 4), ( , ' ',7),..., ( , ' ',300) , ( , ' ',3),..., ( , ' ', 200) ,

( , ' ',5),..., ( , ' ',1000) , ( , ' ',5),..., ( , ' ',150)

m m

m m

a T a T a P b T b T

d P d T g P g T

Group 2

,...

Orgad Keller - Algorithms 2 - Recitation 12 15

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.

ad

( )O m

Orgad Keller - Algorithms 2 - Recitation 12 16

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.

( )O m

( )O m

( )O m

Orgad Keller - Algorithms 2 - Recitation 12 17

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:

1 3

2 4

Group 2

..., ( , ' ',5),..., ( , ' ',1000) , ( , ' ',5),..., ( , ' ',150) ,...

m m

d P d T g P g T

f

Orgad Keller - Algorithms 2 - Recitation 12 18

Abrahamson-Kosaraju Method

The representative of group 2 is , which is smaller than , but the group also contains which is greater than .

1 3

2 4

Group 2

..., ( , ' ',5),..., ( , ' ',1000) , ( , ' ',5),..., ( , ' ',150) ,...

m m

d P d T g P g T

ff

d

g

Orgad Keller - Algorithms 2 - Recitation 12 19

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.

1 3

2 4

Group 2.1 Group 2.2

..., ( , ' ',5),..., ( , ' ',1000) , ( , ' ',5),..., ( , ' ',150) ,...

m m

d P d T g P g T

( )O m

( )O m

Orgad Keller - Algorithms 2 - Recitation 12 20

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.

( log )O mm m

Orgad Keller - Algorithms 2 - Recitation 12 21

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: ( )O m m

( , ' ', )P j ( , ' ', )T k

i k j

ktjp

iT

P

j kp t

i j

Orgad Keller - Algorithms 2 - Recitation 12 22

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: .

( log )O m m

( log )O mm m

( )O m m

( log )O m m m

( log )O n m m

Orgad Keller - Algorithms 2 - Recitation 12 23

Running Time

We can improve to .Left as an exercise.

( log )O n m m