Network Routing Capacity Jillian

∗ Cannons ,

Randy

† Dougherty ,

Chris

‡ Freiling ,

∗

Department of ECE University of California, San Diego {jcannons,zeger}@ucsd.edu

Problem Statement and Assumptions A communications network is a directed, acyclic graph with sources emitting messages and sinks having specific message demands. The messages are drawn from a specified alphabet and, for simplicity, the edges in the graph are assumed to be error-free, cost-free, and of zero-delay. Furthermore, only non-degenerate networks are considered, namely those with at least one directed path to each sink demanding a particular message from the supplying source. The problem is to determine a method of transmitting the messages through the network such that all sink demands are satisfied.

†

‡

Department of Mathematics California State University, San Bernardino [email protected]

Center for Communications Research [email protected]

Fractional Solutions Network messages are fundamentally scalar quantities, but it is also useful to consider blocks of multiple scalar messages as message vectors. Such vectors may correspond to multiple time units in a network. Likewise, the data transmitted on each network edge can also be considered as vectors. However, the message vector dimension and edge vector dimension need not be the same; a (k, n) fractional coding solution uses message dimension k and edge data dimension n. Furthermore, a (k, n) fractional routing solution is a fractional coding solution which copies some input edge components into a single output vector.

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Traditionally, network messages are treated as physical commodities, which are simply transmitted throughout the network without alteration. However, the emerging field of network coding [1, 2] views the messages as information, which can be replicated and transformed by any node within the network. Network coding permits each outgoing edge from a node to carry some function of the data received on the incoming edges of the node. Thus, the goal is to determine a solution (a set of functions) that satisfies all sink demands.

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Figure 1: A network with a scalar linear coding solution.

Routing Capacity Properties A number of properties of the routing capacity have been proven [3]:

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Routing Capacity Definition

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d) Every non-negative rational number is the routing capacity of some network.

Figure 4: A network with a (2, 3) fractional routing solution.

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c) The routing capacity is rational.

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b) The routing capacity is achievable (i.e. there always exists a (k, n) fractional routing solution with k/n = ).

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Figure 4: Note that edges e4,6 and e5,7 must carry all of the information from the sources to the sinks. Thus, 3k ≤ 2n and  ≤ 2/3. The illustrated solution provides a lower bound of  ≥ 2/3. Thus, the routing capacity is 2/3 and is achievable.

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Figure 3: Note that each of the 2k message components must be carried by at least one of the edges e1,2, e1,3, and e1,4. Thus, 2k ≤ 3n and  ≤ 3/2. The depicted fractional routing solution provides a lower bound of  ≥ 3/2. Thus, the routing capacity is 3/2 and is achievable.

a) The routing capacity can be determined by solving a linear programming problem.

Figure 2: A (3, 4) fractional routing solution to the network in Figure 1.

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The network in Figure 1 contains a single source emitting two messages and two sinks each demanding both messages. This network does not have a routing solution (due to the “bottleneck” edge e4,5). However, as depicted, a scalar linear solution using network coding exists.

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Network Coding Example

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Fractional Routing Examples

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and Ken

∗ Zeger

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5 1 2 3 1 2 3 x y Figure 3: A network with a (3, 2) fractional routing solution.

For any non-degenerate network, there exists many fractional routing solutions using different values of k and n. Thus, it is desirable to characterize the largest ratio of message dimension to edge dimension for which a fractional routing solution exists. This ratio represents the percentage of data which can be sent when restricted to routing. If a network has a (k, n) fractional routing solution, then k/n is an achievable routing rate of the network. Let U = {r ∈ Q : r is an achievable routing rate}. Then, the routing capacity of a network is the quantity  = sup U . If U = ∅, then by convention we define  = 0

Routing Capacity Examples The routing capacity of the networks in Figures 2, 3, and 4 can be computed by considering upper and lower bounds on all possible routing rates of the network. Figure 2: Note that each of the 2k message components must be carried by at least two of edges e1,2, e1,3, and e4,5; thus 2(2k) ≤ 3n for all k and n. Consequently,  ≤ 3/4. The illustrated solution provides a lower bound of  ≥ 3/4. Thus, the routing capacity is 3/4 and is achievable.

Center for Wireless Communications Poster Session, April 18, 2005

The framework of network coding allows for the solution of some networks which are unsolvable using traditional routing techniques. The routing capacity of a network provides an indication of performance when only fractional routing is permitted. The routing capacity is rational, achievable, and can be found by solving a linear programming problem.

Future Directions The routing capacity can be generalized to the coding capacity for the case when network coding is permitted. It has been shown that the coding capacity of a network need not be achievable, but is independent of the message alphabet size. However, whether computing the coding capacity is decidable and whether the coding capacity is rational remain open problems. Furthermore, a characterization of networks for which the routing capacity is strictly less than the coding capacity is also unknown.

References [1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Transactions on Information Theory, vol. 46, no. 4, pp. 1204 – 1216, 2000. [2] S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Transactions on Information Theory, vol. IT-49, no. 2, pp. 371 – 381, 2003. [3] J. Cannons, R. Dougherty, C. Freiling, and K. Zeger, “Network routing capacity,” IEEE/ACM Transactions on Networking, submitted October 16, 2004.