Minimum Alphabet Size Research Articles

Latency and Alphabet Size in the Context of Multicast Network Coding

We study the relation between <i>latency</i> and <i>alphabet size</i> in the context of Multicast Network Coding. Given a graph <inline-formula> <tex-math notation="LaTeX">$G = (V, E)$ </tex-math></inline-formula> representing a communication network, a subset <inline-formula> <tex-math notation="LaTeX">$S \subseteq V$ </tex-math></inline-formula> of sources, each of which initially holds a set of information messages, and a set <inline-formula> <tex-math notation="LaTeX">$T \subseteq V$ </tex-math></inline-formula> of terminals; we consider the problem in which one wishes to design a communication scheme that eventually allows all terminals to obtain all the messages held by the sources. In this study we assume that communication is performed in rounds, where in each round each network node may transmit a single (possibly encoded) information packet on any of its outgoing edges. The objective is to minimize the communication latency, i.e., number of communication rounds needed until all terminals have all the messages of the source nodes. For sufficiently large alphabet sizes (i.e., large block length, packet sizes), it is known that traditional linear multicast network coding techniques (such as random linear network coding) minimize latency. In this work we seek to study the task of minimizing latency in the setting of limited alphabet sizes (i.e., finite block length), and alternatively, the task of minimizing the alphabet size in the setting of bounded latency. We focus on the establishing the computation complexity of the problem and present several intractability results. In particular, through reductive arguments, we prove that it is NP-hard to (i) approximate (and in particular to determine) the minimum alphabet size given a latency constraint; (ii) approximate (and in particular to determine) the minimum latency of communication schemes in the setting of limited alphabet sizes.

General Methodology to Identify the Minimum Alphabet Size for Heteropolymer Design

AbstractUnderstanding how to design the structure of heteropolymers through their monomer sequence will have a significant impact on the creation of novel artificial materials. According to mean‐field theories, the minimum number—or alphabet—of distinct monomers necessary to achieve such designability is directly related to the conformational entropy ω of compact polymer structures. Here, a computational strategy to calculate this conformational entropy is introduced and thus predict the minimum alphabet to achieve designability, for a generalized heteropolymer model. The comparison of the predictions with previous results proves the robustness of the approach. It is quantified for the first time how the number of directional interactions is critical for achieving the designability. The methodology that is introduced can be easily generalized to models representing specific polymers. A comparison between conventional polymers monomers are provided, and it is predicted that polyurea, polyamide, and polyurethane residues are optimal candidates to be functionalized for the experimental synthesis of designable heteropolymers. As such, our method can guide the engineering of new types of self‐assembling modular polymers, that will open new possibilities for polymer‐based materials with unmatched versatility and control.

An improved upper bound and algorithm for clique covers

Indeterminate strings have received considerable attention in the recent past; see for example [1] and [3]. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimum alphabet size of indeterminates representing any given undirected graph equals the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in [3]: characterize Θn(m), which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on n vertices and m edges. We provide improvements to the known upper bound for Θn(m) in section 3.3. [3] also presents an algorithm which finds clique covers in polynomial time. We build on this result with a heuristic for vertex sorting which significantly improves their algorithm's results, particularly in dense graphs.

New Bounds and Extended Relations Between Prefix Arrays, Border Arrays, Undirected Graphs, and Indeterminate Strings

We extend earlier works on the relation of prefix arrays of indeterminate strings to undirected graphs and border arrays. If integer array y is the prefix array for indeterminate string w, then we say w satisfies y. We use a graph theoretic approach to construct a string on a minimum alphabet size which satisfies a given prefix array. We relate the problem of finding a minimum alphabet size to finding edge clique covers of a particular graph, allowing us to bound the minimum alphabet size by n+n$n+\sqrt {n}$ for indeterminate strings, where n is the size of the prefix array. When we restrict ourselves to prefix arrays for partial words, we bound the minimum alphabet size by 2n$\left \lceil \sqrt {2n} \right \rceil $. Moreover, we show that this bound is tight up to a constant multiple by using Sidon sets. We also study the relationship between prefix arrays and border arrays. We give necessary and sufficient conditions for a border array and prefix array to be satisfied by the same indeterminate string. We show that the slowly-increasing property completely characterizes border arrays for indeterminate strings, whence there are exactly Cn distinct border arrays of size n for indeterminate strings (here Cn is the nth Catalan number). We give an algorithm to enumerate all prefix arrays for partial words of a given size, n. Our algorithm has a time complexity of n3 times the output size, that is, the number of valid prefix arrays for partial words of length n. We also bound the number of prefix arrays for partial words of a given size using Stirling numbers of the second kind.

Indeterminate strings, prefix arrays & undirected graphs

An integer array y=y[1..n] is said to be feasible if and only if y[1]=n and, for every i∈2..n, i≤i+y[i]≤n+1. A string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater than one of a given alphabet Σ; otherwise it is said to be regular. A feasible array y is said to be regular if and only if it is the prefix array of some regular string. We show using a graph model that every feasible array of integers is a prefix array of some (indeterminate or regular) string, and for regular strings corresponding to y, we use the model to provide a lower bound on the alphabet size. We show further that there is a 1–1 correspondence between labelled simple graphs and indeterminate strings, and we show how to determine the minimum alphabet size σ of an indeterminate string x based on its associated graphGx. Thus, in this sense, indeterminate strings are a more natural object of combinatorial interest than the strings on elements of Σ that have traditionally been studied.

On average throughput and alphabet size in network coding

We examine the throughput benefits that network coding offers with respect to the average throughput achievable by routing, where the average throughput refers to the average of the rates that the individual receivers experience. We relate these benefits to the integrality gap of a standard linear programming formulation for the directed Steiner tree problem. We describe families of configurations over which network coding at most doubles the average throughput, and analyze a class of directed graph configurations with N receivers where network coding offers benefits proportional to /spl radic/N. We also discuss other throughput measures in networks, and show how in certain classes of networks, average throughput bounds can be translated into minimum throughput bounds, by employing vector routing and channel coding. Finally, we show configurations where use of randomized coding may require an alphabet size exponentially larger than the minimum alphabet size required.

Simplicial edge representation of protein structures and alpha contact potential with confidence measure.

Protein representation and potential function are two important ingredients for studying protein folding, equilibrium thermodynamics, and sequence design. We introduce a novel geometric representation of protein contact interactions using the edge simplices from the alpha shape of the protein structure. This representation can eliminate implausible neighbors that are not in physical contact, and can avoid spurious contact between two residues when a third residue is between them. We developed statistical alpha contact potential using an odds-ratio model. A studentized bootstrap method was then introduced to assess the 95% confidence intervals for each of the 210 propensity parameters. We found, with confidence, that there is significant long-range propensity (>30 residues apart) for hydrophobic interactions. We tested alpha contact potential for native structure discrimination using several sets of decoy structures, and found that it often performs comparably with atom-based potentials requiring many more parameters. We also show that accurate geometric representation is important, and that alpha contact potential has better performance than potential defined by cutoff distance between geometric centers of side chains. Hierarchical clustering of alpha contact potentials reveals natural grouping of residues. To explore the relationship between shape and physicochemical representations, we tested the minimum alphabet size necessary for native structure discrimination. We found that there is no significant difference in performance of discrimination when alphabet size varies from 7 to 20, if geometry is represented accurately by alpha simplicial edges. This result suggests that the geometry of packing plays an important role, but the specific residue types are often interchangeable.

Enumeration techniques for best N -phase codes

Efficient implementation techniques for the enumeration of the best N-phase codes are proposed. These techniques allow a single program to simultaneously take advantage of all sidelobe-invariant transformations (i.e. constant and linear phase shift, reversal and conjugation). As an example of application, all Barker codes of length 16 with minimum alphabet size are determined.