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Balanced Allocations
 SIAM Journal on Computing
, 1994
"... Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability (1 + o(1)) ln n/ ln ln n balls in it. Suppose instead that for each ball we choose two boxes at random and place ..."
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Cited by 323 (8 self)
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Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability (1 + o(1)) ln n/ ln ln n balls in it. Suppose instead that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only ln ln n/ ln 2 +O(1) balls  exponentially less than before. Furthermore, we show that a similar gap exists in the infinite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and online load balancing.
Joint TxRx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization
 IEEE TRANS. SIGNAL PROCESSING
, 2003
"... This paper addresses the joint design of transmit and receive beamforming or linear processing (commonly termed linear precoding at the transmitter and equalization at the receiver) for multicarrier multipleinput multipleoutput (MIMO) channels under a variety of design criteria. Instead of consid ..."
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Cited by 290 (19 self)
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This paper addresses the joint design of transmit and receive beamforming or linear processing (commonly termed linear precoding at the transmitter and equalization at the receiver) for multicarrier multipleinput multipleoutput (MIMO) channels under a variety of design criteria. Instead of considering each design criterion in a separate way, we generalize the existing results by developing a unified framework based on considering two families of objective functions that embrace most reasonable criteria to design a communication system: Schurconcave and Schurconvex functions. Once the optimal structure of the transmitreceive processing is known, the design problem simplifies and can be formulated within the powerful framework of convex optimization theory, in which a great number of interesting design criteria can be easily accommodated and efficiently solved, even though closedform expressions may not exist. From this perspective, we analyze a variety of design criteria, and in particular, we derive optimal beamvectors in the sense of having minimum average bit error rate (BER). Additional constraints on the peaktoaverage ratio (PAR) or on the signal dynamic range are easily included in the design. We propose two multilevel waterfilling practical solutions that perform very close to the optimal in terms of average BER with a low implementation complexity. If cooperation among the processing operating at different carriers is allowed, the performance improves significantly. Interestingly, with carrier cooperation, it turns out that the exact optimal solution in terms of average BER can be obtained in closed form.
Supporting Stored Video: Reducing Rate Variability and EndtoEnd Resource Requirements through Optimal Smoothing
 IEEE/ACM Transactions on Networking
, 1998
"... Variablebitrate compressed video can exhibit significant, multipletimescale bit rate variability. In this paper we consider the transmission of stored video from a server to a client across a network, and explore how the client buffer space can be used most effectively toward reducing the variab ..."
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Cited by 268 (18 self)
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Variablebitrate compressed video can exhibit significant, multipletimescale bit rate variability. In this paper we consider the transmission of stored video from a server to a client across a network, and explore how the client buffer space can be used most effectively toward reducing the variability of the transmitted bit rate. Two basic results are presented. First, we show how to achieve the greatest possible reduction in rate variability when sending stored video to a client with given buffer size. We formally establish the optimality of our approach and illustrate its performance over a set of long MPEG1 encoded video traces. Second, we evaluate the impact of optimal smoothing on the network resources needed for video transport, under two network service models: Deterministic Guaranteed service [1, 31] and Renegotiated CBR (RCBR) service [9]. Under both models the impact of optimal smoothing is dramatic. 1 Introduction A broad range of applications is enabled by the capac...
Capacity Scaling in MIMO Wireless Systems Under Correlated Fading
 IEEE TRANS. INFORM. THEORY
, 2002
"... Previous studies have shown that singleuser systems employingelement antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to , assuming independent Rayleigh fading between antenna pairs. In this paper, we explore the capacity of dualantennaarray systems und ..."
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Cited by 256 (2 self)
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Previous studies have shown that singleuser systems employingelement antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to , assuming independent Rayleigh fading between antenna pairs. In this paper, we explore the capacity of dualantennaarray systems under correlated fading via theoretical analysis and raytracing simulations. We derive and compare expressions for the asymptotic growth rate of capacity with antennas for both independent and correlated fading cases; the latter is derived under some assumptions about the scaling of the fading correlation structure. In both cases, the theoretic capacity growth is linear in but the growth rate is 1020% smaller in the presence of correlated fading. We analyze our assumption of separable transmit/receive correlations via simulations based on a raytracing propagation model. Results show that empirical capacities converge to the limit capacity predicted from our asymptotic theory even at moderate n=16. We present results for both the cases when the transmitter does and does not know the channel realization.
Secure Transmission with Multiple Antennas: The MISOME Wiretap Channel
, 2007
"... The role of multiple antennas for secure communication is investigated within the framework of Wyner’s wiretap channel. We characterize the secrecy capacity in terms of generalized eigenvalues when the sender and eavesdropper have multiple antennas, the intended receiver has a single antenna, and t ..."
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Cited by 235 (19 self)
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The role of multiple antennas for secure communication is investigated within the framework of Wyner’s wiretap channel. We characterize the secrecy capacity in terms of generalized eigenvalues when the sender and eavesdropper have multiple antennas, the intended receiver has a single antenna, and the channel matrices are fixed and known to all the terminals, and show that a beamforming strategy is capacityachieving. In addition, we show that in the high signaltonoise (SNR) ratio regime the penalty for not knowing eavesdropper’s channel is small—a simple “secure spacetime code ” that can be thought of as masked beamforming and radiates power isotropically attains nearoptimal performance. In the limit of large number of antennas, we obtain a realizationindependent characterization of the secrecy capacity as a function of the number β: the number of eavesdropper antennas per sender antenna. We show that the eavesdropper is comparatively ineffective when β < 1, but that for β≥2 the eavesdropper can drive the secrecy capacity to zero, thereby blocking secure communication to the intended receiver. Extensions to ergodic fading channels are also provided.
Balancing histogram optimality and practicality for query result size estimation.
 In Proceedings of the 1995 ACM SIGMOD International Conference on Management of Data,
, 1995
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Learning the kernel function via regularization
 Journal of Machine Learning Research
, 2005
"... We study the problem of finding an optimal kernel from a prescribed convex set of kernels K for learning a realvalued function by regularization. We establish for a wide variety of regularization functionals that this leads to a convex optimization problem and, for square loss regularization, we ch ..."
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Cited by 151 (8 self)
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We study the problem of finding an optimal kernel from a prescribed convex set of kernels K for learning a realvalued function by regularization. We establish for a wide variety of regularization functionals that this leads to a convex optimization problem and, for square loss regularization, we characterize the solution of this problem. We show that, although K may be an uncountable set, the optimal kernel is always obtained as a convex combination of at most m+2 basic kernels, where m is the number of data examples. In particular, our results apply to learning the optimal radial kernel or the optimal dot product kernel. 1.
The Critical Transmitting Range for Connectivity in Sparse Wireless Ad Hoc Networks
, 2003
"... In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume n nodes, each capable of communicating with nodes within a radius of r, are randomly and uniformly distributed in a ddimensional re ..."
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Cited by 147 (12 self)
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In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume n nodes, each capable of communicating with nodes within a radius of r, are randomly and uniformly distributed in a ddimensional region with a side of length l; how large must the transmitting range r be to ensure that the resulting network is connected with high probability? First, we consider this problem for stationary networks, and we provide tight upper and lower bounds on the critical transmitting range for onedimensional networks, and nontight bounds for two and threedimensional networks. Due to the presence of the geometric parameter l in the model, our results can be applied to dense as well as sparse ad hoc networks, contrary to existing theoretical results that apply only to dense networks. We also investigate several related questions through extensive simulations. First, we evaluate the relationship between the critical transmitting range and the minimum transmitting range that ensures formation of a connected component containing a large fraction (e.g. 90%) of the nodes. Then, we consider the mobile version of the
Capacity bounds via duality with applications to multipleantenna systems on flatfading channels
 IEEE Trans. Inform. Theory
, 2003
"... A general technique is proposed for the derivation of upper bounds on channel capacity. The technique is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relativ ..."
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Cited by 145 (40 self)
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A general technique is proposed for the derivation of upper bounds on channel capacity. The technique is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relative entropy) over distributions on the channel output alphabet. Every choice of an output distribution — even if not the channel image of some input distribution — leads to an upper bound on mutual information. The proposed approach is used in order to study multiantenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown (or only partially known) at the receiver. It is demonstrated that, for high signaltonoise ratio (SNR), the capacity of such channels typically grows only doublelogarithmically in the SNR. This is in stark contrast to the case with perfect receiver side information where capacity grows logarithmically in the SNR. To better understand this phenomenon