Multidimensional Period Recovery€¦ · aaaaaaaaaa aabbaabbaa aaaaaaaaaa aaabaaabaa aabaaabaaa...

Transcript of Multidimensional Period Recovery€¦ · aaaaaaaaaa aabbaabbaa aaaaaaaaaa aaabaaabaa aabaaabaaa...

Department of Computer Science, Bar Ilan University, IsraelDepartment of Computer Science, Holon Institute of Technology, Israel

Department of Software Engineering, Shenkar College, Ramat Gan, IsraelDepartment of Computer and Information Science, Brooklyn College of the

City University of New York, USA

Multidimensional Period Recovery

Amihood Amir, Ayelet Butman, Eitan Kondratovsky,Avivit Levy, Dina Sokol

SPIRE 2020Orlando, FL, USA
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol

a a a b a a a b a a a

The Problem


The Problem


The Problem

period


The Problem

period


The Problem

period tail


The Problem

period tail

periodic string


The Problem

period must occur at least twice

periodic string


The Problem


The Problem


The Problem

a

a a a b a a a a a a a

The Problem


The Problem


The Problem

a a a b a a a a b a a

The Problem


The Problem


The Problem

a a a b a a a b a a aOriginal text

Candidate text


The Problem

a a a b a a a b a a aOriginal text

Candidate text

The 1D Problem

Input:Text with substitution errors

Output:𝐶 = the set of candidates s.t.

among them should be theoriginal text without the errors

The Multidimensional Problem

Input:d-dimensional text with errors



The 2D Problem

Input:2-dimensional text with errors



The 2D Problem

a a a b a a a b a aa a b a a a b a a aa a a b a a a b a aa a b a a a b a a aa a a b a a a b a aa a b a a a b a a aa a a b a a a b a a

The 2D Problem


The 2D Problem


The 2D Problem

a a a a a a a b a aa a b a a a b b a aa a a a a a a b a aa a b a a a b b a aa a a a a a a b a aa a b a a a b b a aa a a a a a a b a a

The 2D Problem


The 2D Problem


The 2D Problem

a a a a a a a a a aa a b b a a b b a aa a a a a a a a a aa a b b a a b b a aa a a a a a a a a aa a b b a a b b a aa a a a a a a a a a

The 2D Problem


The 2D Problem


The 2D Problem



Original text IndistinguishableCandidate

The 2D Problem




The 2D Problem




The 2D Problem




The 2D Problem


Root

Repetition

The 2D Problem


Root

Repetition

Horizontal Tails

Vertical Tails

The 2D Problem


Root must fully occur atleast twice vertically andat least twice horizontally

The 2D Problem




The ProblemProvide an upper bound on the number of errorss.t. the candidates set size is feasible

1D: 𝑇 ∈ Σ! 𝑝 unknown by the algorithm#𝑒 < 𝑔(𝑛, 𝑝) then #𝑐 = 𝑂 𝑓 𝑛 =

= 𝑂(𝑙𝑜𝑔 𝑛)

2D: 𝑇 ∈ Σ!×# 𝑝×𝑞 unknown by the algorithm#𝑒 < 𝑔(𝑛,𝑚, 𝑝, 𝑞) then #𝑐 = 𝑂 𝑓 𝑛,𝑚 =

= 𝑂(𝑙𝑜𝑔 𝑛𝑚)

Previous WorkOne Dimensional Recovery

Bound on errors Candidates set size

TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛

2 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log! 𝑛)

Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛" log 𝑛)



TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛

2 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 𝑛)

Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")



TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")



TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")

SPIRE 2018[KRRSWZ]

Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 𝑛)



TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")

SPIRE 2018[KRRSWZ]

Edit Dist. <𝑛




TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")

SPIRE 2018[KRRSWZ]

Edit Dist. <𝑛


Our paper Hamming Dist. ≤

12 + 𝜀

𝑛𝑝

Generalization to multi dim. case

𝑂(log 𝑛) 𝑂(𝑛 log 𝑛)

ESA 2020 [ABIK]

Hamming Dist.

≤𝑛2 ⋅ 𝑝

2log" 𝑛 𝑂(𝑛 log 𝑛)



TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")

SPIRE 2018[KRRSWZ]

Edit Dist. <𝑛



12 + 𝜀

𝑛𝑝



ESA 2020 [ABIK]

Hamming Dist.

≤𝑛2 ⋅ 𝑝




TimeComplexity

ACM 2012[AELPS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛


TCS 2018[AALS]

Hamming Dist.

<𝑛


Edit Dist. <𝑛

3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛#/")

SPIRE 2018[KRRSWZ]

Edit Dist. <𝑛



12 + 𝜀

𝑛𝑝



ESA 2020 [ABIK]

Hamming Dist.

≤𝑛2 ⋅ 𝑝


Intuition

Too many corruptions lead to a huge number of indistinguishable candidates

Upper Bound on The Number of Corruptions

S = a2k b a4k+1


S = a2k b a4k+1

n = 6k+2


S = a2k b a4k+1

n = 6k+2

T=

a2k S a4k+1a2k S a4k+1a2k … a4k+1a2k S a4k+1a2k S a4k+1


S = a2k b a4k+1

n = 6k+2

T=

a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1


T=


n = 6k+2

m


T=



T=


=

a2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2k

C1=


a2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2k


T=


=

a2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2k

C1=


a2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2k


T=


=

a2k b a a2k a a a2k-2a2k b a a2k a a a2k-2a2k b a a2k a a a2k-2a2k b a a2k a a a2k-2a2k b a a2k a a a2k-2

C2=

a2k b a a2k b a a2k-2a2k b a a2k b a a2k-2a2k b a a2k b a a2k-2a2k b a a2k b a a2k-2a2k b a a2k b a a2k-2


T=


=


C2=



T=



C2=



T=



C2=


Periodic candidates = n/6

Conclusion

∃ family of 2-dimensional texts s.t.

#𝑒 ≤ !"#$

%&

then #𝑐 = Ω(𝑛)

(For any 𝐷 ≥ 2)

Our Goal

Our Goal is to prove that

#𝑒 ≤ !"'(

#$

%&

then #𝑐 = 𝑂(𝑙𝑜𝑔 𝑛)

The Naïve Approach

By a direct reduction from the previousresults we can prove

If #𝑒 ≤ !"'(

#$

%&

then #𝑐 =

𝑂(𝑙𝑜𝑔 𝑛 𝑙𝑜𝑔𝑚)

In the d-dimensional case

#𝑐 = 𝑂(𝑙𝑜𝑔) 𝑁)𝑑

P

Q

PQ

P

Q

Lemma 4

𝑇* - The 𝑛×𝑚 repetition of 𝑃 (𝑝!×𝑞!)𝑇+ - The 𝑛×𝑚 repetition of 𝑄 (𝑝"×𝑞")

Lemma 4

𝑇* - The 𝑛×𝑚 repetition of 𝑃 (𝑝!×𝑞!)𝑇+ - The 𝑛×𝑚 repetition of 𝑄 (𝑝"×𝑞")

If P is wider than Q, higher than Q, or boththen,

𝐻𝑎𝑚 𝑇*, 𝑇+ ≥#$$

%&$

Symmetrically, If Q wider or higher than P𝐻𝑎𝑚 𝑇*, 𝑇+ ≥

#$%

%&%

First Usage Of Lemma 4Candidates Hierarchical Structure

P

Q

Lemma 5: The case where P widerand Q higher is not possible


We can use Lemma 4 twice

P

Q

𝑚𝑎𝑥𝑛𝑝!

𝑚𝑞!

,𝑛𝑝"

𝑚𝑞"

≤ 𝐻𝑎𝑚 𝑇*, 𝑇+



𝑚𝑞!

,𝑛𝑝"

𝑚𝑞"


≤ 𝐻𝑎𝑚 𝑇*, 𝑇 + 𝐻𝑎𝑚 𝑇+ , 𝑇

≤ 21

2 + 𝜀𝑛𝑝!

𝑚𝑞!

<𝑛𝑝!

𝑚𝑞!

In contradiction

T.I.E



𝑚𝑞!

,𝑛𝑝"

𝑚𝑞"


≤ 𝐻𝑎𝑚 𝑇*, 𝑇 + 𝐻𝑎𝑚 𝑇+ , 𝑇

≤ 21

2 + 𝜀𝑛𝑝!

𝑚𝑞!

<𝑛𝑝!

𝑚𝑞!

In contradiction

T.I.E

Second Usage Of Lemma 4Logarithmic Amount of Candidates

1 + 𝜀 ⋅ 𝑎𝑟𝑒𝑎 𝑄 ≤ 𝑎𝑟𝑒𝑎 𝑃

𝑎𝑟𝑒𝑎 𝑃 ≤ 𝑛𝑚

1 + 𝜀 #- ≤ 𝑛𝑚

#𝑐 ≤ log!'( 𝑛𝑚 = 𝑂(log𝑁)

PQ

T.I.E + Lemma 4




1 + 𝜀 #- ≤ 𝑛𝑚


PQ

T.I.E + Lemma 4




1 + 𝜀 #- ≤ 𝑛𝑚


PQ

T.I.E + Lemma 4

Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P

The algorithm


Proof:#$

%&

is the number of full occ. of P.

𝐻𝑎𝑚 𝑇*, 𝑇 ≤1

2 + 𝜀𝑛𝑝

𝑚𝑞

The algorithm


Proof:#$

%&



2 + 𝜀𝑛𝑝

𝑚𝑞

The algorithmP P P P PP P P P PP P P P PP P P P P

P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’


Proof:#$

%&



2 + 𝜀𝑛𝑝

𝑚𝑞

P P P P PP P P P PP P P P PP P P P P

P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’

𝑛𝑝

𝑚𝑞


Proof:#$

%&



2 + 𝜀𝑛𝑝

𝑚𝑞


P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’

𝑛𝑝

𝑚𝑞


Proof:#$

%&



2 + 𝜀𝑛𝑝

𝑚𝑞


P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’

𝑚𝑞

𝑛𝑝

Conclusion 1:At least half of the full occ. are without

any corruptions

Stage 1P P P P PP P P P PP P P P PP P P P P

P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’

The algorithm at its first stage findsat most one candidate for any dim. 𝑝 × 𝑞


P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’



P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’



P’

P’

P’

P’

P’’ P’’ P’’ P’’ P’’ P’’’

Using renaming scheme with majority alg.

Stage 2

We need to verify for any candidatethat


2 + 𝜀𝑛𝑝

𝑚𝑞

We rely on the kangaroo jump technique

Algorithm Complexity

The overall algorithm complexities are

𝑂(𝑛𝑚 log 𝑛 log𝑚) time and space

And it reports a set of size 𝑂(log 𝑛𝑚)

Open Problems

Consider different definitions of multidimensional repetitions

Observing other distances• Edit distance• Swap distance• Etc.

Multidimensional Period Recovery€¦ · aaaaaaaaaa aabbaabbaa aaaaaaaaaa aaabaaabaa aabaaabaaa...

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Transcript of Multidimensional Period Recovery€¦ · aaaaaaaaaa aabbaabbaa aaaaaaaaaa aaabaaabaa aabaaabaaa...