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
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
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
period
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
period
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
period tail
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
period tail
periodic string
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
period must occur at least twice
periodic string
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
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
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
a
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a a a a
The Problem
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a a a a
The Problem
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a a a a
The Problem
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a b a a
The Problem
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a b a a
The Problem
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a b a a
The Problem
a a a b a a a b a a aOriginal text
Candidate text
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
a a a b a a a a b a a
The Problem
a a a b a a a b a a aOriginal text
Candidate text
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
The Multidimensional Problem
Input:d-dimensional text with errors
Output:𝐶 = the set of candidates s.t.
among them should be theoriginal text without the errors
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
The 2D Problem
Input:2-dimensional text with errors
Output:𝐶 = the set of candidates s.t.
among them should be theoriginal text without the errors
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
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
Original text IndistinguishableCandidate
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
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
Original text IndistinguishableCandidate
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
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
Original text IndistinguishableCandidate
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
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
Original text IndistinguishableCandidate
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Root
Repetition
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Root
Repetition
Horizontal Tails
Vertical Tails
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Root must fully occur atleast twice vertically andat least twice horizontally
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
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
Original text IndistinguishableCandidate
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
SPIRE 2018[KRRSWZ]
Edit Dist. <𝑛
3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 𝑛)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
SPIRE 2018[KRRSWZ]
Edit Dist. <𝑛
3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 𝑛)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
SPIRE 2018[KRRSWZ]
Edit Dist. <𝑛
3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
SPIRE 2018[KRRSWZ]
Edit Dist. <𝑛
3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
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#/")
SPIRE 2018[KRRSWZ]
Edit Dist. <𝑛
3.75 + 𝜀 ⋅ 𝑝𝑂(log 𝑛) 𝑂(𝑛 log 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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Intuition
Too many corruptions lead to a huge number of indistinguishable candidates
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
S = a2k b a4k+1
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
S = a2k b a4k+1
n = 6k+2
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
S = a2k b a4k+1
n = 6k+2
T=
a2k S a4k+1a2k S a4k+1a2k … a4k+1a2k S a4k+1a2k S a4k+1
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
n = 6k+2
m
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
=
a2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2k
C1=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
a2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2k
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
=
a2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2ka2k b a2k a a2k
C1=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
a2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2ka2k b a2k b a2k
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
=
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
=
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Upper Bound on The Number of Corruptions
T=
a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1a2k b a4k+1
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
Periodic candidates = n/6
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Conclusion
∃ family of 2-dimensional texts s.t.
#𝑒 ≤ !"#$
%&
then #𝑐 = Ω(𝑛)
(For any 𝐷 ≥ 2)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Our Goal
Our Goal is to prove that
#𝑒 ≤ !"'(
#$
%&
then #𝑐 = 𝑂(𝑙𝑜𝑔 𝑛)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
P
Q
PQ
P
Q
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Lemma 4
𝑇* - The 𝑛×𝑚 repetition of 𝑃 (𝑝!×𝑞!)𝑇+ - The 𝑛×𝑚 repetition of 𝑄 (𝑝"×𝑞")
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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+ ≥
#$%
%&%
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
First Usage Of Lemma 4Candidates Hierarchical Structure
P
Q
Lemma 5: The case where P widerand Q higher is not possible
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
First Usage Of Lemma 4Candidates Hierarchical Structure
We can use Lemma 4 twice
P
Q
𝑚𝑎𝑥𝑛𝑝!
𝑚𝑞!
,𝑛𝑝"
𝑚𝑞"
≤ 𝐻𝑎𝑚 𝑇*, 𝑇+
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
First Usage Of Lemma 4Candidates Hierarchical Structure
𝑚𝑎𝑥𝑛𝑝!
𝑚𝑞!
,𝑛𝑝"
𝑚𝑞"
≤ 𝐻𝑎𝑚 𝑇*, 𝑇+
≤ 𝐻𝑎𝑚 𝑇*, 𝑇 + 𝐻𝑎𝑚 𝑇+ , 𝑇
≤ 21
2 + 𝜀𝑛𝑝!
𝑚𝑞!
<𝑛𝑝!
𝑚𝑞!
In contradiction
T.I.E
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
First Usage Of Lemma 4Candidates Hierarchical Structure
𝑚𝑎𝑥𝑛𝑝!
𝑚𝑞!
,𝑛𝑝"
𝑚𝑞"
≤ 𝐻𝑎𝑚 𝑇*, 𝑇+
≤ 𝐻𝑎𝑚 𝑇*, 𝑇 + 𝐻𝑎𝑚 𝑇+ , 𝑇
≤ 21
2 + 𝜀𝑛𝑝!
𝑚𝑞!
<𝑛𝑝!
𝑚𝑞!
In contradiction
T.I.E
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P
The algorithm
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P
Proof:#$
%&
is the number of full occ. of P.
𝐻𝑎𝑚 𝑇*, 𝑇 ≤1
2 + 𝜀𝑛𝑝
𝑚𝑞
The algorithm
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P
Proof:#$
%&
is the number of full occ. of P.
𝐻𝑎𝑚 𝑇*, 𝑇 ≤1
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’’’
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P
Proof:#$
%&
is the number of full occ. of P.
𝐻𝑎𝑚 𝑇*, 𝑇 ≤1
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P
Proof:#$
%&
is the number of full occ. of P.
𝐻𝑎𝑚 𝑇*, 𝑇 ≤1
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
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Observation 1:There is at most one candidate for eachdimensions 𝑝 × 𝑞 of P
Proof:#$
%&
is the number of full occ. of P.
𝐻𝑎𝑚 𝑇*, 𝑇 ≤1
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’’’
𝑚𝑞
𝑛𝑝
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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’’’
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
The algorithm at its first stage findsat most one candidate for any dim. 𝑝 × 𝑞
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’’’
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
The algorithm at its first stage findsat most one candidate for any dim. 𝑝 × 𝑞
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’’’
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
The algorithm at its first stage findsat most one candidate for any dim. 𝑝 × 𝑞
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’’’
Using renaming scheme with majority alg.
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Stage 2
We need to verify for any candidatethat
𝐻𝑎𝑚 𝑇*, 𝑇 ≤1
2 + 𝜀𝑛𝑝
𝑚𝑞
We rely on the kangaroo jump technique
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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)
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
Open Problems
Consider different definitions of multidimensional repetitions
Observing other distances• Edit distance• Swap distance• Etc.
Multidimensional Period RecoveryAmihood Amir, Ayelet Butman, Eitan Kondratovsky, Avivit Levy, Dina Sokol
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