Multivariate Analytic Combinatorics for Cost Constrained Channels and Subsequence Enumeration

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. Institute for Communications Engineering, Technical University of Munich, Germany, andreas.lenz@mytum.de. AL has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 801434) Department of Combinatorics & Optimization, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada, smelczer@uwaterloo.ca. SM was supported by an NSERC Discovery Grant. Google Research, cyroid@google.com. Work partially completed while at UCSD. Department of Electrical and Computer Engineering, University of California, San Diego, psiegel@ucsd.edu 1 ar X iv :2 11 1. 06 10 5v 2 [ cs .I T ] 1 4 N ov 2 02 1