Problem
Research questions and friction points this paper is trying to address.
Cost-Constrained Channels
Information Theory
Channel Capacity
Multivariate Analytic Combinatorics for Cost Constrained Channels and Subsequence Enumeration
🤖 AI Summary
This study addresses the enumeration of admissible sequences and capacity characterization for discrete noiseless channels under cost constraints. Methodologically, it models the channel structure as a weighted labeled directed graph and establishes an exact correspondence between this graph and the singularities of a bivariate generating function; asymptotic analysis then leverages multivariate analytic combinatorics, spectral theory of directed graphs, and singularity analysis to derive precise asymptotics for the number of admissible sequences. The key contributions are threefold: (i) it identifies, for the first time, that channel capacity under cost constraints is determined by cost-dependent singularities of the associated generating function; (ii) it provides a novel proof of the equivalence between the combinatorial and probabilistic definitions of cost-constrained capacity; and (iii) it unifies information-theoretic coding theory with enumerative combinatorics, thereby generalizing and extending Shannon’s classical capacity theorem.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Multivariate Analytic Combinatorics
Channel Capacity under Cost Constraints
Special Function Singularities
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