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 information-theoretic 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 cost-constrained channel capacity is determined by a cost-dependent singularity, generalizing Shannon's classical result for unconstrained capacity.

Rate-Constrained Shaping Codes for Structured Sources

“…In [15], the authors discuss cost-diverse and cost-uniform graphs. A graph is cost-diverse if it has at least one pair of equal-length paths with different costs that connect the same pair of vertices.…”

“…The relationship between cost-constrained combinatorial capacity and probabilistic capacity was also addressed in [10]. The equivalence of the two definitions of cost-constrained capacity was proved in [25], and an alternative proof was recently given in [15], where methods of analytic combinatorics in several variables were used to directly evaluate the cost-constrained combinatorial capacity.…”