Mass Error-Correction Codes for Polymer-Based Data Storage Ryan - PowerPoint PPT Presentation

Motivation Problem Statement Construction THANK YOU! Mass Error-Correction Codes for Polymer-Based Data Storage Ryan Gabrys A joint work with S. Pattabiraman and O. Milenkovic ISIT June 8 th , 2020

Motivation Problem Statement Construction THANK YOU! Motivation

Motivation Problem Statement Construction THANK YOU! Protein Sequencing ▸ A protein is a long sequence of amino acids whose composition and order determine the protein’s functionality.

Motivation Problem Statement Construction THANK YOU! Protein Sequencing ▸ A protein is a long sequence of amino acids whose composition and order determine the protein’s functionality. ▸ Mass spectrometry (M/S) has emerged an an important technique for sequencing proteins.

Motivation Problem Statement Construction THANK YOU! Protein Sequencing ▸ A protein is a long sequence of amino acids whose composition and order determine the protein’s functionality. ▸ Mass spectrometry (M/S) has emerged an an important technique for sequencing proteins. ▸ The molecular masses of fragments of the protein sequence are then determined as the output of this mass spectrometry.

Motivation Problem Statement Construction THANK YOU! Protein Sequencing ▸ A protein is a long sequence of amino acids whose composition and order determine the protein’s functionality. ▸ Mass spectrometry (M/S) has emerged an an important technique for sequencing proteins. ▸ The molecular masses of fragments of the protein sequence are then determined as the output of this mass spectrometry. ▸ From these molecular masses, the identities of the corresponding amino acids can be determined.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . ▸ Under this model, previous work in [ADMOP15] studied how to recover a string s [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . ▸ Under this model, previous work in [ADMOP15] studied how to recover a string s [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . ▸ Under this model, previous work in [ADMOP15] studied how to recover a string s provided its composition multi-set C ( s ) . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . ▸ Under this model, previous work in [ADMOP15] studied how to recover a string s provided its composition multi-set C ( s ) . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Model ▸ The composition multi-set C ( s ) of a string s = ( s 1 , . . . , s n ) ∈ { 0 , 1 } n is the multiset C ( s ) = {{ s i , s i + 1 , . . . , s j } ∶ 1 ≤ i ≤ j ≤ n } of all ( n + 1 ) contiguous substrings of s . 2 ▸ As an example, if s = ( 0 , 1 , 0 , 0 ) , then, C ( s ) = {{ 0 } , { 1 } , { 0 } , { 0 } , { 01 } , { 0 , 1 } , { 0 , 0 } , { 0 , 0 , 1 } , { 0 , 0 , 1 } , { 0 , 1 , 0 , 0 }} . ▸ Under this model, previous work in [ADMOP15] studied how to recover a string s provided its composition multi-set C ( s ) . [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Previous Work ▸ The following results are known from [ADMOP15]: [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Previous Work ▸ The following results are known from [ADMOP15]: Theorem All strings of length one less than a prime are reconstructable. [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Previous Work ▸ The following results are known from [ADMOP15]: Theorem All strings of length one less than a prime are reconstructable. Theorem All strings of length one less than twice a prime are reconstructable. [ADMOP15] J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan, “String reconstruction from substring compositions,” SIAM J. Discrete Math , 2015.

Motivation Problem Statement Construction THANK YOU! Problem Statement