In this paper, we identify a fundamental algorithmic problem that we term succinct dynamic covering (SDC), arising in many modern-day web applications, including ad-serving and online recommendation systems in eBay and Netflix. Roughly speaking, SDC applies two restrictions to the well-studied Max-Coverage problem: Given an integer k, X= 1,2,...,n and I= S_1, ..., S_m , S_i a subset of X, find a subset J of I, such that |J| <= k and the union of S in J is as large as possible. The two restrictions applied by SDC are: (1) Dynamic: At query-time, we are given a query Q, a subset of X, and our goal is to find J such that the intersection of Q with the union of S in J is as large as possible; (2) Space-constrained: We don't have enough space to store (and process) the entire input; specifically, we have o(mn), and maybe as little as O((m+n)polylog(mn)) space. The goal of SDC is to maintain a small data structure so as to answer most dynamic queries with high accuracy. We call such a scheme a Coverage Oracle. We present algorithms and complexity results for coverage oracles. We present deterministic and probabilistic near-tight upper and lower bounds on the approximation ratio of SDC as a function of the amount of space available to the oracle. Our lower bound results show that to obtain constant-factor approximations we need Omega(mn) space. Fortunately, our upper bounds present an explicit tradeoff between space and approximation ratio, allowing us to determine the amount of space needed to guarantee certain accuracy.
Previous work has studied the set cover problem under streaming models. One model studied in @cite_7 @cite_13 assumes that the sets are known in advance, only elements arrive online, and, the algorithms do not know in advance which subset of elements will arrive. An alternative model assumes that elements are known in advance and sets arrive in a streaming fashion @cite_9 . Our work differs from these works in that SDC operates under a storage budget, so all sets cannot be stored; moreover, SDC needs to provide a good cover for all possible dynamic query inputs.
@cite_9 @cite_13 @cite_7
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We generalize the graph streaming model to hypergraphs. In this streaming model, hyperedges are arriving online and any computation has to be done on-the-fly using a small amount of space. Each hyperedge can be viewed as a set of elements (nodes), so we refer to our proposed model as the “set-streaming” model of computation. We consider the problem of “maximum coverage”, in which k sets have to be selected that maximize the total weight of the covered elements. In the set-streaming model of computation, we show that our algorithm for maximumcoverage achieves an approximation factor of 14 . When multiple passes are allowed, we also provide a Θ(log n) approximation algorithm for the set-cover. We next consider a multi-topic blog-watch application, an extension of blogalert like applications for handling simultaneous multipletopic requests. We show how the problems of maximumcoverage and set-cover in the set-streaming model can be utilized to give efficient online solutions to this problem. We verify the effectiveness of our methods both on synthetic and real weblog data. We consider the problem of resource allocation in downlink OFDMA systems for multi service and unknown environment. Due to users' mobility and intercell interference, the base station cannot predict neither the Signal to Noise Ratio (SNR) of each user in future time slots nor their probability distribution functions. In addition, the traffic is bursty in general with unknown arrival. The probability distribution functions of the SNR, channel state and traffic arrival density are then unknown. Achieving a multi service Quality of Service (QoS) while optimizing the performance of the system (e.g. total throughput) is a hard and interesting task since it depends on the unknown future traffic and SNR values. In this paper we solve this problem by modeling the multiuser queuing system as a discrete time linear dynamic system. We develop a robust H∞ controller to regulate the queues of different users. The queues and Packet Drop Rates (PDR) are controlled by proposing a minimum data rate according to the demanded service type of each user. The data rate vector proposed by the controller is then fed as a constraint to an instantaneous resource allocation framework. This instantaneous problem is formulated as a convex optimization problem for instantaneous subcarrier and power allocation decisions. Simulation results show small delays and better fairness among users. In the leasing variant of Set Cover presented by [1], elements (U ) arrive over time and must be covered by sets from a family (F ) of subsets of (U ). Each set can be leased for (K ) different periods of time. Let ( | U | = n ) and ( | F | = m ). Leasing a set (S ) for a period (k ) incurs a cost (c_ S ^ k ) and allows (S ) to cover its elements for the next (l_k ) time steps. The objective is to minimize the total cost of the sets leased, such that elements arriving at any time (t ) are covered by sets which contain them and are leased during time (t ). [1] gave an optimal (O( n) )-approximation for the problem in the offline setting, unless ( P = NP ) [22]. In this paper, we give randomized algorithms for variants of Set Cover Leasing in the online setting, including a generalization of Online Set Cover with Repetitions presented by [2], where elements appear multiple times and must be covered by a different set at each arrival. Our results improve the ( O ( ^2 (mn)) ) competitive factor of Online Set Cover with Repetitions [2] to ( O ( d (dn)) = O ( m (mn)) ), where (d ) is the maximum number of sets an element belongs to. We consider the problem of storing segments of encoded versions of content files in a set of base stations located in a communication cell. These base stations work in conjunction with the main base station of the cell. Users move randomly across the space based on a discrete-time Markov chain model. At each time slot each user accesses a single base station based on it's current position and it can download only a part of the content stored in it, depending on the time slot duration. We assume that file requests must be satisfied within a given time deadline in order to be successful. If the amount of the downloaded (encoded) data by the accessed base stations when the time deadline expires does not suffice to recover the requested file, the main base station of the cell serves the request. Our aim is to find the storage allocation that minimizes the probability of using the main base station for file delivery. This problem is intractable in general. However, we show that the optimal solution of the problem can be efficiently attained in case that the time deadline is small. To tackle the general case, we propose a distributed approximation algorithm based on large deviation inequalities. Systematic experiments on a real world data set demonstrate the effectiveness of our proposed algorithms. Index Terms: Mobility-aware Caching, Markov Chain, MDS Coding, Small-cell Networks