Probabilistic Analysis Of Algorithms Research Articles

On a functional contraction method

Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the space $\mathcal{C}[0,1]$ of continuous functions endowed with uniform topology and the space $\mathcal {D}[0,1]$ of c\`{a}dl\`{a}g functions with the Skorokhod topology. The contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach's fixed-point theorem. We develop the use of the Zolotarev metrics on the spaces $\mathcal{C}[0,1]$ and $\mathcal{D}[0,1]$ in this context. Applications are given, in particular, a short proof of Donsker's functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed.

On Tail Bounds for Random Recursive Trees

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

On Tail Bounds for Random Recursive Trees

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of randomb-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

The analysis of a prioritised probabilistic algorithm to find large induced forests in regular graphs with large girth

The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.

On stochastic recursive equations of sum and max type

In this paper we consider stochastic recursive equations of sum type, , and of max type, , where Ai, bi, and b are random, (Xi) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

On stochastic recursive equations of sum and max type

In this paper we consider stochastic recursive equations of sum type,, and of max type,, whereAi,bi, andbare random, (Xi) are independent, identically distributed copies ofX, anddenotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimalLs-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

Probabilistic analysis of algorithms for the Dutch national flag problem

A detailed probabilistic analysis is given of algorithms for the Dutch national flag problem. We derive central and local limit theorems for the cost, as well as probabilities of large deviations. Performance of a related algorithm is also studied.

Recursive random variables with subgaussian distributions

Summary: We consider sequences of random variables with distributions that satisfy recurrences as they appear for quantities on random trees, random combinatorial structures and recursive algorithms. We study the tails of such random variables in cases where after normalization convergence to the normal distribution holds. General theorems implying subgaussian distributions are derived. Also cases are discussed with non-Gaussian tails. Applications to the probabilistic analysis of algorithms and data structures are given.

Density approximation and exact simulation of random variables that are solutions of fixed-point equations

An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.

Density approximation and exact simulation of random variables that are solutions of fixed-point equations

An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.

Experimental Evaluation of Heuristic Optimization Algorithms: A Tutorial

Heuristic optimization algorithms seek good feasible solutions to optimization problems in circumstances where the complexities of the problem or the limited time available for solution do not allow exact solution. Although worst case and probabilistic analysis of algorithms have produced insight on some classic models, most of the heuristics developed for large optimization problem must be evaluated empirically—by applying procedures to a collection of specific instances and comparing the observed solution quality and computational burden. This paper focuses on the methodological issues that must be confronted by researchers undertaking such experimental evaluations of heuristics, including experimental design, sources of test instances, measures of algorithmic performance, analysis of results, and presentation in papers and talks. The questions are difficult, and there are no clear right answers. We seek only to highlight the main issues, present alternative ways of addressing them under different circumstances, and caution about pitfalls to avoid.

Average case results for satisfiability algorithms under the random-clause-width model

In the probabilistic analysis of algorithms for the Satisfiability problem, the randomdclausedwidth model is one of the most popular for generating random formulas. This model is parameterized and it is not difficult to show that virtually the entire parameter space is covered by a collection of polynomial time algorithms that find solutions to random formulas with probability tending to 1 as formula size increases. But finding a collection of polynomial average time algorithms that cover the parameter space has proved much harder and such results have spanned approximately ten years. However, it can now be said that virtually the entire parameter space is covered by polynomial average time algorithms. This paper relates dominant, exploitable properties of random formulas over the parameter space to mechanisms of polynomial average time algorithms. The probabilistic discussion of such properties is news main averagedcase results over the last ten years are reviewed.

Perpetuities with thin tails

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

Perpetuities with thin tails

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

The number of winners in a discrete geometrically distributed sample

round of the game, that is, the ‘‘winners’’ of the game. In 1 the probability distribution of X, the expectation E X of X and the asymptotic behavior of 4 . P X s 1 probability of a single winner for n “ ‘ were to be determined. In w x the solution 2 it was remarked that}surprisingly}this probability does not converge as n “ ‘ but rather has an oscillating behavior. At the same w x time, Eisenberg, Stengle and Strang 5 discussed this problem and related topics, exhibiting the structure of the periodic fluctuation, for which an explicit Fourier expansion was given. Also about this time, Brands, Steutel w x and Wilms 4 came independently to roughly the same results. A recent w x paper by Baryshnikov, Eisenberg and Stengle 3 deals with the existence of the limiting probability of a tie for first place. In fact, a fluctuating behavior in asymptotic expansions is not at all uncommon. There are numerous results of that type, for example, in the w x w x analysis of divide-and-conquer recursions 7, 8, 16 or digital sums 6 , that play a prominent role in the probabilistic analysis of algorithms. Our aim in this paper is to some extent tutorial: the asymptotic technique that yields the Fourier expansions of the fluctuating functions very comfortw x. ably is called ‘‘Rice’s method’’ see the recent survey 9 . In order to convince the reader of the advantages of this method, we will rederive a result on the initially mentioned problem in the sequel, and afterwards present some new results concerning higher moments of distribution as well as the number of . persons reaching a specified level beyond the winner s .

New rigorous results for the Hopfield's neural network model

New asymptotic bounds for the learning capacity of Hopfield's model with random memorized patterns are given. The usual independence assumption is replaced by a different hypothesis on the joint distribution of spins. The proof of the main result uses classical tools of probabilistic analysis of algorithms.

Wafer-Scale Integration of Systolic Arrays

VLSI technologists are fast developing wafer-scale integration. Rather than partitioning a silicon wafer into chips as is usually done, the idea behind wafer-scale integration is to assemble an entire system (or network of chips) on a single wafer, thus avoiding the costs and performance loss associated with individual packaging of chips. A major problem with assembling a large system of microprocessors on a single wafer, however, is that some of the processors, or cells, on the wafer are likely to be defective. In the paper, we describe practical procedures for integrating "around" such faults. The procedures are designed to minimize the length of the longest wire in the system, thus minimizing the communication time between cells. Although the underlying network problems are NP-complete, we prove that the procedures are reliable by assuming a probabilistic model of cell failure. We also discuss applications of the work to problems in VLSI layout theory, graph theory, fault-tolerant systems, planar geometry, and the probabilistic analysis of algorithms.

On some conditioning results in the probabilistic analysis of algorithms

A simple combinatorial approach is given for handling certain conditioning problems that arise in the probabilistic analysis of graph algorithms.