Abstract—Motivated by the study of peer-to-peer file swarming systems à la BitTorrent, we introduce a probabilistic model of coupon replication systems. These systems consist of users, aiming to complete a collection of distinct coupons. Users are characterised by their current collection of coupons, and leave the system once they complete their coupon collection. The system evolution is then specified by describing how users of distinct types meet, and which coupons get replicated upon such encounters. For open systems, with exogenous user arrivals, we derive necessary and sufficient stability conditions in a layered scenario, where encounters are between users holding the same number of coupons. We also consider a system where encounters are between users chosen uniformly at random from the whole population. We show that performance, captured by sojourn time, is asymptotically optimal in both systems as the number of coupon types becomes large. We also consider closed systems with no exogenous user arrivals. In a special scenario where users have only one missing coupon, we evaluate the size of the population ultimately remaining in the system, as the initial number of users, N, goes to infinity. We show that this decreases geometrically with the number of coupons, K. In particular, when the ratio K / log(N) is above a critical threshold, we prove that this number of left-overs is of order log(log(N)). These results suggest that performance of file swarming systems does not depend critically on either altruistic user behavior, or on load balancing strategies such as rarest first. 1.

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We extend Zeilberger's fast algorithm for definite hypergeometric summation to non-hypergeometric holonomic sequences. The algorithm generalizes to differential and q-cases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy.

This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present acerti cate that consists of a pair of functions of two integer variables. To prove the identity, take the two functions that are given, check that condition (1) below is satis ed (a simple mechanical task), and check the equally simple fact that the boundary conditions (F1), (G1), (G2) below

Analysis and experiments have shown that hierarchical problem-solving is most effective when the hierarchy satisfies the downward refinement property (DRP), whereby every abstract solution can be refined to a concrete-level solution without backtracking across abstraction levels. However, the DRP is a strong requirement that is not often met in practice. In this paper we examine the case when the DRP fails, and provide an analytical model of search complexity parameterized by the probability of an abstract solution being refinable. Our model provides a more accurate picture of the effectiveness of hierarchical problem-solving. We then formalize the DRP in Abstrips-style hierarchies, providing a syntactic test that can be applied to determine if a hierarchy satisfies the DRP. Finally, we describe an algorithm called Highpoint that we have developed. This algorithm builds on the Alpine algorithm of Knoblock in that it automatically generates abstraction hierarchies. However, it uses th...

Divisible load applications occur in many fields of science and engineering, can be eas-ily parallelized in a master-worker fashion, but pose several scheduling challenges. While a number of approaches have been proposed that allocate work to workers in a single round, using multiple rounds improves overlap of computation with communication. Unfortunately, multi-round algorithms are difficult to analyze and have thus received only limited attention. In this paper we answer three open questions in the multi-round divisible load scheduling area: (i) How to account for latencies? (ii) How to account for heterogeneous platforms; and (iii) How many rounds should be used? To answer (i), we derive the first closed-form optimal schedule for a homogeneous platform with both computation and communication latencies, for a given number of rounds. To answer (ii) and (iii), we present a novel algorithm, UMR. We use simulation to evaluate UMR in a variety of realistic scenarios.

Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be performed "algorithmically". As a consequence, we can prove any identity of holonomic functions or sequences automatically. Based on this theory, the author implemented a package that contains procedures for automatic manipulations and transformations of univariate holonomic functions and sequences within the computer algebra system Mathematica. This package is introduced in detail. In addition, we describe some different techniques for proving holonomic identities.

The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NPcomplete. Examples are provided that show that 1) the minimal trellis for a nonlinear code may not be observable, i.e., some codewords may be represented by more than one path through the trellis and 2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis product is defined and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and configurations of non-attacking rooks on a triangular chess board is established and used to show that the number of distinct minimal li...

An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.

. We will introduce a new class of erasure codes built from irregular bipartite graphs that have linear time encoding and decoding algorithms and can transmit over an erasure channel at rates arbitrarily close to the channel capacity. We also show that these codes are close to optimal with respect to the trade-off between the proximity to the channel capacity and the running time of the recovery algorithm. 1 Introduction A linear error-correcting code of block length n and dimension k over a finite field IF q ---an [n; k] q -code for short---is a k-dimensional linear subspace of the standard vector space IF n q . The elements of the code are called codewords. To the code C there corresponds an encoding map Enc which is an isomorphism of the vector spaces IF k q and C. A sender, who wishes to transmit a vector of k elements in IF q to a receiver uses the mapping Enc to encode that vector into a codeword. The rate k=n of the code is a measure for the amount of real information...