Multivariate Analytic Combinatorics for Cost Constrained Channels

TL;DR

This paper applies multivariate analytic combinatorics to analyze the asymptotic behavior of sequences in cost-constrained channels, linking graph structures to generating function singularities and generalizing channel capacity results.

Contribution

It introduces a novel approach connecting analytic combinatorics and graph theory to study cost-constrained channels, extending classical capacity results.

Findings

Derived asymptotic formulas for cost-limited sequence counts

Linked channel capacity to singularities of generating functions

Provided a new proof of capacity equivalence

Abstract

Analytic combinatorics in several variables is a branch of mathematics that deals with 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 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 use these new results to invoke theorems from multivariate analytic combinatorics to obtain the asymptotic behavior of the number of cost-limited strings that are admissible by the channel. This builds a new bridge between analytic combinatorics in several variables and labeled weighted graphs, bringing a new perspective and a set of powerful results to the literature of cost-constrained channels. Along the way, we show that the cost-constrained channel capacity is determined by a cost-dependent singularity of the bivariate generating function, generalizing Shannon's classical result for unconstrained capacity, and provide a new proof of the equivalence of the combinatorial and probabilistic definitions of the cost-constrained capacity.