Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1211.2926

Enumeration of sequences with large alphabets

M. Oguzhan Kulekci
View original source ↗Read source document ↗Export metadata ↓

Abstract

This study focuses on efficient schemes for enumerative coding of $σ$--ary sequences by mainly borrowing ideas from Öktem & Astola's \cite{Oktem99} hierarchical enumerative coding and Schalkwijk's \cite{Schalkwijk72} asymptotically optimal combinatorial code on binary sequences. By observing that the number of distinct $σ$--dimensional vectors having an inner sum of $n$, where the values in each dimension are in range $[0...n]$ is $K(σ,n) = \sum_{i=0}^{σ-1} {{n-1} \choose {σ-1-i}} {σ \choose {i}}$, we propose representing $C$ vector via enumeration, and present necessary algorithms to perform this task. We prove $\log K(σ,n)$ requires approximately $ (σ-1) \log (σ-1) $ less bits than the naive $(σ-1)\lceil \log (n+1) \rceil$ representation for relatively large $n$, and examine the results for varying alphabet sizes experimentally. We extend the basic scheme for the enumerative coding of $σ$--ary sequences by introducing a new method for large alphabets. We experimentally show that the newly introduced technique is superior to the basic scheme by providing experiments on DNA sequences.

Explore related subjects

cs.DScs.DMcs.ITmath.IT

Keep this discovery

BibTeXRIS

M. Oguzhan Kulekci. 2012-11-13. Enumeration of sequences with large alphabets. https://arxiv.org/abs/1211.2926

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗