We consider auctions in which the winning bid is the smallest bid that is unique. Only the upper-price limit is given. Neither the number of participants nor the distribution of the offers are known, so that the problem of placing a bid to win with maximum probability looks, a priori, ill-posed. Indeed, the essence of the problem is to inject a (final) minimum into a random subset (of unique offers) of a larger random set. We will see, however, that here no more than two external (and almost compelling) arguments make the problem meaningful. By appropriately modeling the relationship between the number of participants and the distribution of the bids, we can then maximize our chances of winning the auction and propose a computable algorithm for placing our bid.

The goal of this research is threefold: (i) to analyze generalized digital search trees, (ii) to derive the average profile (i.e., phrase length) of a generalization of the well-known parsing algorithm due to Lempel and Ziv, and (iii) to provide analytic tools to analyze asymptotically certain partial differential functional equations often arising in the analysis of digital trees. In the generalized Lempel--Ziv parsing scheme, one partitions a sequence of symbols from a finite alphabet into phrases such that the new phrase is the shortest substring seen in the past by at most b-1 phrases (b=1 corresponds to the original Lempel--Ziv scheme). Such a scheme can be analyzed through a generalized digital search tree in which every node is capable of storing up to b strings. In this paper, we investigate the depth of a randomly selected node in such a tree and the length of a randomly selected phrase in the generalized Lempel--Ziv scheme. These findings and some recent results allow us to compute the average redundancy of the generalized Lempel--Ziv code and compare it to the ordinary Lempel--Ziv code, leading to an optimal value of b. Analytic techniques of (precise) analyses of algorithms are used to establish most of these conclusions.

The purpose of this paper is to analyze the maxima properties (value and position) of some data structures. Our theorems concern the distribution of these random variables. Previously known results usually dealt with the mean and sometimes the variance of the random variables. Many of our results rely on diffusion techniques. This is a very powerful tool that has already been used with some success in algorithm complexity analysis.

In this paper, we settle a long-standing open problem concerning the average redundancy r/sub n/ of the Lempel-Ziv'78 (LZ78) code. We prove that for a memoryless source the average redundancy rate attains asymptotically Er/sub n/=(A+/spl delta/(n))/log n+ O(log log n/log/sup 2/ n), where A is an explicitly given constant that depends on source characteristics, and /spl delta/(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.

The goal of this contribution is twofold: (i) to introduce a generalized Lempel-Ziv parsing scheme, and (ii) to analyze second-order properties of some compression schemes based on the above parsing scheme. We consider a generalized Lempel-Ziv parsing scheme that partitions a sequence of length n into variable phrases (blocks) such that a new block is the longest substring seen in the past by at most b-1 phrases. The case b=1 corresponds to the original Lempel-Ziv scheme. In this paper, we investigate the size of a randomly selected phrase, and the average number of phrases of a given size through analyzing the so called b-digital search tree (b-DST) representation. For a memoryless source, we prove that the size of a typical phrase is asymptotically normally distributed. This result is new even for b=1, and b>1 is a non-trivial extension.

Consider the parsing algorithm developed by Lempel and Ziv (1978) that partitions a sequence of length n into variable phrases (blocks) such that a new block is the shortest substring not seen in the past as a phrase. In practice, the following parameters are of interest: number of phrases, the size of a phrase, the number of phrases of given size, and so forth. In this paper, we focus on the size of a randomly selected phrase, and the average number of phrases of a given size (the so-called average profile of phrase sizes). These parameters can be efficiently analyzed through a digital search tree representation. For a memoryless source with unequal probabilities of symbols generation (the so-called asymmetric Bernoulli model), we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed. In terms of digital search trees, we prove the normal limiting distribution of the typical depth (i.e., the length of a path from the root to a randomly selected node). The latter finding is proved by a technique that belongs to the toolkit of the "analytical analysis of algorithms", and it seems to be novel in the context of data compression.>

When parallel processes are linked in producer-consumer pairs and share a finite buffer where every portion is accessible to each process, it appears that a slow consumer may considerably delay the entire system. Using conditional critical sections, Dijkstra has proposed to reserve for each producer-consumer pair the adequate number of portions for a normal working and to dedicate the rest of the buffer to absorb the production peaks of the various pairs. L. W. Cooprider et al. then developed solutions which avoid systematic inspection and only use the now classical synchronisation primitives P and V. The present paper is devoted to the elaboration of solutions which improve parallelism and, when useful, discharge processes of administrative tasks. We have used and compared four synchronising methods to this aim: conditional critical sections, semaphores, path expressions and monitors.