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The capacity of wireless networks
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2000
"... When n identical randomly located nodes, each capable of transmitting at bits per second and using a fixed range, form a wireless network, the throughput @ A obtainable by each node for a randomly chosen destination is 2 bits per second under a noninterference protocol. If the nodes are optimally p ..."
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Cited by 2220 (32 self)
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When n identical randomly located nodes, each capable of transmitting at bits per second and using a fixed range, form a wireless network, the throughput @ A obtainable by each node for a randomly chosen destination is 2 bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission’s range is optimally chosen, the bit–distance product that can be transported by the network per second is 2 @ A bitmeters per second. Thus even under optimal circumstances, the throughput is only 2 bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signaltointerference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance.
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 958 (14 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
Routing with Guaranteed Delivery in ad hoc Wireless Networks
 WIRELESS NETWORKS
, 2001
"... We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of ..."
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Cited by 648 (74 self)
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We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guarantee that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms.
A quantitative comparison of graphbased models for internet topology
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 1997
"... Graphs are commonly used to model the topological structure of internetworks, to study problems ranging from routing to resource reservation. A variety of graphs are found in the literature, including fixed topologies such as rings or stars, "wellknown" topologies such as the ARPAnet, and randomly ..."
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Cited by 223 (3 self)
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Graphs are commonly used to model the topological structure of internetworks, to study problems ranging from routing to resource reservation. A variety of graphs are found in the literature, including fixed topologies such as rings or stars, "wellknown" topologies such as the ARPAnet, and randomly generated topologies. While many researchers rely upon graphs for analytic and simulation studies, there has been little analysis of the implications of using a particular model, or how the graph generation method may a ect the results of such studies. Further, the selection of one generation method over another is often arbitrary, since the differences and similarities between methods are not well understood. This paper considers the problem of generating and selecting graph models that reflect the properties of real internetworks. We review generation methods in common use, and also propose several new methods. We consider a set of metrics that characterize the graphs produced by a method, and we quantify similarities and differences amongst several generation methods with respect to these metrics. We also consider the effect of the graph model in the context of a speciffic problem, namely multicast routing.
Practical Graph Isomorphism
, 1981
"... We develop an improved algorithm for canonically labelling a graph and finding generators for its automorph.ism grou.p. The emphasis i, on th.e power of the algorithm for,01 fling pr4ctical problem.t, rather than on the theoretical n,icetiu of tJu algo rith.m. Th.e nsult is a.n implementa.tion wh.ic ..."
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Cited by 212 (4 self)
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We develop an improved algorithm for canonically labelling a graph and finding generators for its automorph.ism grou.p. The emphasis i, on th.e power of the algorithm for,01 fling pr4ctical problem.t, rather than on the theoretical n,icetiu of tJu algo rith.m. Th.e nsult is a.n implementa.tion wh.ich011.11. 11I.ccel8/w.ly hll.ndle many grll.ph. & with. II. thot/.,1a.nd or m ore vertice~, a.nd i & ver y likely the most powerful graphisomorphism program currently in use.
Graphs on surfaces
, 2001
"... Let Σ be a surface. We prove that rigidity indices of graphs which admit a polyhedral embedding in Σ and 5connected graphs admitting an embedding in Σ are bounded by a constant depending on Σ. Moreover if the Euler characteristic of Σ is negative, then the separation index of graphs admitting a pol ..."
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Cited by 212 (17 self)
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Let Σ be a surface. We prove that rigidity indices of graphs which admit a polyhedral embedding in Σ and 5connected graphs admitting an embedding in Σ are bounded by a constant depending on Σ. Moreover if the Euler characteristic of Σ is negative, then the separation index of graphs admitting a polyhedral embedding in Σ is also bounded. As a side result we show that distinguishing number of both Σpolyhedral and 5connected graphs which admit and embedding in Σ is also bounded. 1 Introduction and
Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity
, 1988
"... , Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [2 ..."
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Cited by 182 (4 self)
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, Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [24]). The sources considered output strings according to probability distributions in which no single string is too probable. The new model provides a fruitful viewpoint on problems studied previously as: ffl Extracting almost perfect bits from sources of weak randomness: the question of possibility as well as the question of efficiency of such extraction schemes are addressed. ffl Probabilistic Communication Complexity: it is shown that most functions have linear communication complexity in a very strong probabilistic sense. ffl Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [27]). The paper has appeared in SIAM Journal o...
Models of Random Regular Graphs
 In Surveys in combinatorics
, 1999
"... In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular g ..."
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Cited by 157 (33 self)
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In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides. 1
On the ManytoOne Transport Capacity of a Dense Wireless Sensor Network and the Compressibility of Its Data
, 2003
"... In this paper we investigate the capability of largescale sensor networks to measure and transport a twodimensional field. We consider a datagathering wireless sensor network in which densely deployed sensors take periodic samples of the sensed field, and then scalar quan tize, encode and tr ..."
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Cited by 122 (7 self)
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In this paper we investigate the capability of largescale sensor networks to measure and transport a twodimensional field. We consider a datagathering wireless sensor network in which densely deployed sensors take periodic samples of the sensed field, and then scalar quan tize, encode and transmit them to a single receiver/central controller where snapshot images of the sensed field are reconstructed. The quality of the reconstructed field is limited by the ability of the encoder to compress the data to a rate less than the singlereceiver transport capacity of the network. Subject to a constraint on the quality of the reconstructed field, we are interested in how fast data can be collected (or equivalently how closely in time these snapshots can be taken) due to the limitation just mentioned. As the sensor density increases to infinity, more sensors send data to the central controller. However, the data is more correlated, and the encoder can do more compression. The question is: Can the encoder compress sufficiently to meet the limit imposed by the transport capacity? Alternatively, how long does it take to transport one snapshot ? We show that as the density increases to infinity, the total number of bits required to attain a given quality also increases to infinity under any compression scheme. At the same time, the singlereceiver transport capacity of the network remains constant as the density increases. We therefore conclude that for the given scenario, even though the correlation between sensor data increases as the density increases, any data compression scheme is insucient to transport the required amount of data for the given quality. Equivalently, the amount of time it takes to transport one snapshot goes to infinity.
An Optimization Technique for Protocol Conformance Test Generation Based on UIO Sequences and Rural Chinese Postman Tours
, 1991
"... This paper describes a method for generating test sequences for checking the conformance of a protocol implementation to its specification. A Rural Chinese Postman Tour is used to determine a minimumcost tour of the transition graph of a finitestate machine. When used in combination with Unique In ..."
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Cited by 110 (14 self)
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This paper describes a method for generating test sequences for checking the conformance of a protocol implementation to its specification. A Rural Chinese Postman Tour is used to determine a minimumcost tour of the transition graph of a finitestate machine. When used in combination with Unique Input/Output Sequences [18], the technique yields an efficient method for computing a test sequence for protocol conformance testing.