An analysis of a contention resolution algorithm

Summary A single multiaccess channel is studied with the outcome of a transmission being either idle, success, or collision (ternary channel). Packets involved in a collision must be retransmitted, and an efficient way to solve a collision is known in the literature as Gallager-Tsybakov-Mikhailov algorithm. Performance analysis of the algorithm is quite hard. In fact, it bases on a numerical solution of some recurrence equations and on a numerical evaluation of some series. The obvious drawback of it is lack of insight into the behavior of the algorithm. We shall present a new approach of looking at the algorithm and discuss some attempts of analyzing its performance. In particular, expected lengths of a resolution interval and a conflict resolution interval as well as throughput of the algorithm will be discussed using asymptotic approximation and a small input rate approximation.     

Philippe Flajolet's Research in Analysis of Algorithms and Combinatorics

Philippe Flajolet's research in theoretical computer science spans over more than 20 years. He made lasting contributions to the analysis of algorithms and analytic combinatorics. Among many of his results we mention here some in such diversified topics as enumeration, number theory, formal languages, continued fractions, automatic analysis of algorithms, Mellin transform, digital sums, recurrences, trees, random generation of combinatorial objects, random graphs and mappings, polynomial factorization, communications, codes, graphics, etc. This gives only a small snapshot of his work, and we encourage the reader to visit Flajolet's homepage http://www-rocq.inria.fr/algo/flajolet/index.html for a fuller account. The bibliography was taken from his homepage and may not be totally complete, although we added a few items. Our paper was written without giving any prior notice to Philippe Flajolet. It reflects the view of the authors and any misunderstandings and shortcomings should be put on their account.     

On Pattern Frequency Occurrences in a Markovian Sequence

Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for moments (including the variance), and probability of r pattern occurrences for three different regions of r , namely: (i) r=O(1) , (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct certain language expressions that characterize pattern occurrences which are later translated into generating functions. We then use analytical methods to extract asymptotic behaviors of the pattern frequency from the generating functions. These findings are of particular interest to molecular biology problems (e.g., finding patterns with unexpectedly high or low frequencies, and gene recognition), information theory (e.g., second-order properties of the relative frequency), and pattern matching algorithms (e.g., q -gram algorithms).     

Markov types and minimax redundancy for Markov sources

Redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length. In the minimax scenario, one designs the best code for the worst source within the class. Such minimax redundancy comes in two flavors: average minimax or worst case minimax. We study the worst case minimax redundancy of universal block codes for Markovian sources of any order. We prove that the maximal minimax redundancy for Markov sources of order r is asymptotically equal to 1/2m/sup r/(m-1)log/sub 2/n+log/sub 2/A/sub m//sup r/-(lnlnm/sup 1/(m-1)/)/lnm+o(1), where n is the length of a source sequence, m is the size of the alphabet, and A/sub m//sup r/ is an explicit constant (e.g., we find that for a binary alphabet m=2 and Markov of order r=1 the constant A/sub 2//sup 1/=16/spl middot/G/spl ap/14.655449504 where G is the Catalan number). Unlike previous attempts, we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing Markov types. The enumeration of Markov types is accomplished by reducing it to counting Eulerian paths in a multigraph. In particular, we propose exact and asymptotic formulas for the number of strings of a given Markov type. All of these findings are obtained by analytic and combinatorial tools of analysis of algorithms.     

Optimal versus randomized search of fixed length binary words

We consider the search problem in which one finds a binary word among m words chosen randomly from the set of all words of fixed length n. It is well known that the optimal search is equivalent to the Huffman coding that requires on average log/sub 2/ m bits to be checked plus a small additional cost called the average redundancy. The latter is an oscillating function of m and is bounded between zero and 1-(1+lnln2)/In2/spl ap/0.0860713320. As a matter of fact, it is known that finding the optimal strategy for this problem is NP-hard. We propose here several simple randomized search strategies leading, respectively, to the following average redundancies: 1.332746177, 0.6113986565,0.4310617764, and 0.332746177, plus some small oscillations that we precisely characterize. These results should be compared to the optimal, but NP-hard, search algorithm. Our findings extend and make more precise results of Fedotov and Ryabko (2001).     

A universal predictor based on pattern matching

We consider a universal predictor based on pattern matching. Given a sequence X₁, ..., X_n drawn from a stationary mixing source, it predicts the next symbol X_n+1 based on selecting a context of X_n+1. The predictor, called the sampled pattern matching (SPM), is a modification of the Ehrenfeucht-Mycielski (1992) pseudorandom generator algorithm. It predicts the value of the most frequent symbol appearing at the so-called sampled positions. These positions follow the occurrences of a fraction of the longest suffix of the original sequence that has another copy inside X₁X₂···X_n ; that is, in SPM, the context selection consists of taking certain fraction of the longest match. The study of the longest match for lossless data compression was initiated by Wyner and Ziv in their 1989 seminal paper. Here, we estimate the redundancy of the SPM universal predictor, that is, we prove that the probability the SPM predictor makes worse decisions than the optimal predictor is O(n^-ν) for some 0<ν<½ as n→∞. As a matter of fact, we show that we can predict K=O(1) symbols with the same probability of error     

On the average redundancy rate of the Lempel-Ziv code

In this paper, we settle a long-standing open problem concerning the average redundancy r_n of the Lempel-Ziv'78 (LZ78) code. We prove that for a memoryless source the average redundancy rate attains asymptotically Er_n=(A+δ(n))/log n+ O(log log n/log² n), where A is an explicitly given constant that depends on source characteristics, and δ(x) is a fluctuating function with a small amplitude. We also derive the leading term for the kth moment of the number of phrases. We conclude by conjecturing a precise formula on the expected redundancy for a Markovian source. The main result of this paper is a consequence of the second-order properties of the Lempel-Ziv algorithm obtained by Jacquet and Szpankowski (1995). These findings have been established by analytical techniques of the precise analysis of algorithms. We give a brief survey of these results since they are interesting in their own right, and shed some light on the probabilistic behavior of pattern matching based data compression     

Precise minimax redundancy and regret

Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal fixed-to-variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario one finds the best code for the worst source either in the worst case (called also maximal minimax) or on average. We first study the worst case minimax redundancy over a class of stationary ergodic sources and replace Shtarkov's bound by an exact formula. Among others, we prove that a generalized Shannon code minimizes the worst case redundancy, derive asymptotically its redundancy, and establish some general properties. This allows us to obtain precise redundancy for memoryless, Markov, and renewal sources. For example, we present the exact constant of the redundancy for memoryless and Markov sources by showing that the integer nature of coding contributes log(logm/(m-1))/logm+o(1) where m is the size of the alphabet. Then we deal with the average minimax redundancy and regret. Our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources. After formulating some general bounds relating these two redundancies, we prove our assertion for memoryless and Markov sources. Nevertheless, we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy (however, our general results show that maximal and average regrets are asymptotically equivalent).     

Analysis and Stability Considerations in a Reservation Multiaccess System

This paper considers the reservation ALOHA scheme proposed by Roberts. Two random access disciplines for reservation packets are analyzed, random access without retransmission discrimination (ORD) and with retransmission discrimination (WRD). Previous analyses have been approximate. This paper provides an exact numerical analysis of the two-dimension state process and usable computable formulas for delay and throughput in terms of state probabilities, describes the range of parameters for which the system is stable, and presents numerical results.     

Asymptotic average redundancy of Huffman (and other) block codes

We study asymptotically the redundancy of Huffman (and other) codes. It has been known from the inception of the Huffman (1952) code that in the worst case its redundancy-defined as the excess of the code length over the optimal (ideal) code length-is not more than one. However, to the best of our knowledge no precise asymptotic results have been reported in literature thus far. We consider here a memoryless binary source generating a sequence of length n distributed as binomial (n, p) with p being the probability of emitting 0. Based on the results of Stubley (1994), we prove that for p<1/2 the average redundancy R¯_n^H of the Huffman code becomes as n→∞: R¯_n^H={(3/2-(1/ln2+o(1))=0.057304…, α irrational); (3/2-(1/M)(〈βMn〉-½)); (-(1/M(1-2^-1M/))2^-〈nβM〉M/); (+O(ρⁿ), α=N/M rational); where α=log₂ (1-p)/p and β=-log₂(1-p), ρ<1, M, N are integers such that gcd (N, M)=1, and 〈x〉=x-[x] is the fractional part of x. The appearance of the fractal-like function 〈βMn〉 explains the erratic behavior of the Huffman redundancy, and its “resistance” to succumb to a precise analysis. As a side result, we prove that the average redundancy of the Shannon block code is as n→∞: R¯_n^S{(½+o(1), α irrational); (½-1/M (〈Mnβ〉-½)); (+O(ρⁿ), α=N/M rational); where ρ<1. Finally, we derive the redundancy of the Golomb (1966) code (for the geometric distribution) which can be viewed as a special case of the Huffman and Shannon codes, Golomb's code redundancy has only oscillating behavior (i.e., there is not convergent mode). These findings are obtained by analytic methods such as theory of distribution of sequences modulo 1 and Fourier series