Virtual Topologies for Multicasting with Multiple Originators in WDM Networks

Transcription

1 Virtual Topologies for Multicasting with Multiple Originators in WDM Networks Ian Ferrel Adrian Mettler Edward Miller Ran Libeskind-Hadas Department of Computer Science Harvey Mudd College Claremont, California Abstract In this paper we consider the problem of multicasting with multiple originators in WDM optical networks. In this problem, we are given a set S of source nodes and a set D of destination nodes in a network. All source nodes are capable of providing data to any destination node. That is, a destination node may be served by any source node. Our objective is to find a virtual topology in the WDM network which satisfies given constraints on available resources and is optimal with respect to minimizing the maximum hop distance. Although the problem is NP-complete in general, we give polynomial time algorithms for the cases of unidirectional paths and rings. 1 Introduction Wavelength-division multiplexing (WDM) has emerged as a key optical networking technology for realizing low cost, high bandwidth, and scalable data services. Each fiber-optical physical link in a WDM network is partitioned into multiple data channels, each of which operates on a separate wavelength. Thus, WDM permits the use of enormous fiber bandwidth by providing data channels whose individual bandwidths more closely match those of the electronic devices at their endpoints [14]. As WDM technology matures, it is likely to be widely used in systems ranging from local and metropolitan area networks to the backbone of the Next Generation Internet. This work was supported by the National Science Foundation under grant ANI to Harvey Mudd College. The author is now with the Department of Computer Science at UCLA. Corresponding author. 1

2 A communication model of potential importance is that of multicasting with multiple originators. In this communication model there is a designated set of source nodes and a set of destination nodes. All source nodes are capable of providing data to any destination node. Thus, a destination node may be served by any source node. This situation arises, for example, when data is replicated to a number of mirror servers in a network in order to increase reliability and performance [7]. Note that the servers may send the same data to all the destinations (standard multicast) or every destination may receive its own unique data from a server (personalized multicast). This paper addresses problems related to supporting multicasting with multiple originators in WDM networks. Ideally, each message in a WDM network is transmitted from the source to a destination without any optical-to-electronic conversion within the network. Such all-optical communication can be realized by using a single wavelength to establish a connection to each destination, but these connections require many dedicated optical paths which may, in general, be difficult or impossible to find [11]. All-optical wavelength converters may be used to convert from one wavelength to another within the network but converters are likely to be prohibitively expensive for most applications in the foreseeable future [14]. Moreover, in all-optical communication a path is typically dedicated for communication from the source to a specific destination, potentially under-utilizing the bandwidth of the channels on that path. A second approach is to embed a set of light-paths or light-trees in the network. A light-path (light-tree) is a path (tree) comprising channels on a single wavelength. Lighttrees are desirable for multicast communication since a single wavelength may be assigned to transmit data to multiple nodes. Light-trees require multicast-capable switches which can split an incoming signal on a particular wavelength to multiple outputs on the same wavelength. These switches are expensive and are thus frequently assumed to be a limited resource. Data may travel over multiple light-paths or light-trees from the source to a destination node. Within a light-path or light-tree, transmission is entirely optical. At the terminus of a light-path or light-tree the data is converted into electronic form and is delivered to the local node if it has reached its destination; otherwise it is retransmitted on another light-path or light-tree towards its destination. Intermediate nodes on a light-path or light-tree simply allow the data to pass through optically, but do not access the data themselves. Thus, a single light-path from node x to z passing through intermediate node y is different from two light-paths, one from x to y and one from y to z, even if these two light-paths use the same wavelength. The collection of light-paths or light-trees is called the virtual topology. In this paper we consider the problem of finding virtual topologies in WDM networks for multicasting with multiple originators. Given a network, a set of source nodes with identical data, a set of destination nodes, a set of network resource constraints, and an objective function, we wish to find a virtual topology such that for each destination node there is at least one source node that can reach that destination. This virtual topology must satisfy the resource constraints and be optimal with respect to the objective function. Many possible network resource constraints and objective functions could be considered. 2

3 For concreteness, in this paper we restrict our attention to network resource constraints stipulating the number of wavelengths available per channel and the number of transmitters available per node. With respect to receivers, we assume only that each destination node has at least one available receiver. In this paper, we assume that there are no wavelength converters and there are no multicast-capable nodes. Thus, the virtual topology is constructed from light-paths, each of which must be assigned a single wavelength. The objective function considered here is that of minimizing the maximum number of opto-electronic conversions required to reach any destination node in the virtual topology. More explicitly, a path from a source to a destination uses one or more light-paths. Each such light-path is said to incur a hop. In a virtual topology, the hop distance from a source s to a destination d is defined to be the minimum number of hops over all paths from s to d. In the presence of multiple sources, the best hop distance to a destination d is defined to be the minimum hop distance from any source to d. The maximum hop distance of the virtual topology is defined to be the maximum best hop distance over all destination nodes. In other words, the maximum hop distance is the maximum number of hops required from a source to a destination. Each hop incurs latency due to conversion from the optical domain to the electronic domain and back to the optical domain. In addition, each hop requires the use of a transmitter at its head and a receiver at its terminus. Thus, minimizing the maximum hop distance is an objective of interest for both resource usage as well as for performance, particularly in networks in which hop latency dominates channel propagation latency [1, 8, 15, 6]. The problem of broadcasting with multiple originators was studied more than twenty years ago by Farley and Proskurowski [3] and has recently re-emerged as a problem of interest [?]. Some related work has also been reported in the context of placement of mirrors in networks for optimizing web services [7]. Multicasting in WDM networks has received considerable attention in recent years. Thaker and Rouskas [13] and Shen et al. [12] present surveys of results in this area. More closely related to our work is the problem of multiple multicasting, in which multiple distinct multicasts occur concurrently. That is, each source has potentially different data and each destination node indicates the particular source or sources from which it must receive data. Most formulations of the multiple multicast problem are NP-complete and approaches typically use mixed integer linear programming (which is NP-complete itself) or heuristics. For example, multiple multicast problems have been studied by Sahasrabuddhe and Mukherjee [10] who consider constraints on the number of wavelengths, transmitters and receivers, and bandwidth, with the objective of minimizing the number of transmitters and receivers used. Their solution uses a mixed integer linear programming formulation. Shen et al. [11] give heuristics and approximation algorithms for a class of multiple-multicast problems and Mellia et al. [9] give a mixed integer linear programming formulation and several heuristics for problems related to minimizing congestion in multiple-multicast traffic. Although we point out in the next section that the problem of optimal multicasting with multiple originators is NP-complete in general, in this paper we show that the problem can be solved in polynomial time in directed paths and rings. Interestingly, the multiple multicast 3

4 problem (distinct data at the sources and each destination specifies the source or sources from which it must receive data) is NP-complete even in directed rings. 1 Our work also generalizes the results of Hartline et al. [6], although the techniques used are entirely different. Hartline et al. considered the problem of finding optimal virtual topologies, also with respect to minimizing the maximum hop distance, in rings with a single source and constraints only on the number of wavelengths, but not on the number of transmitters. Our work includes this as a special case since we allow any number of sources (possibly one) and any number of transmitters at the nodes (possibly unbounded). The techniques used in this paper may extend to other topologies and for other constraints and objective functions. The remainder of this paper is organized as follows: In Section 2 we give a formal definition of the multicasting with multiple originators problem and preliminaries. In Section 3 we give polynomial time algorithms for paths and rings. We conclude in Section 4. 2 Problem Statement and Preliminaries We begin this section by formally defining the multicasting with multiple originators problem. We then give additional definitions and fundamental results which will be used in the next section. 2.1 Problem Definition An instance of multicasting with multiple originators (MMO) is given by: A directed graph G = (V, E) representing a network, with vertices representing network nodes and directed edges representing optical links between nodes. Henceforth, we use the terms node and vertex interchangeably and similarly for the terms link and edge. A wavelength constraint W, specifying the number of wavelengths available on each edge. All optical links are assumed to have the same number of wavelengths available. A set S V of k source nodes, {s 1, s 2,..., s k }. For each source node s i, a number of available tunable optical transmitters T i. A set D V of destination nodes. D and S are assumed to be disjoint. A uniform number of transmitters T available to each non-source node. Note that we allow each source node to have a different number of transmitters available. However, for simplicity, we assume that all non-source nodes have the same number of transmitters available. 1 This result follows directly from the fact that the problem of determining the chromatic number of a circular arc graph is NP-complete [4]. 4

5 A virtual topology for an instance of MMO is a pair Γ = (P, f) where P is a set of directed paths in G called light-paths and f : P {1,..., W } is an assignment of light-paths to the available wavelengths such that: Any two light-paths in P that share a directed edge are assigned distinct wavelengths under function f. For each d D, there exists a set of light-paths, {p 1,..., p k } P such that p 1 originates at a source s i, p k terminates at d, and for 1 j k 1, the end vertex of p j is the starting vertex of p j+1. For each source s i, there are no more than T i light-paths in P that originate at s i. For each non-source node v, there are no more than T light-paths in P that originate at v. Recall that the hop distance from vertex s to vertex d is defined to be the minimum number of light-paths used over all paths from s to d. The best hop distance to a destination d is the minimum hop distance from any source to d. The maximum hop distance of the virtual topology is defined to be the maximum best hop distance over all destination nodes. In other words, the maximum hop distance is the maximum number of hops required from a source to a destination. Given an instance of MMO, we consider the objective of finding a virtual topology that minimizes the maximum hop distance, henceforth denoted an optimal virtual topology. The problem of finding an optimal virtual topology for an arbitrary MMO problem is known to be NP-complete in arbitrary graphs [2]. However, in Section 3 we show that the problem can be solved in unidirectional paths and rings in time polynomial in the number of vertices, n. More specifically, we show that in a directed path the problem of finding a virtual topology with maximum hop distance H can be solved in O(n) time for any positive integer H. Since the maximum hop distance in a virtual topology cannot exceed n, we can use a binary search process to find the smallest value of H for which there exists a virtual topology. This results in O(n log n) running time. For rings, the problem can be similarly solved in time O(n 2 log n). 2.2 Preliminaries We conclude this section with some preliminaries that will be used in the next section. First, let H denote the maximum hop distance under consideration. A light-path transmitted from a source node is said to be at hopcount H. A non-source node that receives a light-path at hopcount h H may retransmit the data on a new light-path; this light-path has hopcount h 1. No light-path can have hopcount less than 1, since this would imply that more than H hops were incurred. Definition 1 (Misordered Light-Paths and Ordered Virtual Topology). Let l 1 and l 2 be two light-paths on the same edge (u, v) at hopcounts h 1 and h 2, respectively, where 5

6 h 1 < h 2. If l 1 passes through vertex v and l 2 terminates at v then l 1 and l 2 are a misordered pair. A virtual topology with no misordered pairs of light-paths is said to be an ordered virtual topology. Definition 2 (Local Usage Vector). The local usage vector with respect to an edge e in a virtual topology is a vector w = (w H, w H 1,..., w 1 ) such that w k represents the number of light-paths over edge e at hopcount k. Definition 3 (Global Usage Vector). The global usage vector for a virtual topology is a vector w = (w H, w H 1,..., w 1 ) such that w k represents the total number of light-paths in the virtual topology at hopcount k. In a virtual topology, the load on an edge is defined to be the number of light-paths using that edge. The maximum load in a virtual topology is defined to be the maximum load over all edges. Clearly, the number of distinct wavelengths required in a virtual topology is at least as large as the maximum load. The following lemma from [5] states that in a path, the number of wavelengths required is exactly equal to the maximum load. Lemma 1. Let P be a unidirectional path. Any virtual topology on P with maximum load W may be wavelength-assigned using W wavelengths. Such an assignment can be found in time O(n). This result allows us to ignore the wavelength assignment problem in directed paths. Instead, we must ensure only that the load on any edge does not exceed the number of available wavelengths, W. Without loss of generality, in a path or ring we may assume that there are no consecutive source nodes, because two consecutive source nodes s i and s j may be simulated by a single source node with T i +T j transmitters. For simplicity, we will assume that all non-source nodes are destination nodes. That is, we assume that the objective is to broadcast the message. It is not difficult to show that the results in this paper are immediately extendable to the more general case of multicasting to an arbitrary subset of nodes. Finally, without loss of generality we restrict our attention to virtual topologies in which for each destination vertex d there is exactly one path from a source node to d. In other words, the virtual topology is a forest of multicast trees, each rooted at a source node. Thus, each destination node is only required to have one receiver. Note that with respect to minimizing the maximum hop distance, there is no need for a destination vertex to be reachable on more than one path. 3 Optimal Virtual Topologies in Paths Rings In this section we consider the problem of finding optimal virtual topologies in unidirectional paths and rings. Subsection 3.1 establishes an algorithm for paths and Subsection 3.2 uses this algorithm to solve the problem in rings. 6

7 3.1 Paths Paths are assumed to be directed from left to right. Lemma 2 (Ordered Virtual Topology in Paths). If there exists a virtual topology for a path P then there exists an ordered virtual topology for P. Proof. Assume by way of contradiction that there exists a virtual topology for P, but no ordered virtual topology exists. Among the set of virtual topologies for P, let V be a virtual topology that, in order 1. Maximizes the global usage vector with respect to lexicographic ordering. 2. Minimizes the number of misordered pairs of light-paths. 3. Minimizes the smallest distance between the right endpoints of two misordered lightpaths. Let l 1 and l 2 be a misordered pair of light-paths in V, with destination nodes d 1 and d 2 respectively, such that the distance between d 1 and d 2 is minimum. We call this the nearest misordered pair in V. Let l 1 be transmitted at g hops and l 2 be transmitted at h hops, g > h. We show that there are three possible cases, and in each case we construct a new virtual topology V that contradicts the assumptions made in the choice of V. Case 1: Node d 1 has an unused transmitter. In this case, we construct virtual topology V by eliminating light-path l 2 and serving d 2 with l 2, a light-path at g 1 hops transmitted from d 1, as shown in Figure 1. This transformation does not increase the load on any edge and every node in V receives the message with hopcount at least as large as in V. If g 1 > h, then V has a larger global usage vector than V, which contradicts assumption 1. Otherwise g 1 = h. Then V and V have the same global usage vector, but V has fewer misordered pairs. In particular, the light-paths l 1 and l 2, which serve d 1 and d 2 under V, do not form a misordered pair. Because any light-path in a misordered pair with l 2 in V would also be part of a misordered pair with l 2 in V, there are fewer misordered pairs in V. This contradicts 2. Case 2: There is a node v strictly between d 1 and d 2 which is served by a light-path from d 1. This implies that g = h + 1; otherwise, the light-path serving v would be part of a misordered pair with l 2, contradicting the selection of l 1 and l 2 as the nearest misordered pair. In this case, we construct V by terminating l 2 at v to form a new light-path l 2. We serve d 2 with a light-path transmitted from d 1. This transformation is shown in Figure 2. This transformation does not increase the load on any edge and does not change the hopcount at which any which any vertex receives a light-path. Thus, the global usage vectors for V and V are identical and they have the same number of misordered pairs. Light-paths l 1 and l 2 comprise a misordered pair in V, but the distance between their right endpoints is smaller than the distance between the endpoints of l 1 and l 2 in V, which contradicts 3. 7

8 l 2 l 1 g h g l 1 g-1 l 2 (a) d 1 d 2 d 1 d 2 (b) Figure 1: The transformation used in case 1 of Lemma 2. (a) The virtual topology V. (b) The new virtual topology V. h l 2 l 2 h+1 l 1 h l 1 h+1 h h (a) d 1 v d 2 d 1 v d 2 (b) Figure 2: The transformation used in case 2 of Lemma 2. a) The virtual topology V. (b) The new virtual topology V. 8

9 Case 3: If neither of the first two cases apply, then d 1 must transmit T light-paths in V, all of which pass through d 2. Because d 2 can transmit at most T light-paths, we may construct V by terminating l 1 at d 2 and l 2 at d 1, labeling these new light-paths l 1 and l 2, respectively. In V, d 1 serves the nodes that were served by d 2 in V, and vice versa. This is shown in Figure 3. The load over any region in V is at most the load over the corresponding region in V. The global usage vectors for V and V are the same and any misordered pair in V must also be a misordered pair in V : If l 1 is part of a misordered pair in V with no corresponding pair with l 1 in V, there must be a light-path at more than g hops terminating strictly between d 1 and d 2 in V and thus in V, contradicting the selection of l 1 and l 2 as the nearest misordered pair. If l 2 is part of a misordered pair in V with no corresponding pair with l 2 in V, there must be a light-path in V at fewer than h hops terminating strictly between d 1 and d 2, again contradicting the definition of l 1 and l 2. Finally, as l 1 and l 2 do not form a misordered pair in V, and no new misordered pairs were added, V has fewer misordered pairs than V, which contradicts 2. l 2 l 1 g h T l 2 l 1 h g T d 1 d 2 d 1 d 2 (a) (b) Figure 3: The transformation used in case 3 of Lemma 2. (a) The virtual topology V. (b) The new virtual topology V. The next lemma shows that in an ordered virtual topology, the local usage vectors have a particular property. This property will then be exploited in the development of our algorithm. Lemma 3 (Local Usage Vectors in Ordered Virtual Topologies). In an ordered virtual topology on a unidirectional path P the local usage vector for each edge is of the following form. For each edge there is some k, 1 k < H, such that: 1. Element w H is unconstrained. 2. For all i such that k < i < H, w i < T. 3. Element w k T. 4. For all i such that 1 i < k, w i = 0. 9

10 Proof. Assume by way of contradiction that in some ordered virtual topology V for path P, the local usage vector for an edge is not in this form. Let e = (u, v) be the leftmost such edge and let w = (w H, w H 1,..., w 1 ) denote the local usage vector on e. There are two ways that e can violate the above form. Either some element w i > T, 1 i < H, or for some j and k, 1 j < k < H, w k = T and w j > 0. In the first case, node u must transmit a light-path at i hops; otherwise, there would be more than T light-paths at i hops on the edge preceding e, contradicting the assumption that e was the leftmost edge not in the given form. Therefore, u is the destination of a light-path l at i + 1 hops. Because u cannot transmit more than T light-paths, there must also be a light-path passing through u at i hops, forming a misordered pair with l. This contradicts the assumption that V is an ordered virtual topology. In the second case, we examine the hopcount of the light-paths transmitted by u, if any. If u transmits a light-path at k hops, then u is served by a light-path at k + 1 hops. Because a light-path at j hops (j < k) must pass through u, this contradicts the assumption that V is an ordered virtual topology. If u transmits a light-path at j hops, then u is served by a light-path at j + 1 k hops. The local usage vector for the previous edge must therefore be in violation of the given form as it contains the same T light-paths at k hops as does e and also an additional light-path serving u at k or fewer hops. This contradicts the assumption that e was the leftmost edge in violation of the given form. Finally, if u does not transmit a light-path at either j or k hops, then all light-paths at j and k hops on e must also use the previous edge, contradicting the assumption that e was the leftmost edge violating the given form. We now describe a simple algorithm which takes as input an instance of the MMO problem on a unidirectional path and a positive integer H. The algorithm constructs a virtual topology with maximum hop distance at most H or determines that none exists. After describing the algorithm we prove its correctness. Algorithm 1 (Optimal Virtual Topology Algorithm for Paths). INPUT: A unidirectional path P = (V, E), positive integer W indicating the number of available wavelengths, set S of source nodes in P, a positive integer T i for each s i S indicating the number of transmitters available at source s i, set of destination nodes V S, a positive integer T indicating the number of transmitters at each non-source node, and a positive integer H indicating the desired upper-bound on the maximum hop distance. OUTPUT: A virtual topology satisfying all of the given constraints and maximum hop distance at most H or indication that no such virtual topology exists. 1. Let A be a set called the active light path set. A is initially empty. 2. Moving from left to right in the path let v denote the current vertex under consideration. (a) If v is a destination node then i. If A is empty then report that no virtual topology exists satisfying the given constraints and exit. 10

11 ii. Else serve v with any light-path in A with minimum hopcount, h. If h > 1, then v transmits T light-paths at hopcount h 1 and these light-paths are added to A. (b) If v is a source node s i then v transmits T i light-paths at hopcount H and these light-paths are added to A. (c) If the number of light-paths in A exceeds the wavelength constraint W, then the excess light-paths are removed from A in increasing order of hopcount until W light-paths remain. Observe that by construction (specifically, step 2a), any virtual topology constructed by Algorithm 1 is an ordered virtual topology. Therefore, every virtual topology constructed by Algorithm 1 satisfies the properties stipulated in Lemma 3. In particular, for every h < H there are at most T light-paths with hopcount h in such a virtual topology. Algorithm 1 is easily verified to have running time O(n) when W is a constant. Moreover, if a virtual topology is found then, by Lemma 1, it can be wavelength-assigned in time O(n). Therefore, a virtual topology can be found and wavelength-assigned in time O(n). Thus, an optimal virtual topology in a unidirectional path can be found in O(n log n) time using binary search over the n possible values of H. Theorem 1 (Correctness of Algorithm 1). In a unidirectional path, Algorithm 1 finds a virtual topology of maximum hop distance H or less or determines that none exists. Proof. If the algorithm finds a virtual topology then it has maximum hop distance at most H since set A contains only active light-paths which have incurred H or fewer hops. Assume by way of contradiction that there exists a virtual topology V with maximum hop distance at most H but that the algorithm reports that no virtual topology exists. By Lemma 2 we may assume that V is an ordered virtual topology. Let w = (w H,..., w 1 ) and ω = (ω H,..., ω 1 ) denote two local usage vectors. We use the notation w ω and w ω to indicate that w is lexicographically greater than or equal and strictly greater than ω, respectively. Let w(x) and ω(x) denote the local usage vector on an edge x in Algorithm 1 and in ordered virtual topology V, respectively. In the following, we show by induction that for each edge e in P, the local usage vector w(e) on e with respect to Algorithm 1 is lexicographically greater than or equal to the local usage vector ω(e) on e with respect to V. If Algorithm 1 fails to find a virtual topology then at some vertex v there are no active light-paths in the set A. Thus, on the edge e immediately preceding v, w(e) must be the zero vector while ω(e) is not the zero vector. However, w(e) ω(e), yielding a contradiction. Basis: If the leftmost vertex of P is not a source then there exists no virtual topology. Otherwise, Algorithm 1 transmits the maximum number of light-paths possible from the first source. Thus, on the leftmost edge e, w(e) ω(e). 11

12 Induction Hypothesis: Assume that for an edge e, w(e) ω(e). Induction Step: Let f be the edge immediately following e. We wish to show that w(f) ω(f). Let v denote the vertex shared by e and f. We divide our analysis into two cases, one in which v is a source node and one in which it is not. 1. Assume that v is a source node s i. Algorithm 1 adds T i light-paths at hopcount H to the set of active light-paths and then removes excess light-paths beyond the W permitted, in increasing order of hopcount. In V, at most T i light-paths at hopcount H can be introduced at v. Thus, w(f) ω(f). 2. Assume that v is a destination node. In this case Algorithm 1 serves v with the light-path at the lowest available hopcount, h. Node v then transmits T light-paths at hopcount h 1 and removes excess light-paths in increasing order of hopcount. Because V is an ordered virtual topology, it serves v with the lowest hopcount light-path in ω(e). If w(e) = ω(e), then V serves v with a light-path at hopcount h. Therefore, w(f) ω(f). Otherwise, w(e) ω(e) and thus let m, 1 m H, be the first difference between w(e) and ω(e), the largest value for which w(e) m > ω(e) m. (Recall that w(e) and ω(e) are vectors and thus w(e) m and ω(e) m denote the elements at location m in these vectors.) Let h A and h V be the hopcounts at which v is served in A and V, respectively. Our analysis is divided into cases as follows: (a) m > h A and m > h V. Then w(f) m > ω(f) m, no entries with indices greater than m have changed in the local usage vectors, and thus w(f) ω(f). (b) m = h V > h A. No entries with indices greater than m have changed. Also, w(f) m = w(e) m > ω(e) m and ω(e) m > ω(f) m which implies that w(f) m > ω(f) m and thus w(f) ω(f). (c) m = h A > h V. There are two cases to consider: i. w(e) m > ω(e) m + 1. In this case, the first difference has not changed because w(f) m = w(e) m 1 > ω(e) m = ω(f) m. Thus, w(f) ω(f). ii. w(e) m = ω(e) m + 1. In this case, w(f) m = ω(f) m. Because Algorithm 1 transmits the maximum number of light-paths from v, w(f) m 1 = T, or less due to the removal of excess wavelengths in step 2c of the algorithm. If w(f) m 1 = T then ω(f) m 1 < T. Otherwise, on edge e in V there would be T light-paths at m 1 hops and an additional light-path at h V m 1 hops. By Lemma 3, this contradicts the assumption that V is an ordered virtual topology. If w(f) m 1 = t < T, then some of the T light-paths originating at v were removed in step 2c of Algorithm 1. Therefore w(e) contains at most t 1 light-paths with hopcounts m 1 or less. Since the first difference between w(e) and ω(e) is at m and w(e) m = ω(e) m + 1 it follows that ω(e) contains at most t light-paths with hopcounts m 1 or less. Because v is served by 12

13 one of these light-paths in V, ω(f) m 1 < t. Therefore, w(f) m 1 > ω(f) m 1 and thus w(f) ω(f). (d) m = h A = h V. Since w(e) m > ω(e) m it follows that w(f) m > ω(f) m and therefore w(f) ω(f). (e) h V > m h A. In this case the first difference between w(f) and ω(f) occurs at position h V and w(f) ω(f). (f) In all remaining cases, h A > m. Because Algorithm 1 uses the light-path with lowest hopcount to serve v, w(e) m = 0, contradicting the observation the fact that w(e) m > ω(e) m. 3.2 Rings Rings are assumed to be directed clockwise. Lemma 4 (Ordered Virtual Topology in Rings). If there exists a valid virtual topology for a unidirectional ring R, there exists an ordered virtual topology for R. Proof. The proof is identical to the proof of Lemma 2. Definition 4 (H-Nested Pairs of Light-Paths). In a unidirectional ring, let l 1 and l 2 be two light-paths such that both light-paths are at hopcount H, the light-paths have distinct starting nodes and distinct ending nodes, and such that the set of edges traversed by l 1 is a proper subset of the set of edges traversed by l 2. Then l 1 and l 2 are said to be a H-nested pair of light-paths. Definition 5 (Clean Virtual Topology). A virtual topology with no H-nested pairs of light-paths is said to be a clean virtual topology. We begin by assuming that any one node in the ring is capable of complete wavelength conversion; any light-path entering that node on a given wavelength can be optically converted at that node to any wavelength. Such a ring is henceforth denoted a wavelengthconverting ring. The existence of a wavelength converting node is useful in proving several key results, including the next lemma which asserts that in a wavelength-conversion ring a virtual topology with maximum load of W can be wavelength-assigned using W wavelengths. We later show that the assumption that there exists a wavelength converter can be removed. Lemma 5. Let R be a unidirectional ring in which one arbitrary node is capable of complete wavelength conversion. Any virtual topology on R with maximum load W may be wavelengthassigned using W wavelengths. Such an assignment can be found in time O(n). Proof. Let v be the node in R capable of complete wavelength conversion. Let P denote the path induced by removing v from R and adding two new vertices, v exit and v enter at the left and right ends of the path, respectively. Each light-path in R which enters v has a corresponding 13

14 light-path in P which terminates at v enter and each light-path in R which exits v has a corresponding light-path in P which begins at v exit. Thus, a light-path passing through v in R induces two light-paths in P. The maximum load on P remains W. By Lemma 1, P can be wavelength-assigned using W wavelengths. Therefore, R can be wavelength-assigned using the same wavelength-assignment and using wavelength conversion at v. Lemma 6 (Clean Ordered Virtual Topologies). If there exists a virtual topology V on a wavelength-converting ring then there is a clean ordered virtual topology on R. Proof. Assume by way of contradiction that there exists a virtual topology on R, but no clean ordered virtual topology exists. Among the set of virtual topologies for R, let V be an ordered virtual topology that, in order 1. Minimizes the length of the longest light-path in a H-nested pair. 2. Minimizes the number of light-paths in H-nested pairs at that length. In V, let l 1 = (x 1, y 1 ) and l 2 = (x 2, y 2 ) be a H-nested pair, where l 1 is a longest light-path in a H-nested pair. Construct a new virtual topology V identical to V except that l 1 and l 2 are replaced with a new pair of light-paths l 1 = (x 1, y 2 ) and l 2 = (x 2, y 1 ). This does not increase the load on any edge, and all nodes in V are served by light-paths with the same hopcount as in V. Either the number of H-nested pairs of that length is decreased, which contradicts 2, or the length of the longest light-path in a H-nested pair is reduced, which contradicts 1. Definition 6 (Foreign Node). A foreign node with respect to a source node s and a virtual topology V is a destination node that receives the message via a light-path which passes through s in V. Lemma 7 (Foreign Node Position). For any ordered clean virtual topology V on a wavelength-converting ring R, either some source has no light-paths passing through it, or each source has a foreign node immediately following it. Proof. If there is a light-path passing through every source in V, then assume by way of contradiction that there exists a source s that is not followed immediately by a foreign node. Let h be the lowest hopcount at which a light-path passes through s. If h < H, then because V is an ordered virtual topology, a light-path at h hops serves the first destination node following s, and this node is thus a foreign node. Otherwise h = H and because V is a clean virtual topology, one of the light-paths at H hops passing through s serves the node immediately following s. This node is foreign. In both cases, there is a contradiction to the assumption that s is not followed by a foreign node. Lemma 8 (Special Source in Wavelength-Converting Rings). If there exists a virtual topology V on a wavelength-coverting ring R, then there exists a virtual topology V on R such that for some source node s S, no light-paths in V pass through s. 14

15 Proof. Assume by way of contradiction that there is a light-path passing through every source in every virtual topology on R. By Lemmas 6 and 7, there exists a clean ordered virtual topology on R with a foreign node following each source. Among all such topologies, let V be an ordered clean virtual topology that minimizes the sum of the number of foreign nodes with respect to all sources. We create a new topology V with the following transformation: 1. The destination of each light-path is moved to the nearest destination node to its left. 2. The origin of each light-path transmitted from a destination node is moved one destination node to its left. 3. The origin of each light-path transmitted from a source node is unchanged. (a) (b) Figure 4: The transformation used in Lemma 8. (a) The virtual topology V. (b) The virtual topology V. The load on any edge under this new virtual topology is at most the load on the following edge in the original topology, so the maximum load has not been increased. Additionally, each destination node in V is served at the hop at which the following destination node was served in V. This transformation does not introduce any H-nested pairs, so V is also an ordered clean virtual topology. This transformation reduces the number of foreign nodes with respect to every source, which contradicts the assumption that the sum of the number of foreign nodes with respect to every source was minimized. We now show that the requirement that some node have wavelength conversion capability can be removed. Lemma 9 (Rings with no Wavelength Converters). If there exists a virtual topology on a wavelength-converting ring R, then there exists a virtual topology on R with no wavelength converter and such that some source node has no light-paths passing through it. 15

16 Proof. Let V be a virtual topology on wavelength-converting ring R. By Lemma 8 there exists a virtual topology V on R such that for some source vertex s there is no lightpath passing through s. Since the location of the wavelength converter in wavelengthconverting ring R is arbitrary, let the wavelength converter be placed at s. Since no light-path passes through s in V, virtual topology V is unaffected by the removal of the wavelength converter. Lemma 9 permits us to reduce the MMO problem for rings (without wavelength conversion) to the MMO problem for paths. The lemma guarantees that if there exists a virtual topology in a ring, then there exists a virtual topology such that some source has no lightpaths passing through it. Our algorithm for rings, therefore, considers every source node as the candidate for the source node with no light-paths passing through it. Then ring is split into a path at this source node and Algorithm 1 is applied to the path. Algorithm 2 (Optimal Virtual Topology Algorithm for Rings). INPUT: A directed ring R = (V, E), positive integer W indicating number of available wavelengths, set S of source nodes in P, a positive integer T i for each s i S indicating the number of transmitters available at source s i, set of destination nodes V S, a positive integer T indicating the number of transmitters at each non-source node, and a positive integer H indicating the desired upper-bound on maximum hop distance. OUTPUT: A virtual topology satisfying all of the given constraints and maximum hop distance at most H or indication that no such virtual topology exists. 1. For each source vertex s S, consider the path P s induced by removing the edge entering s. (a) Apply Algorithm 1 to P s. If the algorithm succeeds, the virtual topology found for P s is a virtual topology for R. 2. If Algorithm 1 fails for all sources, report that no virtual topology exists satisfying the given constraints. Note that the running time time of this algorithm is O(n 2 ) since there are at most n invocations of Algorithm 1. By using a binary search over the range of possible values of H, an optimal virtual topology in a ring can be found in time O(n 2 log n). 4 Conclusions In this paper we have examined the multicast with multiple originators problem in WDM networks. In particular, we have considered the problem of finding virtual topologies which minimize the maximum hop distance subject to constraints on the number of available wavelengths on the links and transmitters at the nodes. Although the problem is known to be 16

17 NP-complete in general, we have given efficient algorithms for unidirectional paths and rings. Our technique is based on showing that there exist optimal solutions in specific canonical forms and then restricting our attention to solutions in these forms. We believe that this approach may be useful in finding solutions for other physical topologies and also for extending our results to other definitions of optimality. References [1] R. Dutta and G. N. Rouskas. On optical traffic grooming in WDM rings. In Proceedings of the SIGMETRICS/PERFORMANCE, June [2] T. Eilam, M. Flammini, and S. Zaks. A complete characterization of the path layout construction problem for ATM networks with given hop count and load. Parallel Processing Letters, 8(2): , [3] A. Farley and A. Proskurowski. Broadcasting in trees with multiple originators. SIAM Journal of Algebraic and Discrete Methods, 2(4): , December [4] M. Garey, D. Johnson, G. Miller, and C. Papadimitriou. The complexity of coloring circular arc graphs and chords. SIAM Journal of Discrete Mathematics, 1(2): , [5] U. Gupta, D. T. Lee, and J. Y.-T. Leung. An optimal solution for the channelassignment problem. IEEE Transactions on Computers, c-28(11): , November [6] J. R. Hartline, R. Libeskind-Hadas, K. Dresner, E. Drucker, and K. Ray. Optimal virtual topologies for one-to-many communication in WDM paths and rings. IEEE/ACM Transactions on Networking, To appear. [7] S. Jamin, C. Jin, A. Kurc, D. Raz, and Y. Shavitt. Constrained mirror placement on the internet. In Proceedings of the 20th Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), [8] R. Libeskind-Hadas and R. Melhem. Multicast routing and wavelength assignment in multi-hop optical networks. IEEE/ACM Transactions on Networking, 10(5): , October [9] M. Mellia, A. Nucci, E. Leonardi, and M. A. Marsan. Optimal design of logical topologies in wavelength-routed optical networks with multicast traffic. In Proceedings of the IEEE Global Telecommunications Conference GLOBECOM, [10] L. H. Sahasrabudde and B. Mukherjee. Light-trees: Optical multicasting for improved performance in wavelength-routed networks. IEEE Communications Magazine, 37(2):67 73, February

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