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Many-to-many multicast routing can be extensively applied in computer or communication networks supporting various continuous multimedia applications. The paper focuses on the case where all users share a common communication channel while each user is both a sender and a receiver of messages in multicasting as well as an end user. In this case, the multicast tree appears as a terminal Steiner tree (TeST). The problem of finding a TeST with a quality-of-service (QoS) optimization is frequently NP-hard. However, we discover that it is a good idea to find a many-to-many multicast tree with QoS optimization under a fixed topology. In this paper, we are concerned with three kinds of QoS optimization objectives of multicast tree, that is, the minimum cost, minimum diameter, and maximum reliability. All of three optimization problems are distributed into two types, the centralized and decentralized version. This paper uses the dynamic programming method to devise an exact algorithm, respectively, for the centralized and decentralized versions of each optimization problem.

Multicast routing has been increasingly used in computer or communication networks supporting various multimedia applications, such as real-time audio and video conferences, entertainment, and distance learning, [

In a real world, a large number of continuous multimedia applications drive the consumers to advance their

In many practical settings, each destination is a

To the best of our knowledge, there have been a number of studies on the unrestricted many-to-many multicast tree mentioned above (terminal Steiner tree) [

The rest of this paper is organized as follows. In Section

A computer or communication network is frequently modeled as an undirected graph [

Given any two different nodes

In this paper, we are concerned with the centralized and decentralized multicast trees under a fixed topology, namely, the realization of a given rooted and unrooted TeST topology; see Figure

Ignore the dashed lines on the left top subfigure to obtain a sample graph where the number on each edge represents its length. The right top subfigure is a sample rooted TeST and the right bottom is a sample unrooted TeST. An example centralized many-to-many multicast tree under a fixed topology is distinguished by dashed lines on the left top subfigure and an example decentralized one is distinguished on the left bottom subfigure.

Let

In the section, we make some fundamental preliminaries, which will help us to analyze the problems and understand the algorithms proposed in the following.

First of all, we define three kinds of metrics of QoS of tree, including the

We are given an undirected graph

The

Let

We are given an undirected graph

The delay of

Similarly, we have

The maximum delay of leaf-to-leaf path realization of

Let

The maximum delay of root-to-leaf path realization of

Likewise,

Note that realizations with a minimum diameter or radius are both denoted by

We are given an undirected graph

The working probability of

Likewise,

The minimum reliability of leaf-to-leaf path realization of

Let

The minimum reliability of root-to-leaf path realization of

Similarly,

Note that realizations with a maximum diameter or radius reliability are both denoted by

The focus of this paper is three optimization problems of many-to-many multicast tree under a fixed topology from the perspective of three metrics above, which are formally defined as follows.

First of all, we use an INPUT (abbreviated to

Given an INPUT with every edge

Given an INPUT with every edge

Given an INPUT with every edge

Given an undirected graph

When

Next we construct a new graph

When

Next, we construct a complete graph

In this section, we study the centralized and decentralized MCMP, respectively.

According to discussions above in Section

Let

Above analysis leads to algorithm MCCT that can find a minimum cost centralized multicast tree under a TeST, which is denoted as

Given an INPUT as

Step_1 takes

Step_3 only takes

By the discussions above in Section

Given an INPUT as

It is similar to the proof of Theorem

In this section, we study the centralized and decentralized MDMP, respectively.

In the centralized MDMP, to find a minimum delay multicast tree under a given TeST topology is to find a minimum delay realization of the rooted TeST. In this subsection, we design a polynomial-time exact algorithm for the centralized MDMP.

Let

Based on above analysis, we devise algorithm MDCT to find a minimum delay centralized multicast tree under a TeST, which is denoted as

Given an INPUT as

It is similar to the proof of Theorem

The essence of finding a minimum delay multicast tree under a given TeST in the decentralized MDMP is to find a minimum delay realization of the unrooted TeST topology.

First of all, we can always use the method in [

We can use (

Given an INPUT as

It is similar to the proofs of Theorems

In this section, we study the centralized and decentralized MRMP, respectively.

MRPG based on

Firstly, we introduce a fundamental Lemma

Given an undirected graph

On one hand, if

From Lemma

For any edge

We use

Finally,

In essence, above idea of using (

Given an undirected edge-weighted graph

Step_1 spends

In this section, we present an exact algorithm for the centralized and decentralized MRMP, respectively.

The essence of finding a maximum reliability multicast tree under a TeST topology in the centralized MRMP is to find a maximum reliability realization of the rooted TeST.

Let

Above analysis can be described as algorithm MRCT. It can find a maximum reliability centralized multicast tree under a TeST, denoted as

Given an INPUT as

It is similar to the proof of Theorem

To find a maximum reliability multicast tree under a TeST in the decentralized MDMP is essentially to find a maximum reliability realization of the unrooted TeST. We can use the way in [

We can use (

Given an INPUT as

It is similar to the proof of Theorem

In this section, we take the network and the binary tree topology shown in Figure

paths and then obtain

Compute

Compute

Compute

and then obtain

Compute

and then obtain

Compute

Compute

and then obtain

Compute

reliability paths and then obtain

Compute

Compute

reliability paths and then obtain

Compute

Suppose the integer on every link of the network in Figure

For ease of view, we neglect the numbers on edges. The example network for MCMP and the binary tree topology are both the same as those shown in Figure

Suppose the integer on every link of the network in Figure

The example network for MDMP and the binary tree topology are both the same as those shown in Figure

Suppose that every link

The example network for MRMP and the binary tree topology are both the same as those shown in Figure

This paper introduces the architecture of a many-to-many multicast tree with fixed topology, reduces it to a realization of the given TeST topology, and applies the idea of under a fixed topology to deal with three optimization problems, that is, the minimum cost, minimum delay, and maximum reliability multicast tree under a fixed topology problem. Each problem includes the centralized and decentralized versions. For the both versions of each problem, an exact algorithm is devised using the dynamic programming approach, respectively. On the condition that we are given a collection of alternative tree topologies, it is of interests to explore a best topology from all the alternative tree topologies.

Moreover, if we consider two or more weights on every link of a network, it is an interesting and important research topic how to devise an efficient algorithm for the multicast tree problem under a fixed topology with multiple objectives or with a single objective and at least one constraint.

This research was supported in part by the Key Research Grant xkyzd201207 (Ding) and the 2010 Excellent Young Teacher Grant 19093-88 (Ding) of Zhejiang Water Conservancy and Hydropower College.

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