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2017 2nd International Conference on Wireless Communication and Network Engineering (WCNE 2017) ISBN: 978-1-60595-531-5

An Improved Min-cost Max-flow Network Coding Algorithm

Yang-tian-xiu HU

*

, Li-min MENG, Pei-rui JIANG and Yu-zhou SHANG

College of Information Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang, China, 310023

*Corresponding author

Keywords: Network coding, Min-cost Max-flow, The shortest path, Encoding node.

Abstract. Network coding can significantly improve the network communication performance; however, the introduction of encoding nodes will lead to more resource consumption. In this paper, we proposed an improved Min-cost Max-flow network coding algorithm to reduce resource consumption under the new principle of encoding node, which has been augmented until achieves the network maximum flow according to the min-cost path in the shortest paths from the source node to the sink node. Simulation results show that compared with the previous max-flow algorithm, this improved algorithm can not only reduce the total cost, but also reduce the encoding nodes and the network resource overhead when applied to the network coding.

Introduction

Before the network coding was proposed, the network intermediate node had always played a storage-forwarding role. Until 2000, Ahlswede et al. proposed the network coding [1]in a paper entitled " Network Information Flow", proving that intermediate nodes not only act as routing, but also can encode. This kind of linear or nonlinear encoding communication mechanism of intermediate nodes called network coding, which can improve the network throughput and transmission reliability, is widely applied in streaming media, wireless network and P2P network etc.

But the addition of encoding node also brings problems. Because the encoding node must cache the received information, then add the coefficient of linear encoding processing and then forward to the next node. The cache information and linear operation can increase the network nodes' CPU and I/O consumption. Therefore, how to reduce the number of encoding nodes in the case of maximum flow is very important.

Tao et.al [2] proposed a new maximum routing algorithm based on the principle of "Max-flow Min-cut" theorem combined with network augmenting chain and minimal cut set, which can improve the throughput of multicast network. In [3], a minimal cost network coding algorithm based on critical link is proposed. Sandeep Bhadra et.al [4] considered the single-source min-cost multicast problem and presented an allocation rule with the edge cost of each edge being allocated to the flow through the edge. [5-6] is based on the operation of the arc capacity of the wireless network to achieve the least encoding nodes. Sun [7] considered both the arc cost and the number of encoding nodes, so that the algorithm can reduce the cost of network coding and reduce the cost of data transmission.

In this paper, we proposed a Min-cost Max-flow algorithm based on the shortest path, which can find the path of the minimal cost to be augmented in the shortest path cluster from the source node to the sink nodes until the max-flow of the network is realized. The improved algorithm reduces the total network cost and number of the encoding nodes on the basis of achieving maximum flow.

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Related Concepts

Network Model

The communication network is abstracted as a (cost, capacity) network, denoted as

N = (V, A, C, W) [8], V represents the set of nodes (Including the source node set S, sink node set

T, intermediate node set I), and A is the set of the arc between two nodes, and C represents the capacity set and W is the set of cost. For arc ∈ A, () is the flow on the arc . If =

(, ), () is also denoted as . And ∈ W represents the arc's cost.

Feasible Flow

If flow f satisfies:

0 ≤ () = () ≤ () , ∀ ∈ () , ∀ ∈ { , !} (1)

Such a is called a feasible flow of the network, and such a feasible flow always exists. When

∈ , () = ∑$%&$ is the output flow of the node and() = ∑$'&$ is the input flow of the node .

The Shortest Path

In this paper, the shortest path is the path of the minimal number of intermediate nodes from the source node to the sink nodes, because all the arc in this paper units to simplify the condition, to find the shortest path is to find the path of the fewest arcs, and the fewest arcs mean the minimal number of the intermediate nodes.

The Minimal Cost

Each arc in the network has a corresponding cost (The value of cost is positive), the min-cost is minimal sum of the cost on the arc of the path from the source to the sink. The total cost to achieve max-flow in the network can be expressed as: w()*+) = ∑(,)∈,.

Encoding Node

As we all known, there are three kinds of intermediate node including routing node, switching node and encoding node in a network, shown in Figure 1.

In the past, the encoding node is the node that the number of input links is more than the number of output links in the network coding. But the encoding nodes that are defined so may have unnecessary resource consumption due to the encoding of some unnecessary nodes. In this paper, we re-define the encoding node, and if a node is an encoding node then it must satisfy:

1) The number of input links is more than the number of output links. Meanwhile, the transmission information on the input link is different from the transmission information on the output link.

∑ 0&1 ≥ ∑ /0&1 (2)

∑ 0&1 represents the total traffic of all input links, and ∑ /0&1 represents the total capacity of

[image:2.612.221.388.623.700.2]

output links.

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Ford-Fulkerson Algorithm

In the process of dealing with the max-flow in the network, the Ford-Fulkerson algorithm [9] is given in the graph theory. The algorithm achieves the max-flow of the network through the residual network.

In the network, if each sink can reach the maximum capacity on the input links, then the maximum throughput can be achieved accordingly. In graph theory, the Ford-Fulkerson algorithm is proposed to solve the network maximum flow, and the core idea of this algorithm is: starting from a known flow (usually zero flow), it recursively constructs a sequence of its increasing value and terminates it in the maximum flow. After each new stream is found, if there exists augmenting path P of , find out such path, and then the modified flow ∗ based on P is regarded as the next flow of this sequence; if there are no such augmenting path, then the algorithm terminates. So that the final is the maximum flow.

Algorithm Steps

From the above introduction, we can see that the Ford-Fulkerson algorithm can achieve the maximum flow in the network, but this way in the search for an augmenting path when the need to continue to contiguous nodes labeled, so there is a certain blindness. So that there is no guarantee that the number of nodes through augmenting path is the least, then it cannot guarantee the minimum number of coding nodes, but also cannot guarantee the minimum total cost of the network.

The new proposed algorithm is based on the concept of the shortest path and the minimum cost in the basic idea of the Ford-Fulkerson algorithm, which avoids the repeated search by searching for the favorable augmenting path. The total cost of the network is reduced on the basis of the maximum flow of the network.

Here are the steps of the Improved Min-cost Max-flow Algorithm: Input: {the node set V, capacity set C, cost set W, the feasible flow }.

Step 1: For all, ∈ , > 0, > 0. (If there is no arc between and, = 0,

= 0). The value of the feasible flow is initialized to be 0.

Step 2: Set a direction for the arc, relative to this direction, arc set A is divided into two categories , .

n = 1,2,3 …

For !0∈ :, do

( i ). Find the shortest path set P from S to !0.

( ii ). Choose the path whose ∑(,)∈, is minimal in P to augment. (If there are multiple such paths, then choose one to augment arbitrarily).

( iii ). The augmenting flow σ = <=>($%,$'∈,){} .So the flow f? on the arc follows:

f? = @

+ B ∈

− B ∈

∉ A

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( iv ). If ? = , let = 0, = 0, which means that the arc between and is no longer considered.

( v ). And then go to Step 2 ( i ) to find next augmenting path.

Step 3: The above process is iterated until there is no path from S to T.At this time, k augmenting paths are obtained, and BE is the augmenting flow corresponding to each augmenting path. So the max-flow)*+ = ∑ BE E ,so the max-flow min-cost follows:

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Step 4: The linear network coding is performed in the maximum flow network, and the encoding node is found according to the new definition of the encoding node. Record the number of encoding node.

Simulation Results

Before the simulation, we prove the correctness of the algorithm through several examples, but here we give the simulation results directly because of the limitation of space. In this paper, the adopted simulation tool is MATLAB 2015a. In the case where the capacity value and the cost value are given on the arc, the capacity value and the cost value are both randomly generated at the same time in random network [10] and both are positive. (Between 1~20)

[image:4.612.91.526.213.347.2]

Figure 2. The number of nodes and the network total cost. Figure 3. The number of nodes and encoding nodes.

The first set of experiments simulates the relationship between the total cost of the network and the number of network nodes (The network with more than 10 nodes is considered in the simulation), as shown in Figure 2. We can see that, because of the capacity value and cost value is randomly generated, so the total cost of the network is not necessarily with the number of the nodes increases, but by comparing the improved algorithm and Ford-Fulkerson algorithm we can see that the improved algorithm has less total cost than the Ford-Fulkerson algorithm in the case of maximum network flow. This is because the Ford-Fulkerson algorithm is redundant when implementing the maximum flow, so there is a situation where the maximum flow can't be found accurately and the cost of maximum flow is not necessarily minimal.

The second set of experiments simulates the relationship between the number of encoding nodes and the number of network nodes, as shown in Figure 3. We can see that by redefining the encoding nodes, the number of encoding nodes of the improved algorithm is smaller than that of the traditional algorithm, and this advantage is more obvious as the number of network nodes increases. This is because the shortest path leads to the minimal number of nodes passing through, and is more stringent in deciding whether to encode a node, thus reducing the number of unnecessary encoding nodes and reducing the resource consumption in the network.

Conclusion

In this paper, the min-cost max-flow network coding algorithm based on the shortest path can make the network transmit the information at the lowest cost while achieving the maximum throughput. This is because the path is the shortest, and the definition of encoding nodes is improved, which can reduce the number of encoding nodes, reduce the network resources consumption in the encoding nodes. Simulation results show that the algorithm can transmit network information at the minimal cost and improve the transmission efficiency.

Acknowledgement

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References

[1] R. Ahlswede, N. Cai, S. R. Li, and R. W. Yeung, Network information flow, IEEE Trans. Inf. Theory. 46 (2000) 1204-1216.

[2] Shaoguo Tao, Jiaqing Huang, Zongkai Yang, Wenqing Cheng, Rami S. Youail, Maxflow–based Routing Algorithm for Network Coding Multicast, Computer Science. 35 (2008) 107-117.

[3] Shaoguo Tao, Jiaqing Huang, Zongkai Yang, Wenqing Cheng. An improved algorithm for minimal cost network coding, Journal of Huazhong University of Science and Technology (Nature Science Edition). 36 (2008) 1-4.

[4] S. Bhadra, S. Shakkottai, P. Gupta. Min-Cost Selfish Multicast with Network Coding. IEEE Transactions on Information Theory. 52 (2006)5077–5087

[5] Hui Zhang. The Optimization Techniques of Coding Node in Wireless Network, Southwest Jiaotong University. 2015

[6] Dexia Jiang. Research on Network Coding Nodes Selection in Wireless Network, Southwest Jiaotong University. 2014

[7] Ling Sun. Improved network coding algorithm based on minimum cost maximum flow, Xi’an Electronic and Science University. 2010

[8] Ford. L. R, Fulkerson. D. R. Maximal flow through a network, Canadian Journal of Mathematics. 8 (1954)399-404

[9] Haiyin Wang, Qiang Huang, Chuantao Li, Baozeng Chu. Graph algorithm and its MATLAB implementation, first ed., Beijing, 2010.

Figure

Figure 1. The three kinds of intermediate node.
Figure 2. The number of nodes and the network total cost.   Figure 3. The number of nodes and encoding nodes

References

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