Multivariate Analytic Combinatorics for Cost Constrained Channels and Subsequence Enumeration
Abstract
Analytic combinatorics in several variables is a powerful tool for deriving the asymptotic behavior of combinatorial quantities by analyzing multivariate generating functions. We study informationtheoretic questions about sequences in a discrete noiseless channel under cost and forbidden substring constraints. Our main contributions involve the relationship between the graph structure of the channel and the singularities of the bivariate generating function whose coefficients are the number of sequences satisfying the constraints. We combine these new results with methods from multivariate analytic combinatorics to solve questions in many application areas. For example, we determine the optimal coded synthesis rate for DNA data storage when the synthesis supersequence is any periodic string. This follows from a precise characterization of the number of subsequences of an arbitrary periodic strings. Along the way, we provide a new proof of the equivalence of the combinatorial and probabilistic definitions of the costconstrained capacity, and we show that the costconstrained channel capacity is determined by a costdependent singularity, generalizing Shannon's classical result for unconstrained capacity.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.06105
 Bibcode:
 2021arXiv211106105L
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics;
 E.4