IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 2, FEBRUARY 2002
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Multicast Routing, Load Balancing, and Wavelength Assignment on Tree of Rings Xiaohua Jia, Xiao-Dong Hu, Lu Ruan, and Jianhua Sun
Abstract—There are two steps to establish a multicast connection in WDM networks: routing and wavelength assignment. Shortest path tree (SPT) and Minimum spanning tree (MST) are the two widely used multicast routing methods. The SPT method minimizes the delay from the source to every destination along a routing tree, and the MST method is often used to minimize the network cost of the tree. Load balancing is an important objective in multicast routing, which minimizes the maximal link load in the system. The objective of wavelength assignment is to minimize the number of wavelengths used in the system. This paper analyzes the performance of the Sshortest path tree (SPT) and minimum spanning tree (MST) methods in the tree of ring networks, regarding the performance criteria such as the delay and network cost of generated routing trees, load balancing, and the number of wavelengths required in the system. We prove that SPT and MST methods can not only produce routing trees with low network costs and short delays, but also have good competitive ratios for load balancing problem (LBP) and wavelength assignment problem (WAP), respectively. Index Terms—Competitive algorithm, multicast routing, wavelength assignment.
I. INTRODUCTION
M
ULTICAST is a means of group communication that demands simultaneous transmission of messages from a source to a set of destinations [7]. There are two steps to establish a multicast connection in WDM networks [4], [6]: routing and wavelength assignment. Routing finds a directed tree in the given network, which is rooted from the source node and connects all the destinations. Wavelength assignment assigns a wavelength to a routing tree in such a way that two trees must have distinct wavelengths if they share a directed link. There are two popular multicast routing methods, Sshortest path tree (SPT) method and minimum spanning tree (MST) method. An SPT minimizes the delay from the source to every destination. An MST is used as an approximation to the optimal routing tree that minimizes the network cost of a routing tree.
Manuscript received April 19, 2001. The associate editor coordinating the review of this letter and approving it for publication was Prof. K. Park. This work was supported in part by City University of Hong Kong under Grant 7000778 and 973 Information Technology and High-Performance Software Program of China. X. Jia is with the Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]). X.-D. Hu is with the Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China (e-mail:
[email protected]). L. Ruan is with the Department of Computer Science, Iowa State University, Ames, IA, USA (e-mail:
[email protected]). J. Sun is with the Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN, USA (e-mail:
[email protected]). Publisher Item Identifier S 1089-7798(02)01925-7.
The network cost of a tree is defined as the sum of the costs of all the links in the tree. Besides of the delay and network cost of routing trees, load balancing is another objective of multicast routing. The load of a link is the number of routing trees traversing the link. Load balanced routing minimizes the maximal link load in the system. This is known as the load balancing problem (LBP). The objective of wavelength assignment is to minimize the number of wavelengths used in the system. This is called the wavelength assignment problem (WAP). One approach to the WAP is through the load balanced routing, because the number of wavelengths required is at least equal to the maximal link load in the system and the LBP is regarded as easier than the WAP [11]. All of these problems were proved NP-hard even for some simple networks [1]–[3], [9], [11] and were studied separately. In this paper, we analyze the performance of SPT and MST methods in tree of rings networks, regarding the performance criteria such as the delay and network cost of generated routing trees, load balancing, and the number of wavelengths required in the system. We prove that SPT and MST methods can not only produce routing trees with low network costs and short delays, but also have good competitive ratios for LBP and WAP, respectively. II. COMPETITIVE ANALYSIS OF ALGORITHMS The topology of WDM networks that we consider is tree of rings. It can be generated by substituting some of nodes in a tree with some rings. See Fig. 1. Clearly, tree of rings is a general case of star, tree, ring and interconnecting ring networks that are popular in computer networks [10]. with indicating A multicast request is denoted by the set of destination nodes, and a the source node and for the request is represented by . In routing tree and denote two routing trees of particular, produced by MST and SPT algorithms, respectively. is defined as the number of edges it The network cost of . The transmission delay includes and denoted simply by on is measured by from source to a destination from to on , and denoted the cost of the path . by We just present our results on ring networks, as the approach we use is applicable for dealing with general tree of rings networks. be the ring network of nodes. For the simplicity Let of presentation, we assume that can be divided by 4. We label
1089–7798/02$17.00 © 2002 IEEE
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Fig. 1.
IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 2, FEBRUARY 2002
An example of tree of rings.
Fig. 2. (a) A routing tree, (b)
MST (r), and (c) SPT (r).
these nodes clockwise as , where the node label takes module of . Each link between a pair of nodes carries two oppositely-directed fibers for data transmissions in the two on , there directions of the link. Given a request ways to construct routing tree for it. For a simple are , there are example, consider request three routing trees. See Fig. 2, where routing tree (a) has bigger cost than MST (b) and longer delay than SPT (c). The following lemma will be used to prove the subsequent results. on . Lemma 1: For any multicast request contains link i) If path , then path contains link . , for some , then there exii) If on such that the distance ists a node . between and is less than Proof: It is not difficult to prove i). We just consider ii) . Let us divide ring into four and show the case of , for parts . Clearly, , if , , or . Without loss generality, we or . Notice that there exists a node with assume , otherwise consists of a path having distance , this contradicts . Let be the closest node to in . . In this case, consists Case 1: For all . Thus of a single path, which implies . . We assume that is Case 2: For some . If is not on the closest node to in , then , this . If is on , then implies
. This leads to . The following theorem shows that in rings the cost of SPT is less than two times that of MST while the delay of MST is less than two times that of SPT. This is not true in general [5]. on . Theorem 1: For any multicast request . i) , ii) . . i) It can be Proof: It suffices to show the case of verified that the ratio achieves the maximum of when and , for all , or , for all . ii) It can be verified when that the ratio achieves the maximum of or . We first study the performances of the optimal algorithms for LBP and WAP through the competitive analysis proposed in [8]. Theorem 2: The optimal algorithm for LBP has competitive ratio at least 2. be an optimal on-line algorithm, then Proof: Let routes a sequence of requests we consider how algorithm on , which is adversely delivered in the fol. Now suppose, without lowing way: First, deliver clockwise, then lose of generality, that algorithm routes . If algorithm routes clockwise, then deliver has load two. Thus it can not be -competitive link , since optimal off-line algorithm will route these for any two requests one clockwise while the other anticlockwise, and every link has load at most one. If algorithm routes anti-clockwise, then deliver . Hence, no matter how algorithm routes , anticlockwise or clockwise, there always exists a directed link with load two. However, the anti-clockwise while optimal off-line algorithm will route and clockwise. In such way no link has load greater than one. This implies that algorithm has competitive ratio at least 2. By using a similar but more complicated argument, we can prove the follow theorem. has Theorem 3: The optimal algorithm for WAP on . competitive ratio at least We now study the performances of MST and SPT Algorithms for LBP. Theorem 4: SPT algorithm for LBP is 2-competitive on ring networks. Proof: This can be proved by using Lemma 1 i), Theorem 2 and the same adversary argument used in the proof of Theorem 2. Theorem 5: MST algorithm for LBP has competitive ratio at most 4 and at least 3 on ring networks. Proof: The upper bound can be proved by using Lemma 1 and the adversary argument as in the proof of Theorem 2. To show the lower bound, we find three requests such that MST routing produces network load three while the optimal off-line routing produces network load one. In the end we study the performance of SPT-based and MSTbased algorithms for WAP. They first route incoming request by
JIA et al.: MULTICAST ROUTING, LOAD BALANCING, AND WAVELENGTH ASSIGNMENT ON TREE OF RINGS
SPT or MST algorithms, and then assign the request a currently used wavelength if possible. Theorem 6: SPT-based algorithm for WAP has competitive . ratio at most on be a sequence of requests. Proof: Let includes a Suppose, without loss of generality, that , i.e., it starts from clockwise directed path . source node 0 and ends at a destination node , where active requests which are assigned with Thus there are different wavelengths and whose routing trees include at least one directed link in clockwise directed path . This implies that there exists a clockwise directed , for some , which link, say link has load at least
Now applying the same argument used in the proof of Theorem 4, we can deduce that no matter how the optimal off-line algorithm to route (and assign) those active requests whose routing , there exists a directed link having trees traverse link load at least
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TABLE I OBTAINED RESULTS IN THIS PAPER
III. CONCLUSION In Table I we summarize our results on lower bound (LB) and upper bound (UP) of competitive ratios of optimal, SPT, MST, and greedy algorithms for LBP and WAP. The table also gives the worst-case ratios of network cost and delay over the optimal values of the routing trees generated by the listed methods. From the table we can see that SPT and MST algorithms can produce multicast routing trees that have good guaranteed performance for both LBP and WAP in the tree of ring networks. To the knowledge of authors, this is the first theoretical analysis on this multi-criteria optimization problem. ACKNOWLEDGMENT The authors would like to thank the reviewers and the editor for their valuable comments which greatly helped us to improve the paper. REFERENCES
Hence the optimal off-line algorithm needs at least wavelengths. This proves the theorem. By using the same argument we can prove the following theorem. Theorem 7: MST-based algorithm for WAP has competitive on . ratio at most The greedy algorithm for LBP is to route each incoming request without using those most heavily loaded links. The greedy algorithm for WAP is to route each coming request in such a way that a currently used wavelength can be assigned to the request. By using the same technique, we can prove the following two theorems. They show that for ring networks greedy algorithms for LBP and WAP not only may have larger network cost and longer transmission delay but also have larger competitive ratios than SPT and MST algorithms. has comTheorem 8: The greedy algorithm for LBP on . petitive ratio at most and at least Theorem 9: The greedy algorithm for WAP has competitive on . ratio at most
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