Combinatorial Counting Research Articles

New Bounds on the Capacity of Multidimensional Run-Length Constraints

We examine the well-known problem of determining the capacity of multidimensional run-length-limited constrained systems. By recasting the problem, which is essentially a combinatorial counting problem, into a probabilistic setting, we are able to derive new lower and upper bounds on the capacity of (0, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> )-RLL systems. These bounds are better than all previously-known analytical bounds for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ≥ 2, and are tight asymptotically. Thus, we settle the open question: what is the rate at which the capacity of (0, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> )-RLL systems converges to 1 as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> → ∞? We also provide the first nontrivial upper bound on the capacity of general ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> )-RLL systems.

Practical unconditionally secure two-channel message authentication

We investigate unconditional security for message authentication protocols that are designed using two-channel cryptography. (Two-channel cryptography employs a broadband, insecure wireless channel and an authenticated, narrow-band manual channel at the same time.) We study both noninteractive message authentication protocols (NIMAPs) and interactive message authentication protocols (IMAPs) in this setting. First, we provide a new proof of nonexistence of nontrivial unconditionally secure NIMAPs. This proof consists of a combinatorial counting argument and is much shorter than the previous proof by Wang and Safavi-Naini, which was based on probability distribution arguments. We also prove a new result which holds in a weakened attack model. Further, we propose a generalization of an unconditionally secure 3-round IMAP due to Naor, Segev and Smith. The IMAP is based on two ϵ-Δ universal hash families. With a careful choice of parameters, our scheme improves that of Naor et al. Our scheme is very close to optimal for most parameter situations of practical interest. Finally, a variation of the 3-round IMAP is presented, in which only one hash family is required.

Square root singularities of infinite systems of functional equations

Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form $\mathbf{y} (x)=F(x,\mathbf{y} (x))$, where $F(x,\mathbf{y} ):\mathbb{C} \times \ell^p \to \ell^p$ is a positive and nonlinear function, and analyze the behavior of the solution $\mathbf{y} (x)$ at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function $F$ is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of $\mathbf{y} (x)$.

Compound matrices: properties, numerical issues and analytical computations

This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD’s) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD’s of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.

Interrogating domain-domain interactions with parsimony based approaches

BackgroundThe identification and characterization of interacting domain pairs is an important step towards understanding protein interactions. In the last few years, several methods to predict domain interactions have been proposed. Understanding the power and the limitations of these methods is key to the development of improved approaches and better understanding of the nature of these interactions.ResultsBuilding on the previously published Parsimonious Explanation method (PE) to predict domain-domain interactions, we introduced a new Generalized Parsimonious Explanation (GPE) method, which (i) adjusts the granularity of the domain definition to the granularity of the input data set and (ii) permits domain interactions to have different costs. This allowed for preferential selection of the so-called "co-occurring domains" as possible mediators of interactions between proteins. The performance of both variants of the parsimony method are competitive to the performance of the top algorithms for this problem even though parsimony methods use less information than some of the other methods. We also examined possible enrichment of co-occurring domains and homo-domains among domain interactions mediating the interaction of proteins in the network. The corresponding study was performed by surveying domain interactions predicted by the GPE method as well as by using a combinatorial counting approach independent of any prediction method. Our findings indicate that, while there is a considerable propensity towards these special domain pairs among predicted domain interactions, this overrepresentation is significantly lower than in the iPfam dataset.ConclusionThe Generalized Parsimonious Explanation approach provides a new means to predict and study domain-domain interactions. We showed that, under the assumption that all protein interactions in the network are mediated by domain interactions, there exists a significant deviation of the properties of domain interactions mediating interactions in the network from that of iPfam data.

Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems

We present an improved “cooling schedule” for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature (easy region) to the low temperature (difficult region). We present applications of our technique to colorings and the permanent (perfect matchings of bipartite graphs). Moreover, for the permanent, we improve the analysis of the Markov chain underlying the simulated annealing algorithm. This improved analysis, combined with the faster cooling schedule, results in an $O(n^7\log^4{n})$ time algorithm for approximating the permanent of a $0/1$ matrix.

Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace

We consider the problem of efficient integration of an n-variate polynomial with respect to the Gaussian measure in ℝn and related problems of complex integration and optimization of a polynomial on the unit sphere. We identify a class of n-variate polynomials f for which the integral of any positive integer power fp over the whole space is well approximated by a properly scaled integral over a random subspace of dimension O(log n). Consequently, the maximum of f on the unit sphere is well approximated by a properly scaled maximum on the unit sphere in a random subspace of dimension O(log n). We discuss connections with problems of combinatorial counting and applications to efficient approximation of a hafnian of a positive matrix.

The distance approach to approximate combinatorial counting

We develop general methods to obtain fast (polynomial time) estimates of the cardinality of a combinatorially defined set via solving some randomly generated optimization problems on the set. Examples include enumeration of perfect matchings in a graph, linearly independent subsets of a set of vectors and colored spanning subgraphs of a graph. Geometrically, we estimate the cardinality of a subset of the Boolean cube via the average distance from a point in the cube to the subset with respect to some distance function. We derive asymptotically sharp cardinality bounds in the case of the Hamming distance and show that for small subsets a suitably defined “randomized” Hamming distance allows one to get tighter estimates of the cardinality.

Polynomial Time Algorithms to Approximate Permanents and Mixed Discriminants Within a Simply Exponential Factor

We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a nonnegative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high probability, approximates the true value within a factor of O(cn), where n is the size of the matrix (matrices) and where c ≈ 0.28 for the real version, c ≈ 0.56 for the complex version, and c ≈ 0.76 for the quaternionic version. We discuss possible extensions of our method as well as applications of mixed discriminants to problems of combinatorial counting. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 29–61, 1999

Enumerative Questions on Rooted Weighted Necklaces

We develop the language and the tools to create a new family of integer sequences. The concrete motivation for this article came from a combinatorial counting problem first posed by Cipra [Math. Mag.65(1) (1992), 56] and still open in full generality. LetCkbe the cyclic group {q0,q1,…,qk−1} and letZ[Ck] be the algebra of polynomials in the variablesq0,q1,q2,…,qk−1with coefficients fromZ. We introduce weighted rooted necklaces, which give a combinatorial interpretation ofZ[Ck]. Actions onZ[Ck] via a family of operators of the formR(f[q])=qt(f[q]+h[q]), wheret∈Zandh[q],f[q]∈Z[Ck], are introduced. In particular we study the nonlinear, invertible operatorR(f[q])=q−f[0](f[q]−f[0]+∑f[0]i=1σ(f[0])qi), wheref[q]=∑k−1i=0niqi∈Z[Ck] andf[0]=n0. The operatorRcreates equivalence classes onZ[Ck]. We determine a recursive function to count the equivalence class containingG[q]=∑ki=0qi∈Z[Ck] which solves a special case of the aforementioned counting problem posed by Cipra. We give a closed form for such recursive formula in some instances. We also develop a series of related questions.

Theoretical predictions on the equality of radiation-produced dicentrics and translocations detected by chromosome painting

Existing models of chromosome aberrations produced by ionizing-radiation predict equal numbers of dicentrics and translocations if the dose is so low that complex aberrations can be ignored. We show that, for a specific subset of aberrations detected by FISH, dicentric/translocation equality is predicted even at higher doses. Assuming one-colour whole-chromosome painting (with unpainted chromosomes counterstained and centromeres recognizable) the relevant restriction is that the final metaphase pattern be, in the terminology of Simpson and Savage, 'apparently simple'. This means that the painted pattern is required to have the colour/centromere appearance corresponding to a single complete reciprocal exchange but its actual formation, as reflected for example in lengths, is allowed to be more complicated. The restriction to apparent simplicity is significantly less limiting than ignoring all complex aberrations. Our analysis of predicted dicentric/translocation equality in this case uses examples, a combinatorial counting method, Monte Carlo computer programs, and a duality proof. However, we argue that for 'visibly complex' dicentrics or translocations, no similar equality is expected in general. Corresponding experimental results are briefly surveyed. Checking dicentric/translocation equality experimentally can provide a significant test of current chromosome aberration models.

Universal switch modules for FPGA design

A switch module M with W terminals on each side is said to be universal if every set of nets satisfying the dimensional constraint (i.e., the number of nets on each side of M is at most W ) is simultaneously rout able through M . In this article, we present a class of universal switch modules. Each of our switch modules has 6 W switches and switch-module flexibility three (i.e, F s =3). We prove that no switch module with less than 6 W switches can be universal. We also compare our switch modules with those used in the Xilinx XC4000 family FPGAs and the antisymmetric switch modules (with F S =3) suggested by Rose and Brown [1991]. Although these two kinds of switch modules also have F S =3 and 6 W switches, we show that they are not universal. Based on combinatorial counting techniques, we show that each of our universal switch modules can accommodate up to 25% more routing instances, compared with the XC4000-type switch module of the same size. Experimental results demonstrate that our universal switch modules improve routability at the chip level. Finally, our work also provides a theoretical insight into the important observation by Rose and Brown [1991] (based on extensive experiments) that F S =3 is often sufficient to provide high routability.

A loop-free algorithm for generating the linear extensions of a poset

A precise concept of when a combinatorial counting problem is “hard” was first introduced by Valiant (1979) when he defined the notion of a #P-complete problem. Correspondingly, there has been consistent interest in the notion of when a combinatorial listing problem admits a very special regular structure in which transition times between objects being listed are uniformly bounded by a fixed constant. Early descriptions of suchloop-free listing algorithms may be found in the bookAlgorithmic Combinatorics by Even (1973). Recently, the problem of counting all linear extensions of a partially ordered set has received attention with regard to both of these combinatorial concepts. Brightwell and Winkler (1991) have shown, by a very ingenious argument, that the poset-extension counting problem is #P-complete. Pruesse and Ruskey (1992) have shown that the corresponding listing problem can be solved in constant amortized time and have posed the problem of finding a loop-free algorithm for the poset-extension problem. The present paper presents a solution to this latter problem. This sequence of results represents an interesting juxtaposition, in a fixed, naturally-occurring combinatorial problem, of intricate and precisely defined “irregularities” with respect to counting with very strong regularities with respect to listing.

Approximate counting, uniform generation and rapidly mixing Markov chains

The paper studies effective approximate solutions to combinatorial counting and unform generation problems. Using a technique based on the simulation of ergodic Markov chains, it is shown that, for self-reducible structures, almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time. It follows that, for self-reducible structures, polynomial time randomised algorithms for counting to within factors of the form (1 + n−β) are available either for all β ϵ R or for no β ϵ R. A substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic Markov chains, and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established. Finally, the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour.

Level density in large spectroscopic spaces

Statistical spectroscopy methods are used to calculate level densities for one-body Hamiltonians in the space of 440 single particle states. Comparisons are made with combinatorial counting and Fermi gas model. Possibilities to arrive at a more realistic level density formula including two-body effects are discussed.