# Multivariate Analytic Combinatorics for Cost Constrained Channels and Subsequence Enumeration

@article{Lenz2021MultivariateAC, title={Multivariate Analytic Combinatorics for Cost Constrained Channels and Subsequence Enumeration}, author={Andreas Lenz and Stephen Melczer and Cyrus Rashtchian and Paul H. Siegel}, journal={ArXiv}, year={2021}, volume={abs/2111.06105} }

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…

## One Citation

### Rate-Constrained Shaping Codes for Finite-State Channels With Cost

- Computer Science2022 IEEE International Symposium on Information Theory (ISIT)
- 2022

An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. sources.

## References

SHOWING 1-10 OF 93 REFERENCES

### Analytic Combinatorics in Several Variables

- Mathematics
- 2013

This book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective. Analytic combinatorics is a branch of enumeration that uses analytic techniques to…

### Subsequence Combinatorics and Applications to Microarray Production, DNA Sequencing and Chaining Algorithms

- BiologyCPM
- 2006

These considerations allow a detailed analysis of a new DNA sequencing technology (“454 sequencing”) and determine the number of sequences in Σn whose longest strictly increasing subsequence has length k, where 0 ≤k ≤K.

### Performance bounds in constrained sequence coding

- Computer Science
- 1987

This work presents upper bounds on the necessary length of the decoder window for the run-length-limited system for the magnetic recording channel, and bound possible code rates in terms of c.

### Two-dimensional Quantum Random Walk

- MathematicsJournal of Statistical Physics
- 2010

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case…

### Capacity of General Discrete Noiseless Channels

- Computer Science, MathematicsArXiv
- 2008

The capacity of the discrete noiseless channel introduced by Shannon is shown to be well-defined and a generalisation is given for Pringsheim's Theorem and for the Exponential Growth Formula to generating functions of combinatorial structures with non-integer valued symbol weights.

### Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

- MathematicsSIAM Rev.
- 2008

The Morse-theoretic underpinnings of some new asymptotic techniques are described, and the use of these techniques on a variety of problems of combinatorial interest is illustrated.

### Analytic Combinatorics

- Mathematics
- 2009

This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.

### Matching Dyadic Distributions to Channels

- Computer Science2011 Data Compression Conference
- 2011

This work defines a new algorithm called Geometric Huffman Coding (GHC) and proves that GHC finds the optimal dyadic PMF in O(m log m) steps where m is the number of input symbols of the considered channel.

### Minimum-redundancy coding for the discrete noiseless channel

- Computer ScienceIRE Trans. Inf. Theory
- 1961

This paper gives a method for constructing minimum-redundancy prefix codes for the general discrete noiseless channel without constraints and computational experience is presented to demonstrate the practicability of the method.