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## Edge-cut bounds on network coding rates (2006)

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Venue: | J. Network and Systems Management |

Citations: | 47 - 4 self |

### Citations

12169 |
Elements of information theory
- Cover, Thomas
- 2006
(Show Context)
Citation Context ... C21 = C41 = 0 for R1 = 4. Combining these results, we can restrict attention to the graph in Fig. 7. For Fig. 7, we choose Ed = {(2, 3), (4, 3), (2, 5), (4, 5)}, Sd = {1, 2, 3}, [π(1), π(2), π(3)] = =-=[3, 1, 2]-=- and the resulting graph GEd is shown in Fig. 8. We find that R1 + R2 + R3 ≤ C23 + C43 + C25 + C45. (6) Next, in Fig. 7 we choose Ed = {(3, 2), (3, 4), (5, 2), (5, 4)}, Sd = {2, 3}, [π(1), π(2)] = [2,... |

8756 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ...s.com address this problem by developing an information-theoretic counterpart to edge-cut bounds that does apply to network coding. We do this by borrowing from the artificial intelligence literature =-=[14]-=- the concept of d-separation in Bayesian networks. Bayesian networks are graphs whose vertices represent random variables, and d-separation is a graphical procedure that establishes the conditional st... |

1927 | Network information flow
- Ahlswede, Cai, et al.
- 2000
(Show Context)
Citation Context ...immediately applied to communication problems where messages are routed, but not to communication problems permitting the generality of network coding. Network coding has been intensely studied since =-=[1]-=- presented a novel coding scheme that attains a cut-set bound for multicasting in networks. We here study the problem of upper bounding network coding rates for the general case of multi-message multi... |

841 | An algebraic approach to network coding
- Koetter, Médard
- 2003
(Show Context)
Citation Context ...te going from u to v (see, Fig. 1 for illustrations of the capacity region of a directed edge, an undirected edge, and a different type of edge). The terminals can further perform network coding [1], =-=[9]-=-, i.e., each terminal can transmit into each of its TWCs any function of its messages and past received outputs. The standard network cut-set bound gives a useful outer bound on the set of feasible fl... |

450 | Practical network coding
- Chou, Wu, et al.
- 2003
(Show Context)
Citation Context ...several approaches to implementing network codes. One approach designs xed coding functions for each processor based on a centralized knowledge of the network topology. A second approach (see, e.g., =-=[10]-=--[12]) is motivated by issues arising in distributed or dynamic scenarios where centralized control is impractical. This approach has each vertex transmit on its outgoing edges a randomly chosen linea... |

419 |
Maximal Flow Through A Network
- Ford, Fulkerson
- 1956
(Show Context)
Citation Context ...ODUCTION Fifty years ago, several individuals investigated the problem of determining the maximal flow from one vertex to another in a graph subject to capacity limitations on arcs or edges [4], [5], =-=[6]-=-, [16]. The outcome was the celebrated “max-flow min-cut” theorem that states that the maximal flow is the minimum capacity among all edge cuts separating the source and destination vertices (a relate... |

305 |
Multi-Commodity Network Flows
- Hu
- 1963
(Show Context)
Citation Context ... more convenient to draw only the bidirected version of the undirected graph without formally converting it into a line graph. Example 2: Consider the network of Fig. 5 that appeared in a paper by Hu =-=[8]-=-. This network served as an example to show that the vertex-partitioning cut-set bound can be loose for three commodities. We construct the bidirected graph shown in Fig. 6, where the edge from vertex... |

228 | Two-way communication channels
- Shannon
- 1961
(Show Context)
Citation Context ...our results here.s¢¡¤£ undirected directed general form of capacity regions¢£¥¡ Fig. 1. Capacity regions of different types of edges. For networks of bidirectional TWCs in which Shannon’s outer bound =-=[15]-=- for each constituent channel, i.e., edge, is its capacity region, we can reduce the bound of [3, Sec. 14.10] to the following three steps [11]. 1) Convert every network edge into a pair of directed e... |

226 | Combinatorial Optimization - Schrijver - 2003 |

200 | On Randomized Network Coding - Ho, Medard, et al. - 2003 |

166 | Maximal flow through a network, Canadian - Ford, Fulkerson - 1956 |

153 | The switchware active network architecture
- Alexander, Arbaugh, et al.
- 1998
(Show Context)
Citation Context ... = C41 = 0 for R1 = 4. Combining these results, we can restrict attention to the graph in Fig. 9. For Fig. 9, we choose Ed = f(2; 3); (4; 3); (2; 5); (4; 5)g, Sd = f1; 2; 3g, and [(1); (2); (3)] = =-=[3; 1; 2]-=-. The resulting graph GEd is shown in Fig. 10. We nd that R1 + R2 + R3 C23 + C43 + C25 + C45: (7.1) Next, in Fig. 9 we choose Ed = f(3; 2); (3; 4); (5; 2); (5; 4)g, Sd = f2; 3g, and [(1); (2)] = ... |

121 | Minimum-energy multicast in mobile ad hoc networks using network coding
- Wu, Chou, et al.
- 2005
(Show Context)
Citation Context ...t requirements for several network connection problems. A further benet of network coding has been to improve the allocation of physical and medium-access layer resources in wireless ad hoc networks =-=[15]-=-, [16]. For example, suppose one is given a collection of end-to-end communication demands and an objective of minimizing power consumption. It was demonstrated that network codes increase the energy ... |

103 |
A note on the maximum flow through a network
- Elias, Feinstein, et al.
- 1956
(Show Context)
Citation Context ... INTRODUCTION Fifty years ago, several individuals investigated the problem of determining the maximal flow from one vertex to another in a graph subject to capacity limitations on arcs or edges [4], =-=[5]-=-, [6], [16]. The outcome was the celebrated “max-flow min-cut” theorem that states that the maximal flow is the minimum capacity among all edge cuts separating the source and destination vertices (a r... |

91 |
Directed information for channels with feedback
- Kramer
- 1998
(Show Context)
Citation Context ...the calculus of d-separation and fd-separation in FDGs. FDGs are graphs where the vertices represent random variables, and the edges represent the functional dependencies between the random variables =-=[10]-=-, [11]. For instance, suppose we have NRV random variables that are defined by SRV independent (or source) random variables by NRV functions. An FDG G is a directed graph having NRV + SRV vertices rep... |

91 | Network planning in wireless ad hoc networks: A crosslayer approach
- Wu, Chou, et al.
(Show Context)
Citation Context ...irements for several network connection problems. A further benet of network coding has been to improve the allocation of physical and medium-access layer resources in wireless ad hoc networks [15], =-=[16]-=-. For example, suppose one is given a collection of end-to-end communication demands and an objective of minimizing power consumption. It was demonstrated that network codes increase the energy efcie... |

88 |
Multicommodity flows in planar graphs
- Okamura, Seymour
- 1981
(Show Context)
Citation Context ...have R1 + 2(R2 + R3) 8: (7.3) Thus, if R1 = 4 and R2 = 2 we require R3 = 0 with or without network coding. Example 4. Consider the network of Fig. 11 that appeared in a paper by Okamura and Seymour =-=[29]-=-. This network served as an example to show that the vertex-partitioning cut-set bound is not necessarily tight for routing on a planar graph where one cannot draw the graph so that all sources and si... |

86 |
Capacity results for the discrete memoryless network
- Kramer
- 2003
(Show Context)
Citation Context ...rks of bidirectional TWCs in which Shannon’s outer bound [15] for each constituent channel, i.e., edge, is its capacity region, we can reduce the bound of [3, Sec. 14.10] to the following three steps =-=[11]-=-. 1) Convert every network edge into a pair of directed edges whose capacity pair is a boundary point of the capacity region of this edge. 2) Apply the flow cut set bound to get a rate region Rcut. 3)... |

32 |
On the Max-Flow Min-Cut Theorem of Networks
- Dantzig, Fulkerson
- 1956
(Show Context)
Citation Context ...d. I. INTRODUCTION Fifty years ago, several individuals investigated the problem of determining the maximal flow from one vertex to another in a graph subject to capacity limitations on arcs or edges =-=[4]-=-, [5], [6], [16]. The outcome was the celebrated “max-flow min-cut” theorem that states that the maximal flow is the minimum capacity among all edge cuts separating the source and destination vertices... |

25 |
Network Information Flow: Limits and Achievability
- Borade
- 2002
(Show Context)
Citation Context ...ernative to edge-cut bounds that does apply to network coding and further used the bound to derive new capacity theorems for network information flow. A. Network Model We adopted in [13] the model of =-=[2]-=-, [12] where the network is clocked, i.e, there is a universal clock that ticks N times. Vertex u transmits symbols X (n) uv , (u, v) ∈ E, after clock tick n − 1 and before clock tick n for n = 1, 2, ... |

24 | Active Virtual Network Management Prediction
- Bush
- 1999
(Show Context)
Citation Context ... = C41 = 0 for R1 = 4. Combining these results, we can restrict attention to the graph in Fig. 9. For Fig. 9, we choose Ed = f(2; 3); (4; 3); (2; 5); (4; 5)g, Sd = f1; 2; 3g, and [(1); (2); (3)] = =-=[3; 1; 2]-=-. The resulting graph GEd is shown in Fig. 10. We nd that R1 + R2 + R3 C23 + C43 + C25 + C45: (7.1) Next, in Fig. 9 we choose Ed = f(3; 2); (3; 4); (5; 2); (5; 4)g, Sd = f2; 3g, and [(1); (2)] = ... |

17 | An information-theoretic view of network management
- Ho, Médard, et al.
- 2005
(Show Context)
Citation Context ...ging network coding rates according to this tradeoff should prove useful. Yet another application is a new approach to network management for protection from and recovery of link failures (see, e.g., =-=[13]-=-, [14]). Here the network is modeled as a nite-state machine where the operation of a processor is affected by management signals that indicate the current link failures and/or directions for recover... |

15 | On the utility of network coding in dynamic environments
- Ho, Leong, et al.
- 2004
(Show Context)
Citation Context ...al approaches to implementing network codes. One approach designs xed coding functions for each processor based on a centralized knowledge of the network topology. A second approach (see, e.g., [10]-=-=[12]-=-) is motivated by issues arising in distributed or dynamic scenarios where centralized control is impractical. This approach has each vertex transmit on its outgoing edges a randomly chosen linear com... |

11 |
On networks of two-way channels
- Kramer, Savari
- 2003
(Show Context)
Citation Context ...maximal ow is the minimum capacity among all edge cuts separating the source and destination vertices [17]-[20]. A related bound additionally partitions the vertex set into two disjoint sets, and in =-=[21]-=- we developed this latter type of bound for network coding. However, as pointed out in [7, pp. 16-17], sometimes tighter bounds can be found by considering disconnecting edge sets. In this paper, we... |

7 |
Progressive d-separating edge set bounds on network coding rates
- Kramer, Savari
(Show Context)
Citation Context ...f the network in Fig. 11. 8. CONCLUDING REMARKS Upper bounds on network coding rates are currently being developed by other groups [30], [31]. Some of the distinguishing features of our work are (see =-=[33]-=-): the PdE bound applies to EDGE-CUT BOUNDS ON NETWORK CODING RATES: TO APPEAR IN THE JOURNAL OF NETWORK AND SYSTEMS MANAGEMENT, MARCH 2006 7 general multimessage multicast, we have a formal procedure... |

4 |
On Network Theory
- Robacker
- 1955
(Show Context)
Citation Context ...ION Fifty years ago, several individuals investigated the problem of determining the maximal flow from one vertex to another in a graph subject to capacity limitations on arcs or edges [4], [5], [6], =-=[16]-=-. The outcome was the celebrated “max-flow min-cut” theorem that states that the maximal flow is the minimum capacity among all edge cuts separating the source and destination vertices (a related boun... |

4 |
Active Networks and Active Network Management
- Bush, Kulkarni
- 2001
(Show Context)
Citation Context ... literature. Index Terms Network capacity, network coding, active networks, d-separation 1. INTRODUCTION In recent years there has been considerable interest in technologies known as active networks =-=[1]-=- that permit network nodes to execute computations specic to the packets passing through them. The programmability of infrastructure is the key innovation in this approach to network architecture; th... |

2 | PLAN: A Programmable Language for Active Networks - Hicks, Kakkar, et al. - 1999 |

2 |
ANTS: Network Services Without the Red
- Wetherall, Guttag, et al.
- 1999
(Show Context)
Citation Context ... network architecture; the added exibility provides a means to implement novel transmission techniques to improve performance. A small subset of the literature on active networks can be found in [1]-=-=[6]-=-. The optimization of active networks is a critical network management concern. Network optimization has traditionally studied communication networks in the same framework as other types of networks s... |

2 |
On the max-ow, min-cut theorem of networks
- Dantzig, Fulkerson
- 1956
(Show Context)
Citation Context ...n and other individuals discovered the celebrated max-ow min-cut theorem that states that the maximal ow is the minimum capacity among all edge cuts separating the source and destination vertices =-=[17]-=--[20]. A related bound additionally partitions the vertex set into two disjoint sets, and in [21] we developed this latter type of bound for network coding. However, as pointed out in [7, pp. 16-17], ... |

1 |
On networks of two-way channels,” to appear
- Kramer, Savari
(Show Context)
Citation Context ...ut-set bound for multicasting in networks. We here study the problem of upper bounding network coding rates for the general case of multi-message multicast and describe results that first appeared in =-=[12]-=- and [13]. The standard vertex-partitioning cut-set bound of [3, Sec. 14.10] provides one type of bound. We shall discuss a strengthened version of this bound for undirected networks that is taken fro... |

1 |
Network information ow: limits and achievability
- Borade
(Show Context)
Citation Context ...d z with PZ(z) > 0. Alternatively, we say that X ZY forms a Markov chain. We remark that X ZY forms a Markov chain if and only if I(X;Y jZ) = 0: (2.4) 3. NETWORK MODEL We adopt the model of [21], =-=[25]-=- whose components and rules we list for completeness below (see also [26, x III.A-B]). Most of what follows applies to real networks, perhaps with the exception of the clocking described in the rst b... |