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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 3199 (42 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 transmissionâ€™s range is optimally chosen, the bitâ€“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 1200 (13 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 ...
Complexity of finding embeddings in a ktree
 SIAM JOURNAL OF DISCRETE MATHEMATICS
, 1987
"... A ktree is a graph that can be reduced to the kcomplete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a ktree. This problem is motivated by the existence of polynomial time al ..."
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Cited by 382 (1 self)
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A ktree is a graph that can be reduced to the kcomplete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a ktree. This problem is motivated by the existence of polynomial time algorithms for many combinatorial problems on graphs when the graph is constrained to be a partial ktree for fixed k. These algorithms have practical applications in areas such as reliability, concurrent broadcasting and evaluation of queries in a relational database system. We determine the complexity status of two problems related to finding the smallest number k such that a given graph is a partial ktree. First, the corresponding decision problem is NPcomplete. Second, for a fixed (predetermined) value of k, we present an algorithm with polynomially bounded (but exponential in k) worst case time complexity. Previously, this problem had only been solved for k = 1,2,3.
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 329 (7 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.
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 ..."
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Cited by 266 (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.
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 214 (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.
Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity
, 1988
"... ..."
A dynamic survey of graph labellings
 Electron. J. Combin., Dynamic Surveys(6):95pp
, 2001
"... A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done ..."
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Cited by 169 (0 self)
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A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.
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 146 (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
 In Protocol Specification, Testing, and Verification VIII
, 1988
"... ..."