## Theory of Communication Networks (2007)

Citations: | 4 - 2 self |

### BibTeX

@MISC{Pandurangan07theoryof,

author = {Gopal Pandurangan and Maleq Khan},

title = {Theory of Communication Networks},

year = {2007}

}

### OpenURL

### Abstract

### Citations

8530 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...ted Bellman-Ford (DBF) algorithm. It is used in the Routing Information Protocol (RIP) and the Border Gateway Protocol (BGP) of the Internet. We will describe the basic idea behind the DBF algorithm (=-=[15, 2]-=-). Suppose we want to compute the shortest (least-cost) path between s and all other nodes in a given undirected graph G = (V, E, c) with real-valued positive edge weights. During the algorithm each n... |

3460 | Chord: A scalable peer-to-peer lookup service for internet applications
- Stoica, Morris, et al.
- 2001
(Show Context)
Citation Context ... guarantee similar properties [31, 36, 29, 19, 14]. Before we end our discussion on P2P, we briefly mention another important design approach, referred to as a Distributed Hash Table (DHT) (see e.g., =-=[43, 47, 36]-=-). A DHT does two things: (1) creates a fully decentralized index that maps file identifiers to peers and (2) allows a peer to search for a file very efficiently (typically logarithmically in the size... |

2689 | A scalable content addressable network
- Ratnasamy, Francis, et al.
- 2001
(Show Context)
Citation Context ... guarantee similar properties [31, 36, 29, 19, 14]. Before we end our discussion on P2P, we briefly mention another important design approach, referred to as a Distributed Hash Table (DHT) (see e.g., =-=[43, 47, 36]-=-). A DHT does two things: (1) creates a fully decentralized index that maps file identifiers to peers and (2) allows a peer to search for a file very efficiently (typically logarithmically in the size... |

1871 | Randomized Algorithms - MOTWANI, P - 1995 |

1788 | Random Graphs
- Bollobás
- 1985
(Show Context)
Citation Context ...h high probability its diameter is O(log N). The high-level idea of the proof is to show that the network at any time resembles a random graph, i.e., the classical Erdos-Renyi or G(n, p) random graph =-=[7]-=-. The intuition underlying the analysis is that connections are chosen randomly, and thus the topology should resemble a G(n, p) random graph. In 43sa G(n, p) random graph the connectivity threshold i... |

1533 |
Distributed Algorithms
- LYNCH
- 1996
(Show Context)
Citation Context ...rt) and defined by L = maxv∈V L(v), where L(v) = max u∈ΓW (v)(v) l(u, v). Rank selection. The nodes select unique ranks as follows. First a leader is elected by a leader election algorithm (e.g., see =-=[35]-=-). (Or assume that there is one such node, say which initiates the algorithm.) Let s 19sbe the leader node. The leader picks a number p(s) from the range [b − 1, b], where b is a number arbitrarily ch... |

1367 |
Rate control in communication networks: shadow prices, proportional fairness and stability
- Kelly, Maullo, et al.
- 1998
(Show Context)
Citation Context ... S(l) = {s ∈ S|l ∈ L(s)} be the set of sources that use link l. Note that l ∈ L(s) if and only if s ∈ S(l). The resource allocation problem can be formulated as the following nonlinear program (e.g., =-=[23, 32]-=-). The objective is to choose source rates x = {xs, s ∈ S} so as to ∑ maxxs≥0 Us(xs) s subject to ∑ s∈S(l) xs ≤ cl, l = 1, . . . , L. (43) The constraint says that the aggregate source rate at any lin... |

1308 |
Introduction to Parallel Algorithms and Architectures: Arrays
- Leighton
- 1992
(Show Context)
Citation Context ...llection of independent M/M/1 queues, though the arrival process to each of the queues is not necessarily Poisson. An application of Jackson’s theorem can be seen in routing on the butterfly network (=-=[30]-=-[Chapter 3.6]) with Poisson arrivals to the inputs, and exponential transition time through the edges. Assume that the arrival rate of packets to each of the inputs is λ. Then the arrival rate to each... |

1041 | A measurement study of peer-to-peer file sharing systems
- Saroiu, Gummadi, et al.
- 2002
(Show Context)
Citation Context ...alyze for this setting, since such an adversary can make sure that the network is always disconnected.) This also turns out to be a reasonable model (it approximates real-life P2P networks quite well =-=[44]-=-). Let Gt be the network at time t. The evolution of the size of G depends only on the ratio λ/µ = N, the steady state size of the network (see Section 4.1). 42s7.3 Connectivity The following theorem ... |

895 |
Approximation Algorithms
- Vazirani
- 2001
(Show Context)
Citation Context ...e other nodes in the graph (these are called Steiner nodes). Finding the minimum cost Steiner tree in a arbitrary weighted graph is NP-hard. The following approximation algorithm is well-known (e.g., =-=[50]-=-). Transform the given graph G = (V, E, w) as a complete metric graph G ′ = (V, E ′ , w ′ ) where the pairwise edge costs in E ′ are the corresponding pairwise (weighted) shortest path distances in G,... |

603 |
Data Structures and Network Algorithms
- Tarjan
- 1983
(Show Context)
Citation Context ...r is not. The minimum-weight outgoing edge (MOE) of a fragment F is the edge with minimum weight among all outgoing edges of F . As an immediate consequence of the blue rule for MST (see Chapter 6 in =-=[48]-=-), the MOE of a fragment F = (VF , EF ) is an 16sedge of the MST. Consider a cut (VF , V − VF ) of G. The MOE of F is the minimum-weight edge in the cut (VF , V − VF ), and therefore the MOE is an edg... |

536 | Queueing Systems - Kleinrock - 1975 |

341 |
A distributed algorithm for minimumweight spanning trees
- Gallager, Humblet, et al.
- 1983
(Show Context)
Citation Context ...hich is optimal. Here we focus on distributed algorithms for this problem and few more properties of MST. The first distributed algorithm for the MST problem was given by Gallager, Humblet, and Spira =-=[18]-=- in 1983. This algorithm is known as GHS algorithm and will work in an asynchronous model also. GHS Algorithm GHS algorithm assumes that the edge weights are distinct. If all the edges of a connected ... |

191 | Analysis of the evolution of peer-to-peer systems
- Liben-Nowell, Balakrishnan, et al.
- 2002
(Show Context)
Citation Context ...ove protocol is that it takes O(log N) (N is the steady state network size) work per insertion/deletion. In a subsequent paper, using the same stochastic model, Liben-Nowell, Balakrishnan, and Karger =-=[31]-=- show that Ω(log N) work is also required to maintain connectivity. A drawback of the protocol of [38] is that it is not fully decentralized — the entering node needs to connect to a small set of rand... |

165 |
A Minimum Delay Routing Algorithm Using Distributed Computation
- Gallager
- 1977
(Show Context)
Citation Context ...(5) i,k The problem is complicated since the objective function is not linear and the challenge is to come up with a distributed algorithm. We briefly describe a distributed algorithm due to Gallager =-=[17]-=-. Let ti(j) be the total flow (traffic) at node i destined for j; thus ti(j) = ri(j) + � k fki(j). Let φik(j) be the fraction of the node flow ti(j) that is routed over edge (i, k). Since node flow ti... |

147 | A survey and comparison of peer-to-peer overlay network schemes - Lua, Crowcroft, et al. |

126 |
Data network, 2nd Edition
- Bertsekas, Gallager
- 1997
(Show Context)
Citation Context ... to as the “count-to-infinity” problem and shows that the algorithm can be very slow in reacting to a change in an edge cost. Some heuristics have been proposed to alleviate this problem; we refer to =-=[5, 27]-=- for details. An Approximate Distributed Bellman-Ford Algorithm The DBF algorithm can suffer from exponential message complexity in an asynchronous setting. For example, consider the graph shown in Fi... |

123 |
Parallel and Distributed Computation
- Bertsekas, Tsitsiklis
- 1989
(Show Context)
Citation Context ... the Lagrange multipliers. The first term is separable in xs, and hence ⎛ � maxxs ⎝Us(xs) − xs s � l∈L(s) pl ⎞ ⎠ = � ⎛ maxxs ⎝Us(xs) − xs The objective function of the dual problem is thus (e.g., see =-=[6]-=-): minpl≥0D(p) := � ⎛ maxxs≥0 ⎝Us(xs) − xs s s � l∈L(s)pl ⎞ � l∈L(s) pl ⎞ ⎠ . (46) ⎠ + � plcl. (47) The first term of the dual-objective function D(p) is decomposable into S separable subproblems, one... |

89 |
The Probabilistic Method. 2nd Edition
- Alon, Spencer
- 2000
(Show Context)
Citation Context ...xation is the standard multicommodity flow problem where confluence constraints need not be satisfied. Let Di = � v∈V d(i, v), i.e., the total demand for commodity i. Let x : {1, . . . , k} × V × V → =-=[0, 1]-=- and ρ ∈ R + . x(i, u, v) will denote the fraction of the total commodity i that enters node u that leaves for v. If the flow is confluent x(i, u, v) will be 0 or 1 for all i ∈ {1, . . . , k} and u, v... |

75 | Adversarial queuing theory
- Borodin, Kleinberg, et al.
(Show Context)
Citation Context ...etwork there are log n links (queues) from any input node to a output node, the expected time in the system for a customer (packet) is ≤ log N µ−λ . 4.3 Adversarial Queuing Adversarial queuing theory =-=[8]-=- addresses some of the restrictions inherent in probabilistic analysis and queuing theory based on time-invariant stochastic generation. This theory is aimed to systemic study of queuing with little o... |

71 |
Optimization flow control—I: basic algorithm and convergence
- Low, Lapsley
- 1999
(Show Context)
Citation Context ... S(l) = {s ∈ S|l ∈ L(s)} be the set of sources that use link l. Note that l ∈ L(s) if and only if s ∈ S(l). The resource allocation problem can be formulated as the following nonlinear program (e.g., =-=[23, 32]-=-). The objective is to choose source rates x = {xs, s ∈ S} so as to subject to � � maxxs≥0 Us(xs) s∈S(l) s xs ≤ cl, l = 1, . . . , L. (43) The constraint says that the aggregate source rate at any lin... |

70 |
Analysis of backoff protocols for multiple access channels
- Hastad, Leighton, et al.
- 1996
(Show Context)
Citation Context ...otocols have good performance? For example, is exponential backoff better than polynomial backoff? To answer this rigorously we need a model and a performance measure. We consider the following model =-=[21]-=-. Time is partitioned into steps. At a beginning of a step each station receives a new message with probability λi, for 1 ≤ i ≤ N, where N is the number of stations in the system. The arrival of new m... |

65 | A simple local-control approximation algorithm for multicommodity flow
- Awerbuch, Leighton
- 1993
(Show Context)
Citation Context ... fki(j) = � fik(j). (4) k Both the above problems can be solved using linear programming formulations. These algorithms are complex and are not easy to implement distributively. Awerbuch and Leighton =-=[3]-=- gave a local control algorithm for solving the above problems. It gives an (1 + ɛ)-approximation for a desired ɛ > 0. See Chapter 7 for a description of this algorithm. This algorithm can be implemen... |

59 |
Rate Control for Communication Networks
- Kelly, Maullo, et al.
- 1998
(Show Context)
Citation Context ... S(l) = {s ∈ S|l ∈ L(s)} be the set of sources that use link l. Note that l ∈ L(s) if and only if s ∈ S(l). The resource allocation problem can be formulated as the following nonlinear program (e.g., =-=[23, 32]-=-). The objective is to choose source rates x = {xs, s ∈ S} so as to subject to � � maxxs≥0 Us(xs) s∈S(l) s xs ≤ cl, l = 1, . . . , L. (43) The constraint says that the aggregate source rate at any lin... |

54 | Network synchronization with polylogarithmic overhead - Awerbuch, Peleg - 1990 |

43 | Building low-diameter peer-to-peer networks
- Pandurangan, Raghavan, et al.
- 2003
(Show Context)
Citation Context ... rate λ. The duration of time a node stays connected to the network is independently and exponentially distributed with parameter µ. Without loss of generality, let λ = 1 and let N = 1/µ. Theorem 4.1 =-=[38]-=- If t N → ∞ then with high probability |Vt| = N ± o(N). Proof: The number of nodes in the network at time t, X(t), is Poisson distributed with mean λ � t 0 (1 − G(x))dx which is N(1 − e −t/N ). 26 (23... |

40 | A mathematical framework for designing a low-loss low-delay internet
- Low, Srikant
(Show Context)
Citation Context ...e that progressively converges to the optimal. ⎛ pl(t + 1) = max ⎝pl(t) + (γ � ⎞ xs(p(t))) − γcl), 0⎠ . (49) s∈S(l) where γ > 0 is the stepsize parameter (has to be sufficiently small for convergence =-=[33]-=-). The above formula has nice interpretation in the language of economics: if the demand � s∈S(l) xs(p(t)) for bandwidth at link l exceeds the supply cl, then raise the price pl(t); otherwise, reduce ... |

39 | Near-tight Lower Bound on the Time Complexity of Distributed MST Construction
- Peleg, Rubinovich
(Show Context)
Citation Context ...lgorithm takes O(D + √ n) time. Thus the time complexity of Kutten and Peleg’s algorithm is O(D + √ n log ∗ n). The communication complexity of this algorithm is O(|E| + n 1.5 ). Peleg and Rabinovich =-=[41]-=- showed that ˜ Ω(D + √ n) time 1 is required for distributed construction of MST. Elkin [16] showed that even finding an approximate MST in distributed time on graphs of small diameter (e.g., O(log n)... |

39 | Stochastic contention resolution with short delays
- Raghavan, Upfal
- 1999
(Show Context)
Citation Context ...navoidable in backoff protocols: they showed that the delay of any stable exponential or polynomial backoff protocol is at least polynomial in N. We next consider a protocol due to Raghavan and Upfal =-=[42]-=- that gives a protocol that guarantees a delay of O(log N), provided the total arrival rate λ is less than a fixed constant λ ′ ≈ 1/10. The main feature of this protocol is that each sender has a tran... |

37 |
An Unconditional Lower Bound on the Time-Approximation Tradeoff for the Minimum Spanning Tree Problem
- Elkin
(Show Context)
Citation Context ...O(D + √ n log ∗ n). The communication complexity of this algorithm is O(|E| + n 1.5 ). Peleg and Rabinovich [41] showed that ˜ Ω(D + √ n) time 1 is required for distributed construction of MST. Elkin =-=[16]-=- showed that even finding an approximate MST in distributed time on graphs of small diameter (e.g., O(log n)) within a ratio H requires Ω( Khan and Pandurangan’s Approximation Algorithm 1 The ˜ Ω nota... |

36 | Sampling regular graphs and a peer-to-peer network
- Cooper, Greenhill
- 2005
(Show Context)
Citation Context ...at the giant component that emerges from the random process is indeed the whole graph. A number of works have subsequently used random graphs to efficiently design P2P with good properties, see e.g., =-=[29, 14, 19]-=-. In a dynamic network, there is the additional challenge of quantifying the work done by the algorithm to maintain the desired properties. An important advantage of the above protocol is that it take... |

34 | Universal Routing Strategies for Interconnection Networks - Scheideler |

29 |
A last word on
- Stidham
- 1974
(Show Context)
Citation Context ... time spent in the system by customer i. 1 N = lim t→∞ t λ = lim t→∞ T = lim t→∞ � t 0 α(t) t � α(t) i=1 Ti α(t) N(t) (16) Then the following basic equation known as Little’s Lemma holds: N = λT (see =-=[46]-=- for a proof). The M/M/1 System The name M/M/1 reflects the standard queueing theory nomenclature whereby: (1) The first letter indicates the nature of the arrival process. For example, M stands for m... |

28 |
Fast distributed construction of k-dominating sets and applications
- Kutten, Peleg
- 1998
(Show Context)
Citation Context ...es Ω(n lg n + |E|) messages [49]. Thus the communication (message) complexity of GHS algorithm is optimal. However, its time complexity is not optimal. Kutten and Peleg’s Algorithm Kutten and Peleg’s =-=[28]-=- distributed MST algorithm runs in O(D + √ n log ∗ n) time, where D is the diameter of the graph G. The algorithm consists of two parts. In the first part, similar to GHS algorithm, 17sthis algorithm ... |

25 | Contention resolution with constant expected delay
- Goldberg, MacKenzie, et al.
(Show Context)
Citation Context ...ass of protocols that include backoff protocols, if the total arrival rate λ is less than some fixed constant λ1 < 1, then the delay must be Ω(n). Using a more sophisticated protocol, Goldberg et al. =-=[20]-=- are able to show a constant expected delay for arrival rate up to 1/e. This protocol is practical if one assumes that the clocks of the stations are synchronized. If not, it is shown that O(1) delay ... |

23 | A fast distributed approximation algorithm for minimum spanning trees
- KHAN, PANDURANGAN
- 2008
(Show Context)
Citation Context ...f small diameter (e.g., O(log n)) within a ratio H requires Ω( Khan and Pandurangan’s Approximation Algorithm 1 The ˜ Ω notation hides logarithmic factors. � n H log n 18 ) time.sKhan and Pandurangan =-=[24]-=- gave an algorithm for O(log n)-approximate MST that runs in Õ(D + L) time. L is called local shortest path diameter. L can be small in many classes of graphs, especially those of practical interest s... |

14 |
Communication Nets: Stochastic Message Flow and
- Kleinrock
- 1964
(Show Context)
Citation Context ... the second queue than short packets, since their transmission time at the first queue takes longer, thereby giving more time for the second queue to empty out. To deal with this situation, Kleinrock =-=[25]-=- introduced the following independence assumption: 27 (27)sAdopt an M/M/1 model for each communication link regardless the interaction of the traffic on this link with traffic on other links. Although... |

14 |
Computer Networking: A Top-Down Approach, 4th ed
- Kurose, Ross
- 2007
(Show Context)
Citation Context ... to as the “count-to-infinity” problem and shows that the algorithm can be very slow in reacting to a change in an edge cost. Some heuristics have been proposed to alleviate this problem; we refer to =-=[5, 27]-=- for details. An Approximate Distributed Bellman-Ford Algorithm The DBF algorithm can suffer from exponential message complexity in an asynchronous setting. For example, consider the graph shown in Fi... |

14 |
Distributed Computing: A Locality Sensitive Approach
- Peleg
- 2000
(Show Context)
Citation Context ... therefore, the weight of a single edge can be communicated in one time step. This model of the distributed computation is called the (synchronous) CON GEST (log n) model or simply the CON GEST model =-=[40]-=-. The CON GEST model is not very realistic for the Internet. However, it has been widely used model to study distributed algorithms and captures the notion that there is a bound on the amount of messa... |

12 |
Approximate distributed Bellman-Ford algorithms [computer network routing
- Awerbuch, Bar-Noy, et al.
- 1991
(Show Context)
Citation Context ...ted Bellman-Ford (DBF) algorithm. It is used in the Routing Information Protocol (RIP) and the Border Gateway Protocol (BGP) of the Internet. We will describe the basic idea behind the DBF algorithm (=-=[15, 2]-=-). Suppose we want to compute the shortest (least-cost) path between s and all other nodes in a given undirected graph G = (V, E, c) with real-valued positive edge weights. During the algorithm each n... |

11 |
2001] Introduction to Distributed Algorithms, second edition
- Tel
(Show Context)
Citation Context ...nt can reach is lg n, where n = |V |. The algorithm takes O(n lg n + |E|) messages and O(n lg n) time. It can be shown that any distributed algorithm for MST problem requires Ω(n lg n + |E|) messages =-=[49]-=-. Thus the communication (message) complexity of GHS algorithm is optimal. However, its time complexity is not optimal. Kutten and Peleg’s Algorithm Kutten and Peleg’s [28] distributed MST algorithm r... |

10 | Meet and merge: Approximation algorithms for confluent flows
- Chen, Rajaraman, et al.
(Show Context)
Citation Context ...stion ratio among all edges. The minimum congestion ratio problem is to find a confluent flow to satisfy all the demands with minimum congestion ratio. The above problem is NP-hard to solve optimally =-=[12]-=-. Hence the focus is to look for an approximately optimal solution that can be implemented in a distributed fashion. We present an algorithm due to [11] called as the locally independent rounding algo... |

7 | Almost) tight bounds and existence theorems for confluent flows
- Chen, Kleinberg, et al.
(Show Context)
Citation Context ...ore or less uniform and the maximum degree is not large then these conditions are reasonable. Stronger bounds on various special cases of the confluent flow problem have been established. We refer to =-=[12, 10]-=- for details. 3.3 Broadcast Routing Broadcasting is another important communication mode: sending a message from a source node to all other nodes of the network. We will consider two basic broadcastin... |

4 |
Simple distributed algorithms for approximating minimum steiner trees
- Chalermsook, Fakcharoenphol
- 2005
(Show Context)
Citation Context ...uted approximation algorithms have been recently proposed for this problem. Based on the MST approximation described above, an O(n log n)-time 2-approximate distributed algorithm has been proposed in =-=[9]-=-. The NNT approach of [24] described in Section 3.3.2 yields a faster O(log n)-approximation with the same time bounds. Another advantage of the NNT approach is that this algorithm works with virtuall... |

4 |
Viceroy: A Scalable and Dynamic
- Malkhi, Naor, et al.
- 2002
(Show Context)
Citation Context ... random nodes which is assumed to be available in a centralized fashion. A number of subsequent papers have addressed this and devised fully decentralized protocols which guarantee similar properties =-=[31, 36, 29, 19, 14]-=-. Before we end our discussion on P2P, we briefly mention another important design approach, referred to as a Distributed Hash Table (DHT) (see e.g., [43, 47, 36]). A DHT does two things: (1) creates ... |

2 | The confluent capacity of the internet: congestion vs. dilation
- Chen, Marathe, et al.
- 2006
(Show Context)
Citation Context ...that minimize maximum congestion in the network. The congestion of an edge (node) is defined as the total amount of flow going through the edge. Formally, we consider the following problem studied in =-=[11]-=-. Minimum congestion ratio routing: We are given a directed graph G = (V, E) with n nodes, m edges and capacities C(i, j) for each (i, j) ∈ E. Represent the traffic associated with a particular destin... |

2 |
On the Random Walk method
- Gkantsidis, Mihail, et al.
- 2004
(Show Context)
Citation Context ...at the giant component that emerges from the random process is indeed the whole graph. A number of works have subsequently used random graphs to efficiently design P2P with good properties, see e.g., =-=[29, 14, 19]-=-. In a dynamic network, there is the additional challenge of quantifying the work done by the algorithm to maintain the desired properties. An important advantage of the above protocol is that it take... |

2 | The Internet as a Complex System, in The Internet as a Large-Scale Complex System - Park - 2005 |

1 |
Stochastic models of Computer Communication Systems
- Kelley
- 1985
(Show Context)
Citation Context ...nd states that polynomial backoff protocol is stable for any α > 1 and any λ < 1. This was surprising because if the number of station is infinite, it can be shown that polynomial backoff is unstable =-=[22]-=-. The main approach in showing the above results is by analyzing the behavior of the associated Markov chain. For backoff protocols, with finite number of stations, one can associate every possible co... |