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57
Index Coding with Side Information
, 2006
"... Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0, 1} n and wishes to broadcast a single message so that each receiver Ri c ..."
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Cited by 105 (0 self)
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Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0, 1} n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G on n nodes; Ri knows the bits of x in the positions {j  (i, j) is an edge of G}. G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEX codes for {0, 1} n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd antiholes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but is shown not to be tight.
Set Reconciliation with Nearly Optimal Communication Complexity
 in International Symposium on Information Theory
, 2000
"... We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information. We describe an approach to set reconciliation based on the ..."
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Cited by 77 (16 self)
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We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information. We describe an approach to set reconciliation based on the encoding of sets as polynomials. The resulting protocols exhibit tractable computational complexity and nearly optimal communication complexity. Also, these protocols can be adapted to work over a broadcast channel, allowing many clients to reconcile with one host based on a single broadcast, even if each client is missing a different subset.
Distributed functional compression through graph coloring
 In IEEE Data Compression Conf
, 2007
"... We consider the distributed computation of a function of random sources with minimal communication. Specifically, given two discrete memoryless sources, X and Y, a receiver wishes to compute f(X, Y) based on (encoded) information sent from X and Y in a distributed manner. A special case, f(X, Y)=(X, ..."
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Cited by 34 (6 self)
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We consider the distributed computation of a function of random sources with minimal communication. Specifically, given two discrete memoryless sources, X and Y, a receiver wishes to compute f(X, Y) based on (encoded) information sent from X and Y in a distributed manner. A special case, f(X, Y)=(X, Y), is the classical question of distributed source coding considered by Slepian and Wolf (1973). Orlitsky and Roche (2001) considered a somewhat restricted setup when Y is available as side information at the receiver. They characterized the minimal rate at which X needs to transmit data to the receiver, which is equal to the conditional graph entropy of the characteristic graph of X based on f. In our recent work (2006), we further established that this minimal rate can be achieved by means of graph coloring and distributed source coding (e.g. SlepianWolf coding). This characterization allows for the separation between “function coding ” and “correlation coding.” In this paper, we consider a more general setup where X and Y are both encoded (separately). This is a significantly harder setup for which to determine the complete rate region. We find that under a certain condition on the support set of X and Y (called the zigzag condition), it is possible to characterize the whole rate region based on graph colorings at X and Y separately. That is, any achievable pair of rates can be realized by means of first coloring graphs at X and Y separately (function coding) and then using SlepianWolf coding for these colors (correlation coding). We also obtain a characterization of the minimal joint rate under the assumption that there is a unique rate pair that minimizes the joint rate. Finally, we provide simulation results based on graph coloring to establish the rate gains on real sequences. I.
On zeroerror source coding with decoder side information
 IEEE TRANS. INF. THEORY
, 2003
"... Let ( ) be a pair of random variables distributed over a finite product set according to a probability distribution (). The following source coding problem is considered: the encoder knows, while the decoder knows and wants to learn without error. The minimum zeroerror asymptotic rate of transmiss ..."
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Cited by 26 (4 self)
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Let ( ) be a pair of random variables distributed over a finite product set according to a probability distribution (). The following source coding problem is considered: the encoder knows, while the decoder knows and wants to learn without error. The minimum zeroerror asymptotic rate of transmission is shown to be the complementary graph entropy of an associated graph. Thus, previous results in the literature provide upper and lower bounds for this minimum rate (further, these bounds are tight for the important class of perfect graphs). The algorithmic aspects of instantaneous code design are considered next. It is shown that optimal code design ishard. An optimal code design algorithm is derived. Polynomialtime suboptimal algorithms are also presented, and their average and worst case performance guarantees are established.
Lossless and near lossless source coding for multiple access networks
 IEEE Trans. Inform. Theory
, 2003
"... Abstract—A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences ..."
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Cited by 24 (3 self)
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Abstract—A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences
Zeroerror function computation in sensor networks
 In To appear in Proceedings of the 48th IEEE Conference on Decision and Control (CDC
, 2009
"... Abstract — We consider the problem of data harvesting in wireless sensor networks. A designated collector node seeks to compute a function of the sensor measurements. For a directed graph G = (V,E) on the sensor nodes, we wish to determine the optimal encoders on each edge which achieve zeroerror b ..."
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Cited by 22 (8 self)
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Abstract — We consider the problem of data harvesting in wireless sensor networks. A designated collector node seeks to compute a function of the sensor measurements. For a directed graph G = (V,E) on the sensor nodes, we wish to determine the optimal encoders on each edge which achieve zeroerror block computation of the function at the collector node. Our goal is to characterize the rate region in R E . We start with the two node problem, and determine a necessary and sufficient condition for the encoder that yields the optimal alphabet, from which we then calculate the minimum worst case and average case complexity. We then extend this result to trees and derive a necessary and sufficient condition for the encoder on each edge. The further extension of these results to directed acyclic graphs is not immediate. We provide an outer bound on the rate region by finding the disambiguation requirements for each cut, and describe examples where this outer bound is tight. Finally, we consider a collocated network of nodes with a specified order of transmission. We determine a necessary and sufficient condition for each encoder which is based on the transmissions received, and show that the average case complexity of computing a typethreshold function is Θ(1), in comparison to the worst case complexity of Θ(logn). I.
Fundamentals of Large Sensor Networks: Connectivity, Capacity, Clocks and Computation
, 2009
"... Sensor networks potentially feature large numbers of nodes. The nodes can monitor and sense their environment over time, communicate with each other over a wireless network, and process information that they exchange with each other. They differ from data networks in that the network as a whole may ..."
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Cited by 21 (1 self)
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Sensor networks potentially feature large numbers of nodes. The nodes can monitor and sense their environment over time, communicate with each other over a wireless network, and process information that they exchange with each other. They differ from data networks in that the network as a whole may be designed for a specific application. We study the theoretical foundations of such large scale sensor networks. We address four fundamental organizational and operational issues related to large sensor networks connectivity, capacity, clocks and function computation. To begin with, a sensor network must be connected so that information can indeed be exchanged between nodes. The connectivity graph of an adhoc network is modeled as a random graph and the critical range for asymptotic connectivity is determined, as well as the critical number of neighbors that a node needs to connect to. Next, given connectivity, we address the issue of how much data can be transported over the sensor network. We present fundamental bounds on capacity under several models, as well as architectural implications for how wireless communication should be organized. Temporal information is important both for the applications of sensor networks as well as their operation. We present fundamental bounds on the synchronizability of clocks in networks, and also present and analyze algorithms for clock synchronization. Finally we turn to the issue of gathering relevant information, that sensor networks are designed to do. One needs to study optimal strategies for innetwork aggregation of data, in order to reliably compute a composite function of sensor measurements, as well as the complexity of doing so. We address the issue of how such computation can be performed efficiently in a sensor network and the algorithms for doing so, for some classes of functions.
Energy efficient state estimation with wireless sensors through the use of predictive power control and coding
 IEEE Trans. Signal Processing
"... Abstract—We study state estimation via wireless sensors over fading channels. Packet loss probabilities depend upon timevarying channel gains, packet lengths and transmission power levels of the sensors. Measurements are coded into packets by using either independent coding or distributed zeroerro ..."
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Cited by 19 (9 self)
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Abstract—We study state estimation via wireless sensors over fading channels. Packet loss probabilities depend upon timevarying channel gains, packet lengths and transmission power levels of the sensors. Measurements are coded into packets by using either independent coding or distributed zeroerror coding. At the gateway, a timevarying Kalman filter uses the received packets to provide the state estimates. To trade sensor energy expenditure for state estimation accuracy, we develop a predictive control algorithm which, in an online fashion, determines the transmission power levels and codebooks to be used by the sensors. To further conserve sensor energy, the controller is located at the gateway and sends coarsely quantized power increment commands, only whenever deemed necessary. Simulations based on real channel measurements illustrate that the proposed method gives excellent results. Index Terms—Packet loss, power control, scheduling, source coding, state estimation, wireless sensors. I.
Data Verification and Reconciliation With Generalized ErrorControl Codes
 IEEE Trans. on Info. Theory
, 2001
"... We consider the problem of data reconciliation, which we model as two separate multisets of data that must be reconciled with minimum communication. Under this model, we show that the problem of reconciliation is equivalent to a variant of the graph coloring problem and provide consequent upper a ..."
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Cited by 18 (8 self)
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We consider the problem of data reconciliation, which we model as two separate multisets of data that must be reconciled with minimum communication. Under this model, we show that the problem of reconciliation is equivalent to a variant of the graph coloring problem and provide consequent upper and lower bounds on the communication complexity of reconciliation. More interestingly, we show by means of an explicit construction that the problem of reconciliation is, under certain general conditions, equivalent to the problem of finding good errorcorrecting codes. We show analogous results for the problem of multiset verification, in which we wish to determine whether two multisets are equal using minimum communication. As a result, a wide body of literature in coding theory may be applied to the problems of reconciliation and verification.