We study the distribution of envy in random matching markets under the Deferred Acceptance (DA) algorithm. Using tools from applied probability, we compute the expected number of proposing agents whom nobody envies and those who envy nobody. We obtain an exact finite-market expression for the former, based on a connection with the coupon collector problem, and asymptotic bounds for the latter. To put these quantities into perspective, we compare them to their counterparts under Random Serial Dictatorship (RSD): while RSD assigns a constant fraction of agents to their top choice, both DA and RSD leave exactly H n proposing agents unenvied in expectation. Our results show that these clearly unimprovable proposing agents constitute a vanishing fraction of the market. • We study the distribution of envy in random matching markets. • We derive exact expressions under Deferred Acceptance. • The expected number of unenvied students equals the n th harmonic number. • Students receiving top choices follow an approximately geometric rank distribution. • DA and Random Serial Dictatorship produce identical numbers of unenvied students. • Both unenvied and envy-free student fractions vanish asymptotically.

We consider a generalisation of the classical coupon collector problem. We define a super-coupon to be any s-subset of a universe of n coupons. In each round, a random r-subset from the universe is drawn and all its s-subsets are marked as collected. We show that the time to collect all super-coupons is rs-1nslogns[(1+o(1))] on average and has a Gumbel limit after a suitable normalisation. In a similar vein, we show that for any α∈(0,1), the expected time to collect (1-α)-proportion of all super-coupons is rs-1nslog(1α)[(1+o(1))]. The r=s case of this model is equivalent to the classical coupon collector model. We also consider a temporally dependent model where the r-subsets are drawn according to the following Markovian dynamics: the r-subset at round k+1 is formed by replacing a random coupon from the r-subset drawn at round k with another random coupon from outside this r-subset. We link the time it takes to collect all super-coupons in the r=s case of this model to the cover time of random walk on a certain finite regular graph and conjecture that in general, it takes rsrs-1nslogns[(1+o(1))] time on average to collect all super-coupons.

We construct the following cryptographic primitives with unconditional security in a bounded-key model: * One-time public-key encryption, where the public keys are pure quantum states * One-time signatures, where the verification keys are pure quantum states. In our model, the adversary is given a bounded number of copies of the public key. We present efficient constructions and nearly-tight lower bounds for the size of the secret keys. Our security proofs are based on the quantum coupon collector problem, which was originally studied in the context of learning theory. The quantum coupon collector seeks to learn a set of strings (coupons) when given several copies of a superposition over the coupons. We make novel connections between this problem and cryptography. Our main technical ingredient is a family of coupon states, with randomized phases, that come with strong hardness properties. Our analysis improves on prior work by (i) showing that the number of quantum states needed to learn the entire set of coupons is identical to the number of random coupons needed in the classical coupon collector problem. (ii) Furthermore we prove that this result holds for a randomly chosen set of coupons, whereas prior work only lower-bounded the number of coupon states required to learn the worst-case set of coupons.

Abstract We consider a generalization of the coupon collector problem with unequal probabilities, such that there are two additional coupons in the coupon set: one that speeds up the coupon collection process, and the one that slows it down. We derive some upper and lower bounds on the distribution function of the waiting time until a subcollection or a full collection of coupons is sampled.

Waiting Time for a Small Subcollection in the Coupon Collector Problem with Universal Coupon

We consider the following generalization of the classical coupon collector problem. We assume that, in addition to the initial collection of standard coupons, there is one more coupon that acts as a reset button, removing all coupons from the part of the collection that has already been drawn. For the case where standard coupons have unequal probabilities of being drawn, we obtain the distribution of the waiting time until the end of the collection process. For the case where standard coupons have equal probabilities, we derive a simple formula for the expected waiting time in terms of the beta function, and discuss the asymptotic properties of this expected waiting time, when the number of standard coupons tends toward infinity.

The classical coupon collector problem has various modifications and generalizations. One group of generalizations is based on the idea of introducing additional coupons, with special purposes, to the set of available coupons. We consider the case when this set consists of standard coupons (that can belong to the collection), a null coupon (which can be drawn, but does not belong to any collection), and an additional universal coupon, that can replace any of the standard coupons. By employing a Markov chain approach, we derive the exact forms of the k-step transition matrix and the fundamental matrix, which we use to obtain the properties of the waiting time until a subcollection, or a full collection is sampled, and some additional characteristics of the collecting process (probability that the coupon collecting procedure ends in a particular way). We also provide numerical examples and explain possible applications of the variant of the coupon collector problem we considered.

In this paper we consider a generalization of the coupon collector problem where we assume that the set of available coupons consists of standard coupons and an additional penalty coupon, which does not belong to the collection and interferes with collecting standard coupons.Applying Markov chain approach the following problem is solved: how many coupons (on average) one has to purchase in order to complete a collection without interference or to collect n more penalty coupons than standard coupons.Also, we obtain additional results related to the distribution of the waiting time until the collection is sampled without interference or until n more penalty coupons than standard coupons is sampled.

In the authors' previous papers (Ilienko, 2019) and (Ilienko & Stamatiieva, 2021), a novel approach to the classical coupon collector's and birthday problems was introduced, based on the theory of random point measures and their vague convergence. In this paper, we further develop this approach using the birthday problem as a case study, and apply it to establish new limit theorems for a range of nontrivial characteristics of the model. More specifically, we define a point process on a metric space consisting of a countable number of copies of the real line. The atoms of this process correspond to the arrival times of new objects. Owing to the structure of the constructed process, each atom is naturally associated with an integer that indicates the number of times an object of that class has arrived. We demonstrate that, as the number of classes tends to infinity, these processes converge vaguely to a certain Poisson point process on the same space. The application of the continuous mapping theorem to the proven convergence further yields distributional limit theorems for various characteristics of the model. The essentially infinite-dimensional nature of the involved point processes leads to corresponding infinite-dimensional limit theorems for these characteristics, fully revealing their asymptotic structure.

The time for reconstructing the attack graph in DDoS attacks

Recently Chen and Gao \cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over C, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over C by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables.Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over Fq improving on subsequent work by Chen, Gao and Yuan \citeChenGao2018. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.

Abstract Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$ , and that we keep buying boxes until we collect at least m coupons of each type. For $k\geq m$ call a certain coupon a k-ton if we see it k times by the time we have seen m copies of all of the coupons. Here we determine the asymptotic distribution of the number of k-tons after we have collected m copies of each coupon for any k in a restricted range, given any fixed m. We also determine the asymptotic joint probability distribution over such values of k, and the total number of coupons collected.

Metagenomic binning facilitates the reconstruction of genomes and identification of Metagenomic Species Pan-genomes or Metagenomic Assembled Genomes. We propose a method for identifying a set of de novo representative genes, termed signature genes, which can be used to measure the relative abundance and used as markers of each metagenomic species with high accuracy. An initial set of the 100 genes that correlate with the median gene abundance profile of the entity is selected. A variant of the coupon collector's problem was utilized to evaluate the probability of identifying a certain number of unique genes in a sample. This allows us to reject the abundance measurements of strains exhibiting a significantly skewed gene representation. A rank-based negative binomial model is employed to assess the performance of different gene sets across a large set of samples, facilitating identification of an optimal signature gene set for the entity. When benchmarked the method on a synthetic gene catalog, our optimized signature gene sets estimate relative abundance significantly closer to the true relative abundance compared to the starting gene sets extracted from the metagenomic species. The method was able to replicate results from a study with real data and identify around three times as many metagenomic entities. The code used for the analysis is available on GitHub: https://github.com/trinezac/SG_optimization. Supplementary data are available at Bioinformatics Advances online.

Assuming that there are N types of coupons, where the probability that the ith coupon appears is p i 0 for i = 1, . . . , N , with N i=1 p i = 1, we consider the variable T k which represents the number of acquisitions needed to obtain k N different coupons, and the variable Yn which represents the number of different coupons obtained in n acquisitions. In the coupon collector problem it is of interest to obtain the expected value of these random variables, as well as their rth moments. We provide new expressions for the rth moments of T k and Yn, and we give expressions for their moment generating functions. Unlike known formulas, our formula for the rth moment of T k is given in terms of recursive expressions and that of Yn is given in terms of finite sums, so that they can be easily implemented computationally. Furthermore, our formulas allow obtaining simplified expressions of the first few moments of the variables.

We examine the negative occupancy distribution and the coupon-collector distribution, both of which arise as distributions relating to hitting times in the extended occupancy problem. These distributions constitute a full solution to a generalised version of the coupon collector problem, by describing the behaviour of the number of items we need to collect to obtain a full collection or a partial collection of any size. We examine the properties of these distributions and show how they can be computed and approximated. We give some practical guidance on the feasibility of computing large blocks of values from the distributions, and when approximation is required.

Addressing the increasing demand for data exchange has led to the development of data markets that facilitate transactional interactions between data buyers and data sellers. Still, cost-effective and distribution-aware query answering is a substantial challenge in these environments. In this paper, while differentiating different types of data markets, we take the initial steps towards addressing this challenge. In particular, we envision a unified query answering framework and discuss its functionalities. Our framework enables integrating data from different sources in a data market into a dataset that meets user-provided schema and distribution requirements cost-effectively. In order to facilitate consumers' query answering, our system discovers data views in the form of join-paths on relevant data sources, defines a get-next operation to query views, and estimates the cost of get-next on each view. The query answering engine then selects the next views to sample sequentially to collect the output data. Depending on the knowledge of the system from the underlying data sources, the view selection problem can be modeled as an instance of a multi-arm bandit or coupon collector's problem.

Differential fault analysis of NORX using variants of coupon collector problem

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a weak concentration bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the former concerning coverage by i.i.d. random sets with arbitrary distribution, and the latter concerning hitting times for Markov chains with a strong monotonicity property. The growth model seems generally more tractable, and we record some basic results and open problems for that model.

An increasing number of communication and computational schemes with quantum advantages have recently been proposed, which implies that quantum technology has fertile application prospects. However, demonstrating these schemes experimentally continues to be a central challenge because of the difficulty in preparing high-dimensional states or highly entangled states. In this study, we introduce and analyze a quantum coupon collector protocol by employing coherent states and simple linear optical elements, which was successfully demonstrated using realistic experimental equipment. We showed that our protocol can significantly reduce the number of samples needed to learn a specific set compared with the classical limit of the coupon collector problem. We also discuss the potential values and expansions of the quantum coupon collector by constructing a quantum blind box game. The information transmitted by the proposed game also broke the classical limit. These results strongly prove the advantages of quantum mechanics in machine learning and communication complexity.

In combinatorial biotechnology, it is crucial for screening experiments to sufficiently cover the design space. In the BioCCP.jl package (Julia), we provide functions for minimum sample size determination based on the mathematical framework coined the Coupon Collector Problem. BioCCP.jl, including source code, documentation and Pluto notebooks, is available at https://github.com/kirstvh/BioCCP.jl.