Multicast Routing under Delay Constraint in WDM Network with Different Light Splitting

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(1)Multicast Routing under Delay Constraint in WDM Network with Different Light Splitting Ming-Tsung Chen 1, Shian-Shyong Tseng 2 1. Department of Computer and Information Science National Chiao-Tung University Hsinchu 300, Taiwan R.O.C. mtchen.cis88g@nctu.edu.tw. 2. Department of Computer and Information Science National Chiao-Tung University Hsinchu 300, Taiwan R.O.C. sstseng@cis.nctu.edu.tw. Abstract Because the optical WDM network will become a real choice to build up backbone in the future, multicast communications on the WDM network should be supported in communication model for various network applications. In this paper, we define a new multicast problem that is routing a request with delay bound to all destinations in WDM network with different light splitting and propose a new formulation to solve the problem, where the different light splitting means that nodes in the network can transmit one copy or multiple copies to other nodes by using same wavelength. The new problem can be reduced to Minimal Steiner Tree Problem (MSTP) which belongs to NP-Complete problem, and can be solved by an efficient three-phase (Pre-Processing Phase, Generating Phase, and Refining Phase) solution model with Backward Stepwise Sub-path Replacing (BSSR) and Most Cost-Difference First Progressive Replacing (MCDFPR) heuristics to find a feasible light-forest in polynomial time. Finally, experimental results show that our solution model can obtain a near optimal solution. 1. Introduction Optical networks [1] are high-capacity telecommunications networks, based on optical 1.

(2) technologies and optical components, which provide routing, grooming, and restoration at the wavelength level as well as wavelength-services. The technology of WDM (Wavelength Division Multiplexing) network [8], based on optical wavelength-division multiplexing on an optical fiber to form multi-communication channels at different wavelengths, provides connectivity among optical components to make optical communication to meet the increasing demands for high channel bandwidth and low communication delay. To transmit data between source and destination in WDM network, a light-path which connects two nodes should be established. To support multicast communication in WDM network, nodes in WDM network may have the light splitting capacity which is used to split an optical signal of input port to multiple signals of output ports without electrical conversions, and a routing-tree used to transmit a request could be constructed. The light-tree [2] is a special routing-tree by configuring nodes in the physical topology and occupies the same wavelength in tree links. Each branch node of the light-tree is an optical switch had light splitting capacity. The nodes with the light splitting capacity named as MC (multicast capable) node are usually more expensive to build than those without named as MI (multicast incapable) node due to its complexity architecture [1]. Furthermore, two important measurements (communication cost and wavelength usage) for evaluating the performance of routing-tree are usually considered on WDM network for QOS (Quality Of Service). Another measurement, transmission-delay, will come into view in the problem of multicast in WDM network. In order to satisfy the requirement, several protocols and algorithms have been proposed on traditional network or WDM network to solve different problems. Recently, the multicast routing problem in WDM network with sparse light splitting is proposed and solved by X. Zhang, et al. [4]. Besides, the other researches of multicast routing with wavelength conversion [5] or with delay bound [6] but not both have been proposed. 2.

(3) In this paper, two characteristics of WDM network, nodes with different light splitting and a request with delay bounded, are considered simultaneously. A new multicast problem finding a minimal cost light-forest on a WDM network with different light splitting such that the request can be routed under delay bound is proposed. The light-forest is a set of light-trees and whose number is equal to the number of wavelengths used to serve a multicast request. v0 8/0.68. 1/1.52. v8. 3/1.4. v4 1/1.6. v5 1/1.1 v7. 3/1.3. v9. v10 4/1.13. v7. 8/0.5. v6 2/0.4 13/0.23 v1. 4/1.13. v3. 7/0.2. 2/1.5 v10 6/0.5. 14/0.3 v11. 9/0.54. v9 8/0.5. 1/1.1. 4/1.34 v9. 2/1.5. 10/0.4. v5. 1/1.6 v4. 7/0.97. v5 1/1.6. 1/1.52 v0. v12 8/0.68. 2/1.5. v4. v10 3/1.4 v0 8/0.68. 8/0.68. v2 v8. (a). (b). v8. (c). Fig. 1 WDM network and routing-trees for r (v9, {v0, v5, v8, v10},3.3) . For example, the graph in Fig. 1(a) represents a WDM network with 13 nodes, where nodes v7 and v9 are MC nodes. Each link in the graph is associated with a value-pair “a/b”, where a and b are the communication cost and the transmission-delay of link, respectively. For a given request, r (v9, {v0, v5, v8, v10}, 3.3), on the WDM network, the trees shown in Figs. 1(b) and (c) are two possible routing-trees for r, where v9 is the source, v0, v5, v8, and v10 are the destinations, and delay bound is 3.3 time unit. The routing-tree shown in Fig. 1(b) needs 1 wavelength, 17 communication cost units, and 7.53 time unit, and it is not feasible because 7.53 time unit is greater than 3.3. Nevertheless, because v5 is not an MC node and the out-degree of v5 is 2, the routing-tree shown in Fig. 1(c) needs 2 wavelengths for routing the request from v5 to v4 and v5 to v0, 30 communication cost units, and 2.58 time unit, and is feasible because 2.58 time unit is smaller than 3.3. The remainder of this paper is organized as follows. In Section 2, we formally define the problem. In Section 3, a solution model composed of three phases to solve the problem is proposed and each phase of the solution model is described in detail. Section 4 presents the 3.

(4) simulation for solution model, and Section 5 gives some conclusions. 2. Formulation A weighted graph G(V, E) denotes a WDM network, where the node set V represents the optical nodes (switches or routers), and the edge set E represents the optical links between nodes. The numbers of nodes and edges in the WDM network are defined as |V|=n and |E|=l, respectively. Each link is composed of two oppositely directed fibers. For each link e, c(e) and d(e) are associated with edge e to represent the communication cost and transmission-delay, respectively. θ(v) ≥ 1 is used to represent the splitting degree of node v∈V, which is the number of copies that can be forwarded to other nodes. If θ(v) is equal to k, the node v can transmit k copies of request to other nodes concurrently by using same wavelength. In this paper, a multicast request represented by r(s, D, ∆) goes from a certain source s∈V, passes several nodes, arrives at all destinations in set D ⊆ V-{s} finally, where |D| = m, and the transmission-delays of light-paths between s and any destination di in D expect to be bounded by ∆. Assume there are q paths, Pi(u,v) = < eu , wi , ewi , wi , ... ewi 1. 1. ki. 2. ,v. > (i.e. < u , w1i , w2i , ..., wki i , v > ),. between two nodes u and v, where ewi , wi is a link of nodes wij and wij +1 , ki is the number of j. j +1. internal nodes in Pi(u,v), and 1 ≤ i ≤ q. The P(u, v)={Pi(u,v) | 1≤i≤q} is used to represent a set of all light-paths between two nodes u and v. The notations of V(Pi(u,v)) and E(Pi(u,v)) are used to represent nodes and links in path Pi(u,v). The communication cost and transmission-delay of path Pi(u, v) described by c(Pi(u,v)) and d(Pi(u,v)) are represented as. communication cost : c( Pi (u, v)) =. ∑ c (e ) = c ( e. e∈Pi ( u , v ). transmission-delay : d ( Pi (u, v)) =. u , w1i. ∑ d (e) = d (e. e∈Pi ( u ,v ). ) + c(ewi , wi ) + ... + c(ewi ,v ). u , w1i. 1. ki. 2. ) + d (ewi , wi ) + ... + d (ewi ,v ) 1. 2. ki. Among light-paths in P(u,v), two critical paths, critical-cost path (CCP) whose 4.

(5) communication cost is minimal and critical-delay path (CDP) whose transmission-delay is minimal, can be represented as Pc(u, v) and Pd(u, v) : P c (u, v) = min c( Pi (u, v)) 1≤i ≤ q. P d (u, v) = min d ( Pi (u, v)) 1≤i ≤ q. Given k light-paths, P (u1, v1), P (u2, v2), …, and P (uk, vk), they can be combined into a k. graph,. U P(u , v ) . Applying Prim’s MSpT (Minimal SPanning Tree) algorithm for this graph, i. i. i =1. k. k. i =1. i =1. two routing-trees (spanning trees), MSpTc( U P(ui , vi ) ) and MSpTd( U P(ui , vi ) ), can be obtained ,where the MSpTc is used to find an MSpT whose multicast cost is minimal and MSpTd is used to find an MSpT whose transmission-delay is minimal. Given a routing-tree T for r(s, D, ∆), root s of T has k sub-trees, T1, T2, …, Tk. Assume that the splitting degree of s is θ(s) which can route θ(s) copies to other nodes by using a wavelength. The lower bound of required wavelength of T is the maximum of required wavelengths of sub-trees. ω(T) and ϖ(T) representing the number and the lower bound of required wavelengths of T. can be defined as : ϖ(T)= max ω (Ti ) 1≤i ≤ k.  ∑ ω (Ti )  ,ϖ (T )) ω(T) = max( 1≤i≤k  θ (s)   . Because communication cost of an edge depends on the number of wavelengths passing through the edge, the total communication cost of the edge is directly proportional to the number of passing wavelengths. That is, because Ti needs ω(Ti) wavelengths, the total communication cost of es ,si for routing a request from s to si should be ω(Ti)⋅c(es,si), where. si is the root of Ti. The communication cost and transmission-delay of T is described as : c(T ) =. ∑ (ω (T ) ⋅ c(e. 1≤ i ≤ k. i. s , si. ) + c(Ti )). d (T ) = max(d (Ti ) + d (es ,si )) = max d ( PT ( s, d i )) 1≤i ≤ k. 1≤i ≤ m. 5.

(6) Because communication cost and wavelength are critical resources usually in WDM network, α is defined as the ratio of the weight between two measures. If α effects how to choice a routing-tree that needs more wavelengths and lower communication cost or a routing-tree that needs fewer wavelengths and more communication cost. The multicast cost. function f is defined as f (T)= c(T) +αω(T) In our problem, the network does not provide wavelength conversion between different wavelengths and a light-tree can be routed by using a wavelength, so a routing-tree T need to be separated into ω(T) light-trees, TL1, TL2, …, TL ω(T), whose number of needed wavelengths is equal to 1; that is, T =. ω (T ). i L. U (T. ω (T ). ) , f(T)=. ∑ f (T i =1. i =1. i L. ) , and ω (TLi)=1 for 1≤i≤ ω (T). A. light-forest Γ={ TL1, TL2, …, TL ω(T)} is a set of light-trees TLi which route a request to the partial destinations in D, where Di can be used to represent the set of partial destinations. A light-forest Γ is feasible if it satisfies three constraints, destination constraint, delay. constraint, and degree constraint formulated as i =k. (1) destination constraint : D = U Di i =1. (2) delay constraint : d(TLi) ≤∆ , where TLi ∈Γ, ∀i, 1≤i≤ω(T) (3) degree constraint : ω(TLi)=1 , where TLi ∈Γ, ∀i, 1≤i≤ω(T) A routing-tree is a candidate if it satisfies delay and destination constraints. An efficient candidate means that the candidate needs lower multicast cost. It should be noted that a candidate does not necessarily satisfy degree constraints and that a candidate could be separated into a feasible light-forest. Therefore, in our solution model, once an efficient candidate is found, a feasible light-forest can be obtained easily by separating this candidate. The multicast costs of light-forest and candidate are equivalent. Two special cases, a network with no light splitting, which let all splitting degrees of nodes be equal to 1, and a network with sparse light splitting, which let all splitting degrees of nodes be equal to 1 or ∞, were proposed [6], [4]. In this paper, we first define the following 6.

(7) generalized problem, given a WDM network G(V, E) with different light splitting and a request r(s, D, ∆), find a feasible light-forest Γ such that f(Γ) is minimal. We then propose a three-phase solution model (Pre-Processing Phase, Generating Phase, and Refining Phase) with several heuristics to find a light-forest to route the request. The detailed description will be depicted as follows. 3. Solution Model. This new problem is NP-Complete because this problem can be reduced to the MSTP which is NP-Complete [9]. It is difficult to find an optimal light-forest in polynomial time. Therefore, Backward Stepwise Sub-path Replacing (BSSR) and Most Cost-Difference First. Progressive Replacing (MCDFPR) heuristics are proposed to find a near optimal light-forest in polynomial time. The BSSR heuristic is backward stepwise tracing reversely from di to s in order to find a node v such that d(PT(s, v)) + d(Pd(v, di)) ≤ ∆, and replacing the sub-path PT(v,. di) of PT(v, di) with the corresponding CDP Pd(v, di)). The MCDFPR heuristic is replacing the most cost-difference path PT(u,v) with maximizing of c(PT(u,v))- c(PT(u,v)) for any two nodes. u and v in V(T). The solution model consisting of the following three phases is shown in Fig. 2. (1). Pre-processing Phase – Construct the information matrices, Critical-Cost Path. Matrix (CCPM) which is the matrix of Pc(vi, vj) for vi, vj ∈V and 1≤i, j≤n, and Critical-Delay Path Matrix (CDPM) which is the matrix of Pd(vi, vj) for all vi, vj ∈V and 1≤i, j≤n, by using Dijkstra’s Shortest Path algorithm.. (2). Generating Phase – Use Backward Stepwise Sub-path Replacing (BSSR). heuristic to find a candidate; otherwise, return a null-tree to indicate no candidate.. 7.

(8) Network Topology Information. Pre-Processing Phase. Request r (s, D, ∆). Information Matrices. Generating Phase. Light-Tree. Refining Phase. Light-Forest. Fig. 2 Solution Model. (3). Refining Phase – Use Most Cost-Difference First Progressive Replacing. (MCDFPR) heuristic to refine the candidate to decrease multicast cost, and then separate the candidate into a feasible light-forest. Because the WDM network topology is statically pre-constructed, the pre-processing for all critical-cost paths, all critical-delay paths, and corresponding communication costs and transmission-delays between any pair of nodes can reduce the computation time. By using the two matrices, Generating Phase can find a candidate for r(s, D, ∆). The third phase, the. Refining Phase, can be used to refine the candidate to gain a near optimal candidate, and to separate the near optimal candidate into a feasible light-forest. 3.1. Pre-processing Phase. When the WDM network is constructed, the Pre-processing Phase is performed one time by applying the Dijkstra’s Shortest Path algorithm which requires O(n2) time in linear array structure to construct and save two matrices (CCPM and CDPM ) in disk or media for reusing.  n Because there are   pairs of nodes for finding CCP and CDP, the time complexity of  2. Pre-processing phase is O(n4). 3.2. Generating Phase. The Generating Phase consists of three steps : 8.

(9) (1). Checking Step – Check the multicast request whether a candidate exists or not to. avoid non-necessary computation. (2). Finding-MST Step – Apply the Minimal Distance Network Heuristic (MDNH). [3] to find a Minimal Steiner Tree (MST). (3). Rerouting Step – Refine the MST to gain a candidate.. Checking Step. The Checking Step with O(|D|) = O(m) is used to check c(Pd(s, di)) ≤ ∆ for all di in D. After the step, a Boolean value “TRUE” is returned, if it is positive; “FALSE” is returned, otherwise. The processing continues unless a FALSE-result is obtained. In Fig.1 (a), assume a request r (v9, {v0, v5, v8, v10}, 3.3) is given. Because d(Pd(v9,. v0))=1.9, d(Pd(v9, v5))=0.5, d(Pd(v9, v8))=2.58, and d(Pd(v9, v10))=1.5 all are smaller than 3.3, the TRUE-value will be returned. Finding-MST Step. The Finding-MST Step applying the Minimal Distance Network Heuristic (MDNH) [3] needs O(mn2) to find the MST (Minimal Steiner Tree) which is a routing-tree in WDM, where the MDNH algorithm only considers communication cost and destination constraint. Hence, the found MST may not be a candidate or a light-tree.. Finding_MST(r(s, D, ∆)) { 1. G1=(V’, E’), where V’= s∪D, E’={ evi,vj| Pc(vi, vj)≠∅, ∀ vi, vj∈ V’ }, and c(evi,vj)= c(Pc(vi, vj)) 2. T1 = MSpTc (G1) 3. T = MSpT c (. UP. c. ( vi , v j ) ). evi ,v j ∈T1. 4. delete all leaf nodes of T which do not belong to D 9.

(10) }. In Fig. 1, the sub-graph G1 which covers source and destinations and whose MSpT with minimal communication cost are shown in Figs. 3(a) and (b), respectively. Each edge in Figs. 3(a) and (b) represents a path kept in CCPM. The corresponding paths of edges in Fig. 3(b) are shown in Fig. 3(c), and then these paths can be merged into a graph. After applying Prim’s. MSpT algorithm to this graph, a routing-tree T shown in Fig. 3(d) can be obtained. In this step, a routing-tree T with c(T)=17 and d(T) =7.53 for a request r(v9, {v0, v5, v8,. v10}, 3.3) with near optimal communication cost can be found, but it does not satisfy the bounded delay 3.3. Vdelay(T) is used to represent the subset of destinations which violates the. delay constraint in T. Because (d(Pd(v9, v8))=7.7, d(Pd(v9, v0))=7.02, and d(Pd(v9, v5))=3.73), Vdelay(T) is equal to {v8, v0, v5}. T will be adjusted to become a candidate in the Rerouting. Step. v9 v9. 6/3.3. 6/3.22 14/3.9. v0. 2/3.12. v5. 2/1.5. v0. 2/3.12. v5. 2/1.5. 7/5.35 8/0.68. 5/2.23. 8/0.68. 5/2.23. v10. v8. v10. 10/3.8 v8. 14/1.78. (a) G’. (b) T1 v9 2/1.5 v10 4/1.13. v0. v5 1/1.1 v7 v9. 2/1.5 v10. 4/1.13 v10. 1/1.1. v5. 1/1.52. v0. v4. v7. 1/1.6. 8/0.68 v8. 1/1.6 v5. v4. v8 1/1.52. 8/0.68 v0. (c) Corresponding path of edges in T1. (d) Light-tree T. Fig.3 Processes of Finding-MST Step. Rerouting Step. In Rerouting Step, the paths between s and di in Vdelay(T) must be rerouted and replaced 10.

(11) with the shorter-delay path that is found by different heuristic algorithms. The path PT(v, dmax) is rerouted first by using the Backward Stepwise Sub-path Replacing (BSSR) heuristic, where. dmax∈Vdelay(T) and d ( P T ( s, d max )) = max d ( P T ( s, d i )) . The skeleton of this step is d i ∈Vdelay (T ). described as follows.. Rerouting-Step (T, r(s, D, ∆)) /* T is a light-tree*/ { 1. while(TRUE) 2.. choose dmax∈Vdelay(T), such that d ( P T ( s, d max )) = max d ( P T ( s, d i )). 3.. if (d(PT(s, dmax)) ≤ ∆) return TRUE // satisfy the delay constraint T = BSSR (T, r(s, D, ∆), dmax, Vdelay(T)). 4.. d i ∈Vdelay (T ). 5. end while-loop }. Because each edge in the routing-tree T must be traveled one time to find dmax and compute d(PT(s, v)) for all v ∈ V(T), the computation of Sub-step 2 requires O(lT), where lT is the number of edge in routing-tree T. The time of while-loop from sub-steps 1 to 5 can be limited by the number of destinations in Vdelay(T). The sub-step 4, BSSR heuristic, is described as follows.. BSSR (T, r(s, D, ∆), dmax, Vdelay(T)) /*T is a routing-tree and dmax is a heavy-delay destination*/ { 1. u = Parent(dmax) 2. while(u is not empty) 3. 4.. // parent of dmax in T // if u is empty, u is a root of routing-tree T. if (d(PT(s, u)) + d(Pd(u, dmax)) ≤ ∆) return Reconstruct_tree(T, r(s, D, ∆), u, dmax, Vdelay(T)). 5. u = Parent(u) 6. end white-loop 7. return Reconstruct_tree(T, r(s, D, ∆), s, dmax, Vdelay(T)) }. 11.

(12) Reconstruct_tree(T, r(s, D, ∆), u, dmax, Vdelay(T)) { 1. DN = {x| x ∈ V(PT(u, dmax))∩D} // V(PT(u, dmax)) is a set of nodes in path PT(u,. dmax) 2. T = T - PT(u, dmax) 3. T = MST d (T U. UP. d. (u, v)). v∈DN. 4. return T }. We know that the merging of T and Pd(u,v) in Reconstruct_tree dominates the time complexity of BSSR heuristic and needs O(n2). v0 8/0.68. v8. v8. 10/0.4. v9. 10/0.4. v3. 2/1.5. v3. 4/1.34. v10. 4/1.34. v10. (a) The remnant. v10. (b) CDP between v10 and v8. (c) CDP between v10 andv0 v9. v9 8/0.5. 8/0.5. 2/1.5 v10. v5. v2 v10 8/0.68. 3/1.4. v3. v12. 10/0.4. 8/0.5. 6/0.5. v5. 4/1.34. v5. 2/1.5. v0. 7/0.97. v8. v9 2/1.5. v0. v10. (d) CDP betweenv10 and v5. 8/0.68. (e) After replacing of v8. v3 v8. 10/0.4. (f) After replacing of v0. Fig. 4 Processes of Rerouting Step. The routing-tree in Fig. 3 (d) is used in this phase as an input data and α =5, and dmax =. v8 are chosen first. Using BSSR heuristic, nodes in T would be checked reversely from v8 to v9 till the node v10 is found because d(PT(v9, v10)) + d(Pd(v10, v8)) ≤ 3.3. So the path, PT (v10, v8) = < ev10,v7, ev7,v5, ev5,v4, ev4,v0, ev0,v8>, must be rerouted ; the remnant, the routing-tree in Fig. 3 (d) removed PT (v10, v8), which is shown in Fig. 4 (a). Nevertheless, v8, v0, and v5 are destinations, 12.

(13) so the paths of Pd(v10, v8), Pd(v10, v0), and Pd(v10, v5) shown in Figs. 4 (b)~(d) need be merged into the remnant to obtain a new routing-tree shown in Fig. 4 (e). After the step, we can obtain a candidate shown in Fig. 4 (f), which satisfy delay and destination constraints and whose communication cost, transmission-delay, wavelengths, multicast cost are 44, 2.55, 1, 49, respectively. The complexities of three steps (Checking Step, Finding-MST Step, and Rerouting Step) which are O(m), O(mn2), and O(n2) can be summarized into O(mn2). 3.3. Refining Phase. Because the candidate may be not efficient, the Refining Phase with the Most. Cost-Difference First Progressive Replacing (MCDFPR) heuristic is used to obtain a near optimal candidate, and then the Separating Step is used to split the candidate into a light-forest. The detail of MCDFPR heuristic consisting of 9 Steps is described as follows.. MCDFPR (T) { 1.While computation is not exhausted 2. for each u, v ∈V(T) Heapify(K, (u, v , c(PT(u, v))- c(Pc(u, v)))) 3. (u’,v’) = Pop(K) // K need re-heapify. 4. While((u’,v’)≠∅) 5. T’= MSpTc ( (T- PT(u’,v’)) ∪Pc(u’,v’) )// u’ is predecessor of v’ if (d(T’)≤ ∆) if (f(T’) < f(T)) T= T’ goto step 1 8. end-while loop 9. end-while loop 6. 7.. // // PT(u’,v’) will be replaced with Pc(u’,v’). }. In this procedure, Heapify(K, (u, v , c(PT(u, v))−c(Pc(u, v)))) [10], is used to push the data structure (u,v, c(PT(u, v))−c(Pc(u, v))) into a heap K in decreasing order by the value of c(PT(u, 13.

(14) l  v))−c(Pc(u, v)). If mT is the number of leaves and lT is the depth of T, mT ⋅  T  pairs of data 2. structures will be pushed into K. Hence, the time complexity of Step 2 needs O(mT lT2log mT. lT2) to build a heap K. The Step 3 needs O(log mT lT2) to pop root (u’,v’) from K and to tune K. Because Step 5 needs O(n2) at most to merge (T- PT(u’,v’)) and Pc(u’,v’). The time complexity of this phase needs O(a(mT lT2log mT lT2 + b⋅ n2)), where b is the average loop time between. Steps 4 to 8 to find a better candidate and the refining time, a, is the loop time between Steps 1 to 9 . Because b = O(mTlT2) in the worst case, the complexity needs O(a(mT lT2log mT lT2 +. mTlT2 n2)) and is reduced to O(a mTlT2 n2) due to n > lT. Because the refined candidate does not satisfy degree constraint, the Separating Step will be used to separate the candidate into a light-forest. The primary concept in Separating Step is to separate branches whose out-degree is greater than splitting degree ( θ T (v) > θ v ) into a set of light-trees in order to satisfy degree constraint. v9. v9 2/0.4. 8/0.5. v6. v10. 8/0.68. (a) Pc(v8,v9). v10 3/1.4. v0. v0. 2/1.5. v5. 3/1.4. 1/1.52. 8/0.5. v6. v5. 3/1.3 v4. v8. v9. 2/0.4 2/1.5. v0 8/0.68. 8/0.68 v8. v8. (b) after replacing. (c) final candidate. Fig. 5 Processes in Refining Phase. The candidate in Fig. 4 (f) is an input data of this phase; the MCDFPR heuristic is used to separate the path PT(v8,v9) = <ev9,v2, ev2,v12, ev12,v3, ev3,v8>, because c(PT(v9, v8))−c(Pd(v9, v8))≥. c(PT(vi, vj))−c(Pd(vi, vj)) for each vi, vj ∈V(T)-{v8, v9}. i.e., PT(v8,v9) is the most cost-difference path in T (Fig. 4 (f)). After the first iteration of MCDFPR heuristic, PT(v8,v9) is replaced with. Pc(v8,v9) shown in Fig. 5 (a) to gain a refined candidate in Fig. 5 (b). The near optimal better candidate shown in Fig. 5 (c) can be gained finally after MCDFPR heuristic is processed two 14.

(15) times. In Fig. 5 (c), the communication cost, transmission-delay, wavelengths, and multicast are 21, 2.58, 1, 26, respectively. Hence, the multicast cost will be reduced from 44 to 26 successfully. Finally, because the Pre-Processing Phase with complexity O(n4) is performed only one time for initializing WDM network, we can ignore the time complexity of the phase. Hence, the complexity of our solution model needs O(mTlT2 n2) for finding a near optimal light-forest. 4. Simulation. Our work focuses on how to find a near optimal light-forest such that destination, delay, and degree constraints are satisfied. We simulate the solution model proposed in previous sections to evaluate the performance of our heuristics. The approach used in this simulation can be referred in Waxman [7]. In the approach, there are n nodes in the networks, these nodes are distributed randomly over a rectangular grid, and are placed on an integer coordinates. Each link between two nodes u and v is added with the probability function. P(u, v) = λ exp(− p(u, v) / γδ ) , where p(u,v) is the distance between u and v, δ is the maximum distance between any two nodes, and 0 < λ, γ ≤ 1. In the probability function, larger value of λ produce networks with high link densities, while small value of γ increases the densities of short links relative longer ones. In our simulations, we use λ=0.7, γ=0.9, size of rectangular grid = 50, and n = 100. To reduce the complexity of problem, the cost function c of link (u, v) in the network is the distance between u and v on the rectangular coordinated grid and delay function d of link (u, v) is generated randomly. For each request r(s, D, ∆), source s, destinations D, and delay constraint ∆ are generated randomly. Nevertheless, the delay constraint given by a request must be reasonable; otherwise, the light-forest cannot be found. In. our. simulation,. the. network. topology,. communication. costs. of. edges,. transmission-delays of edges, and a request are generated randomly. Therefore, we discuss several comparisons consisted of execution time, multicast cost, transmission-delay, 15.

(16) wavelengths on different number of destinations. Besides, the efficiency of Refining Phase will be discussed in our simulation because Refining Phase can improve utility of network. The execution time of Processing Phase, execution time of Refining Phase, improved ratio of multicast cost in Refining Phase, overhead ratio of transmission-delay in Refining Phase, and comparison of wavelengths for different number of destinations are shown in Table. 1. We observe that the execution time rises moderately and is directly proportional to the number of destinations m. However, the growth of execution time in Refining Phase is unstable because the loop in the MCDFPR heuristic can be executed several times to improve multicast cost until it can not be improved anymore. Hence, the computation time of m=26 needs 1900 milliseconds. Let Γbefore _ refining and Γafter _ refining be light-forests of before Refining Phase and after Refining Phase, respectively. The average improvement ratio of multicast cost and the average overhead ratio of transmission-delay are 20.83% and 12.13% respectively, where by. the. improvement. c(Γbefore _ refining ) − c(Γafter _ refining ) c(Γbefore _ refining ). ratio and. and. overhead. ratio. d (Γafter _ refining ) − d (Γbefore _ refining ) d (Γbefore _ refining ). are. computed. .. The simulation results of α about comparisons of improvement of multicast cost, overhead of transmission-delay, and usage of wavelength are shown in Figs. 6 to 8, where m is number of destinations, variation of wavelength is equal to the wavelength improvement of Refining Phase, respectively. Although the value of α is increasing, the improvement ratio of multicast cost shown in Fig. 6 will hold on 12% at least. Furthermore, the overhead ratio of transmission-delay is almost constant shown in Fig. 7 and the numbers of wavelength is inverse proportional to α shown in Fig. 8. For the improvement of multicast cost, overhead of transmission-delay, and variation of wavelength on Refining Phase under different network sizes (n=40, 50, …, 100) and different numbers of destinations (m=5, 10, 20, 30) are shown from Figs. 9 to 11, respectively. The average improvement ratios of m=5, 10, 20, and 30 under different network sizes shown in 16.

(17) Fig. 9 are 23.28%, 21.51%, 22.69%, and 19.50%, respectively. The Fig. 10 is described the comparison of overhead ration of transmission-delay and whose average overhead ratios of m=5, 10, 20, and 30 under different network sizes are 7.50%, 4.05%, 7.05%, and 5.76%. The average variation of wavelengths shown in Fig. 11 are 0.46, 0.81, 1.91, and 2.46 for m=5, 10, 20, and 30; the average improvement ratios of wavelength are 14.48%, 12.12%, 16.81%, and 14.76%. Finally, we choose different requests with different number of destinations randomly and compare the results between using our solution model (3-Phase) and using genetic algorithm (GA) on network with 30 nodes. The GA in our simulation choices population size = 500, generation size = 200, crossover rate = 0.8, and mutation rate = 0.1. The comparisons of computation time, multicast cost, and transmission-delay are shown in the Table. 2. Because the average computation times of our model and GA model are 0.1024 seconds and 5818.688 seconds, we can find that the computation time of GA model is far greater than our model. Therefore, we can make sure that the three-phase solution model can obtain a near optimal light-forest in polynomial time. 5. Conclusion. In this paper, a new formulation and a new multicast routing problem under delay constraint in WDM network with different light splitting are studied. A three-phase solution model consists of Pre-Processing Phase, Generating Phase, and Refining Phase, where Backward Stepwise Sub-path Replacing (BSSR) and Most Cost-Difference First Progressive Replacing (MCDFPR) heuristics have been used in Processing Phase and Refining Phase to improve the efficiency, respectively. Using the solution model, a light-forest that needs fewer wavelengths and lower multicast cost to transmit a request to meet delay constraint can be found in polynomial time. To evaluate the performance of our solution model, an experiment has been made. The experimental result shows the average improvement ratio of multicast cost, the average 17.

(18) overhead ratio of transmission-delay, and average improved ratio of wavelength are 20.83%, 12.13%, and 8.1% in Refining Phase. For different values of α, different network sizes, and different numbers of destinations, the benefits of three-phase solution model will hold still. Finally, compared with GA, the three-phase solution mode is more efficient and better than GA. For future research, it is important to discuss the multicast routing and wavelength assignment on-line or to accommodate multiple multicast requests currently in WDM network with different light splitting. We are now trying to refine our solution model to solve the problem of multi-hop system. Acknowledgement. This work was partially supported by Ministry of Education and National Science Council of the Republic of China under Grand No. 89-E-FA04-1-4, High Confidence Information Systems and by Telecommunication Lab., Chunghwa Telecom Co., Ltd. References. [1] P. E. Green, “Fiber-Optic Networks”, Cambridge, MA : Prentice-Hall 1992 [2] L.H. Sahasrabuddhe and B. Mukherjee, “Light-trees : Optical multicasting for imporoved performance in wavelength-routed networks”, IEEE Commun. Mag., vol 37, pp 67-73, Feb, 1999. [3] L. Kou, G. Markowsky, L. Berman, “A Fast Algorithm for Steiner Trees,” Acta Informatica, Vol. 15, pp. 141-145,1981 [4] X. Zhang, J, Wei, and C. Qiao “Constrained Multicast Routing in WDM Networks with Sparse Light Splitting”, Proceedings of INFOCOM 2000, May [5] N. Sreenath, N. Krishna Mohan Reddy, G. Mohan, C. Siva Ram Murthy, “ Virtual Source Based Multicast Routing in DWM Networks with Sparse Light Splitting” (IEEE 2001) [6] Xiao-Hua Jia, Ding-Zhu Du, Xiao-Dong Hu “Integrated algorithm for delay bounded 18.

(19) multicast routing and wavelength assignment in all optical networks”, Computer Communications 24 (2001) p.1390-1399 [7] B. M. Waxman, “Routing of multipoint connections”, IEEE JSAC, 6 (1988), pp 333-342 [8] E. Lowe, “Current European WDM Development Trends”, IEEE Communications Magazine 36 (2) (1998) 46-50 [9] R. M. Karp, “Reducibility among Combinational Problems,” In R.E. Miller, J. W. Thatcher(Eds.), Complexity of Computer Computations, Plenum Press, New York, pp. 85-103, 1972 [10] T. H. Cormen. et al. ”Introduction to Algorithm”, McGraw-Hill Book Company, pp. 142-152. 19.

(20) Table. 1. Wavelengths Before Refining 3 3.4 4.6 4.2 4.6 5.8 4.8 7 5.2 6.6 5.6 7.4 8.2 7.6 8.4 10.8 7.6 9.4 9.6 13 11 7.4 7.2 10.6 10.6 9.4. Overhead Ratio(%) 7.56 9.46 13.59 12.6 12.63 13.83 8.31 7.86 16.33 10.04 16.03 11.02 14.1 13.98 13.9 16.06 12.48 9.49 8.7 10.93 14.81 10.85 13.6 9.36 13.64 14.18. Wavelengths After Refining 3 3.2 4.6 4.2 4.6 4.8 4.6 7.2 4.6 6.2 4.8 7 9.2 5.8 8 11 6.6 8.4 10.4 12.6 8.4 6 5.8 7.4 9.2 8.4. Variation of wavelenth 0 -0.2 0 0 0 -1 -0.2 0.2 -0.6 -0.4 -0.8 -0.4 1 -1.8 -0.4 0.2 -1 -1 0.8 -0.4 -2.6 -1.4 -1.4 -3.2 -1.4 -1. m 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20. Execution Execution Time of Time of 3-Phase(sec.) GA(sec.) 0.001 3494 0.02 3322 0.03 3799 0.05 4385 0.04 4552 0.03 4786 0.08 6085 0.06 5365 0.211 6558 0.361 7711 0.071 5923 0.015 7235 0.18 7734 0.1 7724 0.11 7008 0.28 7418. Multicast Cost of 3-Phase 41.652 47.571 63.844 76.403 80.852 77.801 106.506 97.203 99.065 119.658 99.587 96.539 123.66 159.509 121.632 119.62. Multicast Cost of GA 36.768 42.75 57.116 58.889 75.548 73.184 93.896 97.029 102.771 112.977 96.401 116.361 124.476 120.805 118.821 126.739. Trans. Delay of 3-Phase 2 2.268 1.821 2.622 1.904 1.867 2.368 2.451 2.916 3.387 3.11 2.38 2.231 2.333 2.325 3.115. Trans. Delay of GA 2.302 2.676 2.046 3.098 1.948 2.22 2.563 2.355 3.333 3.194 2.716 2.737 2.554 2.529 2.623 3.133. 18 16. Overhead Ratio(%). Improvment Ratio(%). 35 30 25. m =5 m =10. 20. m =20. 15. m =30. 10. 14 12. m=5. 10. m=10. 8. m=20. 6. m=30. 4 2. 5. 0 0.1. 0 0 .1. 1. 5. 10. 20. 30. 40. α. 50. 60. 70. 80. 90. 5. 10. 20. 30. 40. 50. α. 60. 70. 80. 90 100. Fig. 7 Overhead ratio for different α.. 1.5 45. 1. Improvment Ratio(%). Variation of Wavelength. 1. 100. Fig. 6 Improvement ratio for different α.. 0.5. m=5. 0 0.1. -0.5. 1. 5. 10 20 30 40 50 60 70 80 90 100. m=10. -1. m=20. -1.5. m=30. -2 -2.5. 40 35 30. m=5. 25. m=10. 20. m=20. 15. m=30. 10 5 0. -3. 40. α. 15. m=5. 10. m=10. 5. m=20. 0. m=30. 50. 60. 70. 80. 90. 100. -10. Variation of Wavelength. 20. 40. 60. 70. 80. 90. 100. Fig. 9 Improvement Ratio of multicast cost.. 25. -5. 50. Num. of Nodes( n ). Fig. 8 Variations of wavelength for α.. Overhead Ratio(%). m 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30. Execution Execution ImprovTime of Time of ment Gent. Phase(msRef. Phase(ms Ratio(%) 40 26.2 27.19 36 84.4 17.66 38 46 18.55 46.4 142.2 20.58 42 160 21.82 42 142.2 24.81 48.2 192.4 24.88 50 312.6 20.53 54 304.2 20.35 52 419 19.59 60 354.4 19.23 60 497 19.75 62 559 21.94 76.2 594.6 19.37 78.2 959.4 24.36 72 647.2 18.93 78 905.4 20.59 76 615.2 25.05 82.4 654.4 20.9 92 741 19.52 86 785.2 22.84 92.4 1854.6 21.84 92 795.2 16.06 104.4 1384 18.3 108.6 1173.4 18.65 84 1206 18.36. Table. 2 Comparisons of 3-Phase and GA.. 1 0 -1. 40. 50. 60. 70. 80. 90. 100. m=20. -3. m=30. -4 -5. Num. of Nodes(n ). Num. of Nodes(n ). Fig. 10 Overhead Ration of Trans. delay. 20. m=5 m=10. -2. Fig. 11 Variations of wavelength..

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