overlap matching

Overlap Overlap MatchingMatching

By Itamar NabriskiBy Itamar Nabriski

A. Amir, R. Cole, G. Landau, R. Hariharan, M. Lewenstein, E. Porat, Overlap Matching, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms (2001) 279-288,.

Lecture StructureLecture Structure

Discrete ConvolutionsDiscrete Convolutions Overlap Matching Problem Overlap Matching Problem

DefinitionDefinition Overlap Matching AlgorithmOverlap Matching Algorithm Reduction from Swap MatchingReduction from Swap Matching

Discrete Discrete ConvolutionsConvolutions

Definition 1Definition 1Let T be a function whose domain is {0,…,n-1}Let T be a function whose domain is {0,…,n-1}

Let P be a function whose domain is {0,…,m-1}Let P be a function whose domain is {0,…,m-1}

(we’ll view them as arrays of numbers of length n and m (we’ll view them as arrays of numbers of length n and m respectively)respectively)

][][])[(m

iPijTjPT

},...,0{ mnj

The Convolution of T and P at index j is defined as follows:

Exempli GratiaExempli Gratia

TT=Ronaldinho =Ronaldinho

(n=10)(n=10)PP=Deco =Deco

(m=4)(m=4)

(TxP)[2]=

nnDD+a+aee+l+lcc+d+doo

(assume, for now, each letter represents a number)

Thus Naïve Computional Time is O(m)

RRoonnaallddiinnhhoo

DDeeccoo

(TxP)[2]=

number of possible convolutions is O(n)

Computing All The ConvolutionsComputing All The Convolutions

Naïve ApproachNaïve Approach ::

},...,0{ mnj

For each j pay O(m) time, for a total of O(nm)

Devious ApproachDevious Approach ::Using the “Fast Fourier Transform” (FFT) For each j pay O(log m) time, for a total of O(n log m)

(machine word size must be O(log m) though)

Using ConvolutionsUsing Convolutions

Before we perform convolutions on T and P we preprocess each letter using a constant number of constant time functions (total O(n)), retaining the running time of O(nlogm)

For each index j we can use a constant time function f to format the output of the constant number of convolutions, retaining a running time of O(nlogm)

For each index j we can preform a constant amount of convolutions, retaining the running time of O(nlogm)

Preprocessing

Using several Convolutions

Postprocessing

Using Convolutions - Using Convolutions - ExampleExample

∑ = {a,b}

T = ababbaaa

P = abba

Thus, for example, T and P exactly match at index j = 2 :

aabbaabbbbaaaaaa

aabbbbaa

Testing For Exact Matching at index j

otherwise 0

xif 1prep(x)

Preprocessing functions (x is a letter):

otherwise 0

xif 1(x)prep

When we write prep(S), S being a string, we mean we apply prep to all characters of S

])[()(

jPprepTprep

otherwise 0

0y xif 1),( yxf

Convolutions we will use:

Postprocessing function f :

For every index j, 1]))[()(],)[()(( jTprepPprepjTprepPprepf bbaa

Iff there is an exact matching of P and T at index j.

T = ababbaaa

prepa(T)=10100111

prepb(T)=01011000

P = abba

prepa(P)=0110

prepb(T)=1001

1100110000111111

00111100j =2

prepa(T) X prepa(P)[2]=

= 1*0+0*1+0*1+1*0=0

0011001111000000

11000011

prepb(T) X prepb(P)[2]=

= 0*1+1*0+1*0+0*1=0

F(0,0) = 1 = exact match at 2

Overlap Overlap Matching Matching Problem Problem

DefinitionDefinition

Stractural StringStractural String Linear structure made of segmentsLinear structure made of segments Each segment can be marked or Each segment can be marked or

unmarkedunmarked A Stractural String is a A Stractural String is a

concatenation of segmentsconcatenation of segments

Exampli Gratia

TThhee__WWaallllss__OOff__JJeerriicchhoo

001122334455667788991010111112121212141415151616171718181919

Thus, the actual characters are not

important

Defintion of Overlap MatchingDefintion of Overlap Matching

Stractural String T (text) of length nStractural String T (text) of length n Stractural String P (pattern) of length mStractural String P (pattern) of length m m≤nm≤n Both T and P have some marked Both T and P have some marked

segmentssegments

InputInput

All locations k in T where if P All locations k in T where if P is aligned at k, all marked is aligned at k, all marked segments have overlaps of segments have overlaps of

even lengtheven length

OutputOutput

ExampleExampleT = Franz_Beckenbauer

P = The_Kaiser

FFrraannzz__BBeecckkeennbbaauueerr

TThhee__KKaaiisseerr

FFrraannzz__BBeecckkeennbbaauueerr

TThhee__KKaaiisseerr

Overlaps (j=3 is not a valid overlap match – has an odd overlap):

Overlap Overlap Matching Matching AlgorithmAlgorithm

General PreprocessingGeneral PreprocessingEach segment can Each segment can startstart at either an at either an eveneven or or oddodd index and index and endend at either an at either an eveneven or or oddodd indexindex

We will produce from We will produce from TT four new four new segmentssegments TToooo,T,Teeee,T,Toeoe andand TTeoeo

TToooo will have 1’s in the place of all will have 1’s in the place of all characters belonging to segments that characters belonging to segments that start start andand end at an odd index end at an odd index andand 0’s 0’s otherwiseotherwise , for example:, for example:

TToooo = = 0001111100 0001111100

And analgously for the other segment And analgously for the other segment types …types …

General PreprocessingGeneral PreprocessingSince the pattern Since the pattern PP tends to move around tends to move around we will need to treat its we will need to treat its segment indexessegment indexes a a bit differentlybit differently

We will produce from We will produce from PP eight new eight new segmentssegments POPOoooo,PO,POeeee,PO,POoeoe,PO,POeoeo, , PEPEoooo,PE,PEeeee,PE,PEoeoe andand PE PEeoeo

The bigThe big ‘O’ ‘O’ inin PO POeeee meansmeans the all the all segments in P that start and end at an segments in P that start and end at an even location even location relative to T’s indexrelative to T’s index, , when when PP is aligned to an odd index of is aligned to an odd index of TT

And analgously for the other segment And analgously for the other segment types …types …

(don’t worry there is an example in the next slide …)

General PreprocessingGeneral Preprocessing

001122334455667788T=

000000000000000000Too=

000000001100000000Tee=

000000000000111100Teo=

001111000000000000Toe=

00112233445566P =

Since P is always aligned to T at some index j we treat’s P’s indexes

relative to T, thus:j+j+00

j+j+11

j+j+22

j+j+33

j+j+44

j+j+55

j+j+66

Assume j is now odd, then for that location we will use the four PO’s:

00000000000000POoo =

00000011111100POee =

00000000000000POeo =

11110000000000POoe =

Thus for every location j we have 16 (a constant) possible number of Text-Pattern pairings:

{Too,Tee,Toe,Tee} × {PXoo,PXee,PXoe,PXee}

If we can determine, using convolutions, for each pairing

if it only contains even overlaps we can solve the

Overlap Matching problem in O(n log m) time

X=parity(j)

Case 1Case 1

Case 1 occurs when for Tab, Pcd either a=c or b=d

This covers 12 of the 16 cases.

We now show a solution for when a=c. This covers 8 cases, we use the solution on the reverse* strings of T and P (thus ‘a’ becomes ‘c’ and ‘b’ becomes ‘d’) to solve the 4 remaining cases.* Computing the reverse strings does not alter the run time (do it during general preprocessing)

Case 1 (a=c)Case 1 (a=c)

For every two marked segments St in Tab starting at index x and Sp in Pcd starting at index y:

|x-y| is always even

(since even-even = even and odd-odd =

We now create a convolution that will return 0 for index j iff there is no odd overlap at j

Case 1 (a=c)Case 1 (a=c)

For every segment in Tab we replace the 1’s by an alrenating series of 1’s and -1’s beginning with 1.

00000011111111 00000011--1111--11

000011--1111--1111

001111 = 1 - 1 = 0

In case where we have only even (and/or no) overlaps:

00000011--1111--11

001111111111 = 1 – 1 + 1 = 1 > 0

In case where we have at least one odd overlap:

31 5 6

Case 2Case 2

Case 2 occurs when Toe, Peo (Teo, Poe is symmetric)

If a segment in Toe is contained in a segment in Peo or vice versa then the overlap is even, otherwise overlap is odd.

00111111110000111111111111111100

111111110000111111110000

2 4 8 11

Case 2Case 2Containment Elimination Property

Convolution at index j gives zero if all overlaps are containments, otherwise it gives a positive result .

To achieve this we will actually use 3 convolutions, a combination of their output will give us the desired answer.

Case 2Case 2Fleshing Out The Solution

For each segment St in Toe that starts at index st, replace the segment’s 1’s by st,1…1,-stFor each segment Sp in Peo that starts at index sp, replace the segment’s 1’s by sp,1…1,-sp

00000011111111 000000331111--33

Case 2Case 2Containment:

St, 1, 1, ……………. ………………….1, -St, 1, 1, ……………. ………………….1, -StSt

Sp, 1, 1, …………….1, Sp, 1, 1, …………….1, -Sp-Sp

St, 1, 1, …………..1, -St, 1, 1, …………..1, -StSt

Sp, 1, 1, ……………………….……….1, -Sp, 1, 1, ……………………….……….1, -SpSp

sp + (len(Sp)-2) + -sp =

len(Sp)-2st + (len(St)-2) + -st =

len(St)-2

St, 1, 1, ……………. St, 1, 1, ……………. ………….1, -St………….1, -St

Sp, 1, 1, ………………….1, Sp, 1, 1, ………………….1, -Sp-Sp

sp + (k-2) + -st

k-2St, 1, 1, ……………. St, 1, 1, …………….

………….1, -St………….1, -St

Sp, 1, 1, ………………….1, Sp, 1, 1, ………………….1, -Sp-Sp

st + (k-2) + -sp

No Containment (overlap of length k):

Case 2Case 2Problem 1The indexes of the pattern Peo change for each index j , raising the preprocessing time to O(m) for each convolution!Problem 2We need to find a way to remove “The size of the overlap -2” from the resulting convolution.

len(Sp)-2len(St)-2

sp + (k-2) + -stst + (k-2) + -sp

sp - st 0<

st - sp 0<

Containment

No ContainmentRemove

“overlap - 2”

Case 2Case 2Solving Problem 2

Perform another convolution, The “Overlap Length Convolution” subtract its value from the main convolution.

Every segment both Toe and Peo is replaced by 0,1,1,….1,0 giving us “size of overlap -2” for each overlap. 00000011111111 00000000111100

00000000111100

00111111110000

Overlap of length 4 :

= 0+1+1+0 = 2 = “overlap -2”

The trouble is with the pattern Peo segments whose indexes change in each index j. Instead treat the pattern segments relative to Peo. (“Zero Containment Convolution”)

000000331111--33

0000441111--4400

000000331111--33

0000221111--2200

We created a new problem, overlap convolutions can be negative and thus the overall convolution at index j can turn out to be zero when there is an odd overlap.

00000000001111771111--770000

0000221111--220000

= 2+1-7 = -4

We want to get the benefits of both worlds. Towards that end we’ll add to the result a third convolution “The Shifting Convolution”. This simply corrects the problem caused by using the pattern indexes.

Every segment in T is replaced by 1,0…0,1 and every segment in P is replaced by 0,1…,1,0 and the result is multiplied by index j.

0011000011000011000011

001111111111110000

2 * j = 2 * 2 = 4

This replenishes our “losses”

0011000011000000000000

00111111000000

Thus, the convolution gives 0 for each containment overlap and 1 for each non-containment overlap.

0011000000000000110000

00111111000000

Thus multiplying by j we return “one j” to each non-containment

overlap

Case 2Case 2Final Algorithm

Thus we implement the “Containment Elimination Property” by:

Zero Containment Convolution

Shifting Convolution

Overlap Length Convolution

Containment Elimination Property

Very Powerful Technique!

Amazing! He’s a master of the “Shifting Convolution”

Case 3Case 3

Case 3 occurs when Too, Pee (Tee, Poo is symmetric)

If a segment in Too is contained in a segment in Pee or vice versa then the overlap is odd, otherwise overlap is even.

001111110000001111111111110000

111111000000111111000000

2 3 8 107 13

Case 3 - Using Case 2Case 3 - Using Case 2Containment:

St, 1, 1, ……………. ………………….1, -St, 1, 1, ……………. ………………….1, -StSt

Sp, 1, 1, …………….1, Sp, 1, 1, …………….1, -Sp-Sp

St, 1, 1, …………..1, -St, 1, 1, …………..1, -StSt

Sp, 1, 1, ……………………….……….1, -Sp, 1, 1, ……………………….……….1, -SpSp

sp + (len(Sp)-2) + -sp =

len(Sp)-2st + (len(St)-2) + -st =

len(St)-2

St, 1, 1, ……………. St, 1, 1, ……………. ………….1, -St………….1, -St

Sp, 1, 1, ………………….1, Sp, 1, 1, ………………….1, -Sp-Sp

sp + (k-2) + -st

k-2St, 1, 1, ……………. St, 1, 1, …………….

………….1, -St………….1, -St

Sp, 1, 1, ………………….1, Sp, 1, 1, ………………….1, -Sp-Sp

st + (k-2) + -sp

No Containment (overlap of length k):

Case 3Case 3We’ll use the same convolution as in Case 2 and two additional ones:

Conv1: Every segment in Too of length len replace by 0,1,2,…,len-1.

Replace Pee segments by 1,0,…,0.

00111111111100 00001122334400

Conv2: (Opposite of 1) Every segment in Pee of length len replace by 0,1,2,…,len-1.

Replace Too segments by 1,0,…,0.

0011111100 0011000000

Case 3Case 3

000011223344000000

0011000000= 3

The first convolution gives us the length of all areas like the one marked in green:

00110000000000

0000112200 = 0

If, for some overlap, the first convolution is positive the second will be zero, and vice versa.

It gives us for every two overlapping segments which St is “ahead” of Sp

Case 3Case 3

000000110000000000

00112233440000= 3

The reverse case (second convolution):

And the first one is now zero:

000000001122000000

11000000000000= 0

Case 3Case 3

000011223344556600

0011000000= 3

This is true also for containments:

The convolution from Case 2 gives the same

value for non containments and zero for

containments.

Case 3Case 3

Conv1 + Conv2 – ConvCase2 = positive

= containments = odd overlap

Conv1 + Conv2 – ConvCase2 = 0

= no containments = only even overlaps

Overlap MatchingOverlap Matching

Each Case (1,2,3) takes O(n log m) :

1. A constant number of preprocessing functions O(n)

2. A constant number of convolutions O(n log m)

3. A constant time computable function O(1)

for a total runtime of O(n log m)

Algorithm Final Outcome

Swap MatchingSwap Matching

CCOONNNNEERR__MMAACCLLEEOODD

CCNNOONNRREE__MMAALLCCEEDDOO

Swap MatchingSwap MatchingFormal Definition

Let S =s1,…,s2 be a string over alphabet ∑

A swap permutation for S is a permutation

π : {1,…,n} → {1,…,n} such that:1.If π(i) = j then π(j) = I

2.For all i, π(i) member of { i-1 , i , i+1 }

3. If π(i) ≠ i then sπ(i) ≠ si

Lemma (will not be proven):

A solution to swap matching over alphabet {a,b} of time O(f(n,m)) implies a solution of time O(log|∑|f(n,m)) over alphabet ∑.

And there exists an algorithm to do so.A. Amir, Y. Aumann, G. Landau, M. Lewensten, N. Lewenstein, Pattern matching with swaps, J. Algorithms 37 (2) (2000) 247-266.

aabbbbbbbbaaaaaaaabbaabbbbbb

Maximal Alternating Segment (MAS)

Lemma:

The pattern P does NOT match in a particular alignment iff there exists a MAS A in T and MAS B in P such that:

1.The characters of A and B misalign in the overlap

2.The overlap is of odd length

Swap MatchingSwap MatchingLemma Intuition

aaaaaabbaabbbbbbaabbaa

aaaaaabbaabbbbbbaabbaabb

Even overlap mismatch

Odd overlap mismatch

Swap MatchingSwap MatchingProof →(by contradiction):

Assume P is aligned to T at index j and we can’t swap match and the two MAS A,B do not exist:1.All MAS’ overlaps match exactly –

contradiction – we don’t even need to swap.

2.There exists at least one pair A,B that do not match exactly in an even overlap:w.l.o.g overlapA=(ab)* overlapB=(ba)*

we can swap within the overlap boundaries and get the desired result - contradictionThus, there must be one MAS A,B that have

a misaligned odd overlap

Swap MatchingSwap MatchingProof ←(by contradiction):Assume there exist MAS A,B that misalign in an odd overlap and P and T swap match at index j :

w.l.o.g overlapA=(ab)*a overlapB=(ba)*b

The we must swap with letters outside of the overlap but by definition of MAS this will not help and we can’t swap match. Contradiction.

Swap MatchingSwap MatchingAlgorithm

Construct from T:

1.Teven-a where all MAS with a’s on even indexes are marked segments.

2.Todd-a where all MAS with a’s on odd indexes are marked segments.

aabbaabbbbbbbbaaaaaabbT =

aabbaabbbbbbaaaaaaaabb

aabbaabbbbbbaaaaaaaabbTodd-a =

Teven-a =

2 6 87 9

We provide a similar construction for P :

Peven-a ,Podd-a using P’s index !

When matching, if the index j of T is odd we will use one for the other (Peven-a becomes Podd-a and vice versa)

aabbbbbbaaPeven-a =

Aligned at T ’s index 3 it becomes Podd-a:

aabbbbbbaa

If index j is even, T swap matches P iff Teven-a overlap matches Podd-a at j and Todd-a overlap matches Peven-a at j.

If index j is odd, T swap matches P iff Teven-a overlap matches Peven-a at j and Todd-a overlap matches Podd-a at j.

Swap MatchingSwap MatchingAlgorithm – Why does it work?

An even-a MAS and an odd-a MAS will never exactly match:

By the lemma if their overlap is odd then swap matching is not possible and this is exactly what we examine using the Overlap Matching method

aabbaabbaa

bbaabbaabb

even-a MASodd-a MAS

Runtime O(n log m):

1.We pay O(n) to segmentize to MAS.

2.We pay O(n log m) to run overlap matching.Thus, for an alphabet ∑ we can swap match at O(n log m log|∑|)

Improvement over previous deterministic upper bound of O(nm1/3log m log|∑|)

A. Amir, Y. Aumann, G. Landau, M. Lewensten, N. Lewenstein, Pattern matching with swaps, J. Algorithms 37 (2) (2000) 247-266.

overlap matching

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