Distributed Scheduling Algorithms for Wavelength Convertible WDM Optical Interconnects

Transcription

1 Distriuted Scheduling Algorithms for Wavelength Convertile WDM Optical Interconnects Zhenghao Zhang and Yuanyuan Yang Department of Electrical and Computer Engineering State University of New York, Stony Brook, NY 79, USA Astract Optical communication is attracting more and more attentions ecause of its huge andwidth to meet the ever increasing demand of emerging computing/networking applications. In this paper we study distriuted scheduling algorithms to resolve output contentions in WD- M optical interconnects with wavelength conversion aility. We consider the general case of limited range wavelength conversion, including the full range wavelength conversion. Two types of limited range wavelength conversions, circular symmetrical and non circular symmetrical, are studied. We introduce the request graph and show that finding the largest group of contention-free connection requests to achieve maximum network throughput is equivalent to finding a maximum matching in the request graph. Compared with the existing algorithm for finding a maximum matching in an aritrary ipartite graph with time complexity O(N k d), the algorithms we present have time complexity of O(k) and O(dk) (independent of interconnect size N) for non-circular symmetrical and circular symmetrical wavelength conversion, respectively, where k is the numer of wavelengths per fier and d is the conversion degree. In addition, our algorithms can e easily implemented in hardware, and used for time slotted WDM optical interconnects where connections hold for different numer of time slots. I. INTRODUCTION AND BACKGROUND Many emerging computing/networking applications, such as data-rowsing in the world wide we, video conferencing, video on demand, E-commerce and image distriuting, require very high network andwidth often far eyond that today s high-speed networks can offer. Optical networking is a promising solution to this prolem ecause of the huge andwidth of optics: a single fier has a andwidth of nearly 0 THz [6]. To fully utilize the andwidth, a fier is divided into a numer of independent channels, with each channel on a different wavelength. This is referred to as wavelength-division-multiplexing (WDM). In a WDM all optical network, data is modulated on a selected wavelength channel and this information-earing signal remains in the optical domain throughout the path from source to destination. In the asence of wavelength conversion aility, the signal is required to e on the same wavelength from hop to hop, which is referred to as the wavelength continuity constraint. This constraint can e removed when wavelength converters are employed in the network. Wavelength converter converts a signal on one wavelength to another wavelength, and makes the network more flexile for satisfying various connection requests. Studies show that network performance is greatly improved y using wavelength converters [6]. If a wavelength converter can convert a wavelength to any other wavelength in the optical system, it is called full range wavelength con- Research supported in part y the U.S. National Science Foundation under grant numers CCR and CCR verter. However, this type of wavelength conversion is quite difficult and expensive to implement due to technological limitations [], []. A realistic all-optical wavelength converter may only e ale to convert to a limited numer of wavelengths for any given wavelength, which is called limited range wavelength converter. In general, a limited range wavelength converter can convert a wavelength to a set of adjacent wavelengths, and the numer of wavelengths in the set is called conversion degree. Research [], [], [] shows that limited range wavelength converters can achieve network performance similar to full range wavelength converters even when the conversion degree is very small. Thus, limited range converters are considered as a practical, cost-effective choice for providing wavelength conversion aility in WDM networks, which will e the main focus of this paper. A WDM optical interconnect (also called WDM switch in some literature) provides interconnections etween a group of input fier links and a group of output fier links with each fier link carrying multiple wavelength channels. Such an optical interconnect can e used to serve as a crossconnect (OXC) in a wide-area communication network or to provide high-speed interconnections among a group of processors in a parallel and distriuted computing system. The WDM interconnect we consider is an N N switch, i.e., there are N fiers on the input side of the switch and N fiers on the output side of the switch. On each fier there are k wavelengths that carry independent data. Thus, there are a total of Nk input wavelength channels and Nk output wavelength channels in the interconnect. Any input wavelength channel can e connected to any output fier. In addition, there are limited range wavelength converters with conversion degree d (d» k) equipped on the output side of the switch. With limited range wavelength conversion, an input wavelength channel may e connected to d adjacent channels on an output fier. Clearly, full range wavelength conversion is a special case of limited range conversion when d = k. Figure shows an N N WDM interconnect with k wavelengths on each fier, in which each wavelength can e converted to d adjacent wavelengths. It can e seen from the figure that an input fier is first fed into a demultiplexer, where different wavelength channels are separated one from another. The separated wavelength channels are the input of a switching faric, through which an input wavelength channel is connected to d adjacent wavelength channels on an output fier. At the output side of the switching faric, there are optical cominers for each output wavelength channel which is used to comine /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

2 inputs N all input channels with wavelength convertile to 0 0 κ 0 κ DMUX DMUX 0 κ 0 κ Nk Nk Switching Faric Cominer Wavelength converter 0 κ 0 κ MUX MUX 0 κ 0 κ Fig.. A wavelength convertile WDM optical interconnect. outputs N the optical signals that are ale to access this output wavelength channel. Thus, there are Ndinputs to a cominer, ut only one of them may carry signal at a time. The output of the cominer is then fed into a wavelength converter, which converts the wavelength of the signal to the desired wavelength. Then the k output wavelengths are multiplexed into an output fier. The WDM optical interconnect can e operated either asynchronously or synchronously. The former case applies to WD- M wavelength routing networks that are similar to electrical circuit switching networks. A typical scenario for the latter case would e an optical WDM packet switching network where information is carried in optical packets that arrive to the interconnect at the eginning of each time slot. The duration of an optical packet is usually assumed to e one time slot. However, we also consider the general case of multiple time slot duration in this paper. All connections or optical packets are assumed to e of the same priority. Since optical uffers are currently made of fier delay lines and are still very expensive [], we also assume that there are no uffers in the WDM optical interconnect. The traffic pattern considered in this paper is unicast, i.e., each connection request is destined for only one output fier. The connection request does not specify which wavelength channel on the destination fier it should e connected to, and we can assign to it any free wavelength channel accessile to this connection. Under these assumptions, the connection requests arrived at the interconnect in one time slot can e partitioned into N susets according to their destinations. The decision of accepting a request or not in one suset does not affect the decisions in other susets, as no connection request elongs to two susets. This provides a asis for a fast distriuted scheduling algorithm, as the scheduling for each output fier can e done independently of other output fiers. As will e seen later, the time complexity of the distriuted scheduling algorithms we propose is only in terms of the numer of wavelengths per fier and the conversion degree, and is independent of the interconnect size. A gloal scheduling algorithm, on the other hand, will have a time complexity at least linear to the size of the interconnect. Thus, in the following we will focus on the scheduling algorithms that have a distriuted nature and can e run independently for each output fier. The input to the scheduling algorithm is the connection requests destined to this fier. The output of the algorithm is the decision whether a request is granted or not, and if granted, which wavelength channel it is assigned to. Note that if the wavelength conversion is full range the scheduling is trivial: if no more than k connection requests arrived at this output fier, grant all; if more than k arrived, aritrarily pick k out of them. This is ecause full range wavelength converters are capale of converting any incoming wavelength to any outgoing wavelength, which makes the connection requests indistinguishale in the wavelength domain. However, the scheduling ecomes more complex when we use limited range wavelength converters, as the wavelengths of connection requests can no longer e considered indistinguishale. For example, in the interconnect in Figure, if k = 6, and d =, consider the case where two connections on, three connections on and one connection on have arrived and all want to destine to the same output fier with six wavelengths 0 ; ;::: ;. If the wavelength conversion is full range, all of them can e satisfied ecause there are totally six requests which does not exceed the total numer of wavelengths on a fier. However, since the conversion degree is in this interconnect, not all can e satisfied. This is ecause that there are five connection requests arrived on and, ut there are only four output wavelengths accessile to and : f 0 ; ; ; g with conversion degree. If not all connection requests can e satisfied, we say there is output contention among the connection requests. When a contention occurs, the scheduling algorithm needs to select some of the connection requests to transmit while reject others. To maximize network throughput, the algorithm should find the largest group of requests that are contention-free. In this example, we can drop one request coming on or and realize the rest of connection requests. Extensive research has een conducted on scheduling algorithms for various electronic switches (which can e considered as a single wavelength switch). For example, [7] and [8] considered scheduling algorithms in input-uffered electronic switches under unicast traffic. Scheduling algorithms for WD- M roadcast and select networks were also well studied in recent years, see, for example, [7], [8]. In this type of network, the source node roadcasts its information to all other nodes via a selected wavelength, and only the destination node tunes into this wavelength to get the message, so that only one wavelength on the fier is used at a time, oth for the source and the destination node. Note that this is a quite different type of network from the WDM interconnect considered in this paper. We consider a space-division switch where all wavelengths on a fier can e utilized simultaneously. There has also een some work in the literature on the performance analysis of WDM optical interconnects with limited range wavelength conversion /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

3 in WDM wavelength routing networks, e.g., [], [], []. In these studies, the packet arrivals at the optical interconnect were assumed to e asynchronous, thus eliminates the need for a scheduling algorithm since the requests have a natural order and are assumed to e served according to the first come first served rule. However, future WDM optical interconnects are more likely to e operated synchronously [9] [0], for example, optical packet switching networks [], where connection requests arrive in time slots. This type of interconnect is also more suitale for interconnecting processors in a parallel or distriuted computing system as synchronization is required a- mong collaorating processors. In such an interconnect, a general method for resolving output contention has to e studied. In this paper, we will focus on contention-free scheduling algorithms for such synchronized WDM optical interconnects. II. PRELIMINARIES A. Wavelength Conversion As mentioned earlier, in a WDM optical interconnect with limited range wavelength conversion, an incoming wavelength may e converted to a set of adjacent outgoing wavelengths, and the numer of wavelengths convertile is equal to the conversion degree. We can use a conversion graph to visualize the wavelength conversion. A conversion graph is a ipartite graph, in which each of the left side vertex represents an input wavelength and each of the right side vertex represents an output wavelength. Thus, if there are k wavelengths on each fier, there are k vertices on each side of the conversion graph. If input wavelength i can e converted to output wavelength j, we draw an edge etween them. Figure (a) is the conversion graph for k = 6 and conversion degree Φ d =, in which, wavelength Ψ i may e converted to (i ) mod 6; i ; (i+) mod 6,fori f0; ;::: ;g. In general, we assume i may e converted to f (i e) mod k; (i e+) mod k;::: ; i ;::: ; (i+f ) mod kg, wheree and f are the numers of wavelengths that i may e converted to on its minus and plus side, respectively. Clearly, e+f + = d. In the example aove e = f =. We call the wavelengths that i may e converted to as the adjacency set of i in the conversion graph, and represent the adjacency set as an interval of integers: [i e; i + f ]. Note that all the numers inside this interval are in mod k. In other words, interval [x; y] represents numers fx mod k; (x +) mod k; : : : ; y mod kg. This notation is purely for presentational convenience, since the adjacency set of some wavelength may not e an interval. For example, the adjacency set of 0 is f ; 0 ; g and apparently ; 0; and are not an interval, ut since ( mod 6) = ( mod 6), we can represent it as [ ; ]. The conversion scheme discussed aove is the common assumption aout limited range wavelength conversion in the literature. The resulting conversion graph is circular symmetrical. There are also other types of assumptions aout wavelength conversion which is not circular symmetrical. For example, some authors considered the adjacency set of any i is interval [u; v], where u = max f0;i eg and v = inputs 0 (a) outputs 0 0 inputs () outputs Fig.. Conversion graphs of two types of wavelength conversion six wavelengths and conversion degree three. (a) Circular symmetrical. () Noncircular symmetrical. min fk ;i+ f g, which means that the wavelengths near one end cannot e converted to the wavelengths on the other end. For example, if k = 6 and e = f =, 0 can only e converted to 0 and, and it cannot e converted to.figure () shows the conversion graph of this type of conversion when k =6and e = f =. Note that the adjacency set of this type of conversion is indeed an interval. We will consider oth types of wavelength conversions in this paper. B. Prolem Formalization As descried in Section I, we consider an N N WDM optical interconnect with k wavelength on each fier. Any input wavelength channel can e connected to its adjacency set on any output fier according to its conversion degree. We depict the relationship etween the connection requests destined for an output fier and the availale wavelength channels on that output fier y a ipartite graph, called request graph. Onthe left side of the request graph, each node represents a connection request and on the right side of the graph, each node represents an output wavelength. We use A to represent the set of the left side vertices and B for the right side vertices. Let the vertices on the right side e in the same order as their corresponding wavelength indexes. For example, 0 is aove, is aove and so on. The vertices on the left side are also ordered according to their wavelength indexes (requests with the same wavelength are in an aritrary order). There is an edge etween a left side node a and a right side node if the wavelength of connection request a can e converted to output wavelength. Thus the conversion graph discussed earlier can e simply considered as a special case of the request graph when there is exactly one connection request coming on each wavelength. For convenience we also define the request vector. A request vector is a k row vector, with the i th element representing the numer of connection requests arrived on wavelength i. Figure shows the request graphs of the two types of conversions when the request vector is [; ; 0; ; ; ]. In addition, for any a i A, letw (i) denote the wavelength index of connection request a, W (i) [0;k ]. For example, in Figure (a), W (0) = W () = 0,andW () =. When no confusions arise, we will use it to represent the wavelength of connection request a i as well. A request graph of an output fier can e easily formed and stored in hardware. For example, the left side vertices of the request graph can e implemented y an Nk inary vector /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

4 0 connection requests a a a a a a 6 (a) output wavelengths 0 0 connection requests a a a a a a 6 () output wavelengths Fig.. Request graphs of two types of conversions when the request vector is [; ; 0; ; ; ] in a 6-wavelength interconnect with conversion degree. (a) Circular symmetrical. () Non-circular symmetrical. 0 (an Nk it register), with element (i )Λk + j eing means j on the i th input fier is destined for this output fier and 0 otherwise. The Nk inary vector is set at the eginning of each time slot. The right side vertices of the request graph can e implemented y a k vector with each element storing the decision of which input wavelength channel it is assigned to. The edges that represent the relationship etween the inputs and outputs do not need to e implemented explicitly and can e considered emedded in the circuit. In a request graph G, lete denote the set of edges. Any wavelength assignment can e represented y a suset of E, E, where edge a E if wavelength channel is assigned to connection request a. Under unicast traffic, any connection request needs only one output channel and an output channel can e assigned to only one connection request. It follows that the edges in E are vertex disjoint, ecause that if two edges share a vertex, either one connection request is assigned two wavelength channels or one wavelength channel is assigned to two connection requests. Thus, E is a matching in G. Foragiven set of connection requests, to maximize network throughput, we should find a maximum matching in the request graph. In the example aove, not all left side vertices can e matched ecause there are seven vertices on the left (connection requests) while there are only six vertices on the right (availale wavelength channels). The maximum matchings for oth types of conversions are shown in Figure, and in this case they are identical. The est known algorithm for finding maximum matching in an aritrary ipartite graph was given in [], and has time complexity O(n (m + n)), wheren and m are the numer of vertices and edges in the ipartite graph, respectively. If we directly adopt this algorithm in our scheduling algorithm, the time complexity would e as high as O(N k d), since the left side vertices in a request graph alone could e as large as Nk and each left side vertex is adjacent to d right side vertices. However, faster algorithms are required for scheduling in WDM optical interconnects as the decision has to e made in real-time within a time slot, which is in the order of μs []. In the rest of the paper, we will show that the request graph for limited range wavelength conversion exhiits some nice properties so that faster algorithms are possile. We will present two fast scheduling algorithms with time complexity O(k) and O(dk), respectively, for two types of wavelength conversion- 0 a a a a a a 6 (a) 0 0 a a a a a a 6 Fig.. Maximum matchings in two request graphs in Figure. (a) Circularsymmetrical. () Non-circular-symmetrical. s, where k is the numer of wavelengths and d (» k) is the conversion degree. Note that the time complexity of our algorithms is independent of the interconnect size N, and can e easily implemented in hardware also. Because finding the largest group of contention-free connection requests is equivalent to finding a maximum matching in the request graph, in the following we will descrie these algorithms in the form of finding a maximum matching in ipartite graphs. We first consider the case that all output wavelength channels are availale at the eginning of time slots in Section III and Section IV. Then in Section V, we show that our algorithms can e easily extended to the case that some of the output wavelength channels are occupied. III. SCHEDULING ALGORITHM FOR NON-CIRCULAR SYMMETRICAL WAVELENGTH CONVERSION First we consider non-circular symmetrical wavelength conversion. In this case, the request graph is a convex ipartite graph as defined in []. A ipartite graph G is convex if there exists an ordering» of the right side vertices B such that for any a A and distinct ; B (with» ), edge a E and a E ) a E for any B and»»,wheree is the set of edges in G. Inotherwords, if we use B(a) to denote the set of vertices in B adjacent to a, B(a) is an interval for any a A in this ordering. With noncircular symmetrical wavelength conversion, a request graph is convex ecause for every left side node a, B(a) is the set of wavelengths that wavelength W (a) may e converted to, and they form an interval if the right side vertices are ordered according to the wavelength indexes. For example, in Figure (), if we check left side vertex a, the set of vertices adjacent to it is B(a )=f 0 ; ; g, which can e represented y interval [0; ]. And so do other left side vertices. Simpler algorithms exist for finding a maximum matching in a convex ipartite graph. The algorithm shown in Tale was descried in []. The input to this algorithm is: () The left side vertex set A and the right side vertex set B; () For each left side vertex a, the set of vertices adjacent to it denoted y interval [BEGIN (a);end(a)]. We call BEGIN (a) and END(a) the BEGIN value and END value of a, respectively. The output of the algorithm is array MATCH[]. MATCH[i] = j means that the i th right side vertex is matched to the j th left side vertex. MATCH[i] =Λ if the i th right side vertex is not matched to any left side vertex. We can see that in this algorithm, the i th vertex in B is () /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

5 TABLE GLOVER S ALGORITHM FOR FINDING A MAXIMUM MATCHING IN A CONVEX BIPARTITE GRAPH Glover s Algorithm for i := to j B j do U:=fk : k A; (i; k) Eg if U = ; MATCH[i] :=Λ else j := element in U with minimum END value MATCH[i] :=j delete j from A end if end for TABLE FIRSTAVAILABLE ALGORITHM FOR FINDING A MAXIMUM MATCHING IN A REQUEST GRAPH OF NON-CIRCULAR SYMMETRICAL WAVELENGTH CONVERSION First Availale Algorithm for i := to k do let a j e the first vertex in A adjacent to i if no such a j exists MATCH[i] :=Λ else MATCH[i] :=j delete a j from A end if end for matched to an adjacent vertex in A whose interval ends closest to it. The algorithm has time complexity O(jEj) =O(Nkd), and the time consuming part is the formation of set U and finding the smallest-ending vertex j. Glover s algorithm has a lower time complexity than the algorithm for general ipartite graphs in [] since the ipartite graph is convex. Our request graph is a special type of convex ipartite graphs. Hence Glover s algorithm can e further simplified into First Availale Algorithm as descried in Tale. In this algorithm, we match right side vertex i with the first left side vertex that is adjacent to it. In other words, we choose the top edge in the request graph and add it to the matching in each iteration. Notice that here, for a right side vertex, instead of finding all the adjacent left side vertices and then selecting among them the one with the smallest END value, all we need to do is to find the first adjacent left side vertex. We have the following result concerning this algorithm. Theorem : First Availale Algorithm finds a maximum matching in a request graph of non-circular symmetrical wavelength conversion. Proof. In a request graph of non-circular symmetrical wavelength conversion, we notice that for left side vertices a j and a l,ifj» l, thenbegin(j)» BEGIN (l) and END(j)» END(l). If we use Glover s Algorithm to find the maximum matching for the request graph, in each iteration, when we are searching a matching for the i th right side vertex, we can simply match it to the vertex with smallest index on the left side that is adjacent to it. This is ecause that the END value of any other vertex with larger index cannot e smaller than it. Thus, Glover s Algorithm can e simplified to First Availale Algorithm when applied to a request graph of non-circular symmetrical wavelength conversion. Now we discuss the implementation of this algorithm. From Tale, we should find the first adjacent vertex to a right side vertex in each step. Since the packets on the same wavelength are identical for the purpose of maximizing the matching size, we need only to find the first input wavelength that has at least one packet and can e converted to current output wavelength. After that if there are more than one packets on this input wavelength, to ensure fairness, a random selecting or a round-roin scheduling procedure should e adopted as suggested in [7] [8]. All this can e implemented in hardware and the execution time of each step would e a constant. Thus, the time complexity of this algorithm is O(k). Note that the time complexity is independent of the interconnect size and conversion degree. Of course, if less complex hardware is used it may not e the case. However, since timing is critical here, the method descried aove is preferred. IV. SCHEDULING ALGORITHM FOR CIRCULAR SYMMETRICAL WAVELENGTH CONVERSION When wavelength conversion is circular symmetrical, the request graph is no longer a convex ipartite graph, ecause that the left side vertices near one end will have links to the right side vertices on the other end (e.g., edge and a 6 0 in Figure ), and the adjacency set is not an interval. This makes the prolem more complicated. However, in the following discussion we show that we can still apply the First Availale Algorithm y reaking the request graph. A. Breaking the request graph First, we introduce some useful definitions in our discussion. Definition : Edge a j v crosses a i u when Case. W (j) 6= W (i).. W (j) [u f +;W(i) ] and v [u +;W(j)+f ].. W (j) [W (i)+;u +e] and v [W (j) e; u ] Case. W (j) =W (i).. j<iand v [u +;W(j) +f ].. j>iand v [W (j) e; u ] For example, in Figure () edges and a 0 cross each other, edge a crosses a, ut edge and a, though intersecting with each other in the figure, are not a pair of crossing edges. Definition : For request graph G, leta i and u e two vertices such that a i u E. LetG 0 e the sugraph of G otained y removing vertices a i and u, all the edges incident to them, and all edges that cross edge a i u in G. We call G 0 the reduced graph of G. By doing this, we say we reak G at a i u. Edge a i u is called the reaking edge /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

6 We have the following result. Lemma : There exists a maximum matching in the request graph where there are no crossing edges. Proof. We prove this lemma y showing that every pair of crossing edges in the maximum matching can e replaced y two non-crossing edges. Let G e the request graph and E e the edge set of G. Suppose edge a i u and a j v are in a maximum matching M and cross each other. If W (j) =W (i), when edge a i u E and a j v E, wehavea i v E and a j u E. Edge a i v and a j u are not in M, ecause if one of them is then there would e two edges sharing one vertex in M. They also do not cross each other. In Case. when j<i, from the definition we know that v [u +;W(j)+f ], thus [v +;W(j) +f ] is a suset of [u +;W(j) +f ]. But u = [u +;W(j) +f ], thenu = [v +;W(j) +f ]. Thus, y definition a i v and a j u do not cross each other. A similar proof can e given for Case.. Therefore, we can remove a i u and a j v from M and add a i v and a j u to M. The new matching is still a maximum matching. For W (j) 6= W (i), we prove Case. when W (j) [u f +;W(i) ] and v [u +;W(j) +f ]. The proof for Case. is similar. Since a i u E, thesetof left side vertices with wavelengths [u f;w(i)] are adjacent to u. Also, since W (j) [u f +;W(i) ], a j u E. We now show that a i v E. The adjacency set of wavelength W (i) is I = [W (i) e; W (i) +f ]. If (W (j)+ff) mod k = W (i), the adjacency set of wavelength W (j) is J = [W (i) e ff;w(i) +f ff]. By definition, v [u +;W(j) +f ] = [u +;W(i) +f ff] ρ [u; W (i) +f ]. Sinceu I and [u; W (i) +f ] I, it follows that v I, or,a i v E. Edge a i v does not cross a j u ecause W (j) [v f +;W(i) ] ut u= [v +;W(j)+f ]. Hence, for W (j) 6= W (i) the pair of crossing edges can also e replaced y two non-crossing edges not in M. We can apply the same procedure to every pair of crossing edges in M and otain a new maximum matching without crossing edges. As an example, in Figure (), edge and a 0 are a pair of crossing edges. If they are in a matching, we can replace them with edge 0 and a. Similarly, edge a and a are a pair of crossing edges. If they are in a matching, we can replace them with edge a and a. For left side vertex a j, the adjacency set in G can e represented as [W (j) e; W (j) +f ]. When G is roken at edge a i u, a j s adjacency set in G 0 will e in one of the following three forms: [W (j) e; u ], [u +;W(j) +f ], or [W (j) e; W (j)+f]. This is ecause when we reak G at edge a i u,ifw(j) [u f +;W(i) ], the links to wavelengths [u +;W(j) +f ] would have een deleted and the adjacency set in G 0 is [W (j) e; u ]. IfW (j) [W (i)+;u +e], the links to wavelengths [W (j) e; u ] would have een deleted and the adjacency set in G 0 is [u +;W(j) +f ]. If W (j) = W (i), whenj > i, the adjacency set in G 0 is [u +;W(i) +f ]; whenj < i, the adjacency set in G 0 is [W (i) e; u ]. OtherwiseW (j) = [u f;u + e], i.e., a j is not adjacent to u in G, and the adjacency set in G 0 is still 0 a a a a 6 a (a) 0 a a Fig.. Breaking at edge a in Figure (a) (a) Deleting vertex a and and all the incident edges and crossing edges. () Reordering a and to the top. [W (j) e; W (j) +f ]. Now when G is roken, the order of vertices in G 0 can e represented as: 0 a a 6 a () ;a ;::: ;a i ;a i+ ::: ;a jaj ; and 0 ; ;::: ; u ; u+ ;::: ; jbj We now left shift these vertices until a i+ and u+ are at the left end: a i+ ;a i+ ;::: ;a jaj ; ;a ;::: ;a i ; and u+ ; u+ ;::: ; jbj ; 0 ; ;::: ; u : In the aove shifting, only the ordering of vertices is changed, ut no edges are deleted or added. It is not difficult to verify that all three types of adjacency sets are intervals in the new ordering: [u +;W(j) +f ] represents the left most W (j) + f u vertices, [W (j) e; u ] represents the right most u W (j) +e vertices, and [W (j) e; W (j) +f ] represents d consecutive vertices in the middle. Hence G 0 in is a convex ipartite graph. Moreover, for vertices a j and a l,ifvertexa j appears on the left of a l in the new ordering, the first and the last vertex adjacent to a j will also e on the left of the first and the last vertex adjacent to a l, respectively. This means that in the new vertex ordering of G 0, for left side vertices a j and a l,ifj» l, thenbegin (j)» BEGIN(l), END(j)» END(l). Thus we have: Lemma : The vertices in the reduced graph G 0 can e ordered in a way such that the First Availale Algorithm can e used to find the maximum matching for G 0. Figure shows an example of reaking the request graph in Figure (a) at edge a. Now the question is whether a maximum matching in G 0 along with the reaking edge is a maximum matching of G. The following lemma shows this is true if the reaking edge is in a no-crossing-edge maximum matching of G. Lemma : If edge a i u is in a no-crossing-edge maximum matching M of request graph G, then edge a i u along with a maximum matching of graph G 0 which is otained y reaking G at edge a i u is a maximum matching of G. Proof. If edge a i u is in a no-crossing-edge maximum matching M of G, then edges in L = M nfa i u g are all in G 0. It follows that L is a matching of G 0. Itisalsotruethatany matching of G 0 along with edge a i u is a matching of G. Now L is a maximum matching of G 0, since if it is not, the maximum /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

7 TABLE BREAK AND FIRST AVAILABLE ALGORITHM FOR FINDING A MAXIMUM MATCHING IN A REQUEST GRAPH OF CIRCULAR SYMMETRICAL WAVELENGTH CONVERSION Break and First Availale Algorithm aritrary choose one left side vertex a i do for all right side vertex u adjacent to a i egin reak G at edge a i u, let the reduced graph e G u. apply First Availale Algorithm to G u to find a maximum matching M u. end let M w e the matching with max cardinality among all M u return M w [ a i u matching of G 0 written as L 0 contains more edges than L. Then L 0 [fa i u g which is also a matching of G contains more edges than M. This contradicts with the fact that M is a maximum matching. Therefore, any maximum matching of G 0 contains the same numer of edges as L. Hence, any maximum matching of G 0 together with edge a i u contains the same numer of edges as M. The lemma follows. Finding an edge in G that is in a no-crossing-edge maximum matching is not trivial. Examples can e found that none of the d edges incident to a left side vertex can e assumed to e in a no-crossing-edge maximum matching. However, we can prove that there must e at least one such an edge among the d edges. Lemma : For any left side vertex a i in the request graph G, at least one of the d edges incident to it is in some no-crossingedge maximum matching. Proof. We say a i is saturated in matching M if there is an edge in M incident to a i. It is ovious that there must e a maximum matching in which a i is saturated. Since given any maximum matching M 0,ifa i is saturated, the claim is true. If a i is not saturated in M 0, all the right side vertices adjacent to a i must e saturated. Now we can aritrarily match a i to one of them, say, u, and remove the edge in M 0 incident to u,by doing this we otain a new maximum matching in which a i is saturated. Now suppose M 00 is a maximum matching in which a i is saturated. If M 00 is already a no-crossing-edge matching, then M 00 is the maximum matching we need. If there are crossing edges in M 00, we can apply the procedure descried in Lemma to otain a no-crossing-edge maximum matching M. Notice that any vertex saturated in M 00 is still saturated in M. The lemma follows. From this lemma we know that if we try all d edges, we can otain a maximum matching. B. The Scheduling Algorithm Now we are in the position to present the Break and First Availale Algorithm as descried in Tale. By comining Lemmas, and, we can otain the following result concerning the Break and First Availale Algorithm. Theorem : The Break and First Availale Algorithm gives the maximum matching of a request graph of circular symmetrical wavelength conversion. The time complexity of the Break and First Availale Algorithm is O(dk), ecause we need to try all d reduced graphs, and each reduced graph has k right side vertices. Again note that the time complexity is independent of network size N. We can also implement this algorithm in parallel and time complexity could e reduced to O(k), ut we then need d units of hardware. If the conversion degree d is large, the time complexity or the hardware cost will e high. However, as we mentioned in Section I, in WDM networks, it has een shown that with very small conversion degree, the network performance can e close to the case of full conversion (d = k), []. Most of the proposed WDM optical interconnects with limited range wavelength conversion have a conversion degree of d = or d =. Thus, Break and First Availale Algorithm is suitale for hardware implementation in this case. C. Discussions on the Approximation Algorithm In the Break and First Availale Algorithm we try all d reduced graphs ecause we do not know efore hand which edge elongs to a no-crossing-edge maximum matching. In some applications, if the speed of the scheduling algorithm is of more concern than achieving the maximum network throughput, we can trade-off the time complexity of the algorithm with the network throughput (the size of the matching). Now we discuss such an approximation algorithm. The question is: if we want to try only one reduced graph to save time or hardware cost, which reduced graph we should choose such that the matching we otain will still e close to a maximum matching. Lemma : If edges a j v and a l w crosses edge a i u and W (j) [W (i)+;u +e], W (l) [u f +;W(i) ], then edge a j v and a l w cross each other. Proof. We have v [W (j) e; u ] and w [u +;W(l)+ f ]. Hence v [W (j) e; w ]. IfW (j) [W (l) +;w +e], edge a j v and a l w cross each other. W (j) [W (i) + ;u +e] and W (l) [u f +;W(i) ] imply that W (j) [W (l)+;u +e]. Similarly, since w [u +;W(l)+f ], we have W (j) [W (l) +;w +e]. Now the d right side vertices adjacent to a i are fw (i) e; W (i) e +;::: ;W(i);::: ;W(i) +fg. If edge a i u E, thenu is in this set. Suppose u is the ffi(u) th element in the set when counted from the left to the right. For example, if u = (W (i) e) mod k, thenffi(u) = ; if u = (W (i) e +) mod k, thenffi(u) =. We have the following lemma. Lemma 6: Edge a i u may cross up to max fffi(u) ;d ffi(u)g edges in a no-crossing-edge maximum matching. Proof. Given a no-crossing-edge maximum matching M, if a i u M, it does no cross any edge in M. If a i u = M, we first consider the requests coming in the wavelength range [u f +;W(i)]. The set of right side vertices adjacent to them and that may constitute a crossing edge of a i u can e represented as [u +;W(i) +f ]. There are totally d ffi(u) elements in this set. It follows that in any matching there are at most this numer of edges crossing a i u for the left side /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)

8 vertices in the wavelength range [u f +;W(i)]. By a similar argument, there are at most ffi(u) edges crossing a i u with left side vertices in the wavelength range [W (i);u +e]. From Lemma, any edge in one group crosses all the edges in the other group. Thus, if the matching has no crossing edges, only one group may e in that matching. Hence, the maximum numer of edges crossing a i u is the larger of the two. Theorem : By reaking the request graph at edge a i u, the difference etween the matching we find and a maximum matching cannot e greater than max fffi(u) ;d ffi(u)g, where u is the ffi(u) th vertex adjacent to a i when counting from the end of the minus side. Proof. Given any no-crossing-edge maximum matching M, y reaking the request graph at edge edge a i u, G 0 contains at least j M j max fffi(u) ;d ffi(u)g edges in M. These edges are also a matching in G 0. Hence the cardinality of the maximum matching of G 0 cannot e less that that. Thus y reaking the request graph at edge a i u we find a matching with at least j M j max fffi(u) ;d ffi(u)g edges. Corollary : The upper ound on the difference etween a maximum matching of G and the matching given y reaking G at edge a i u achieves the smallest value (d )= when ffi(u) =(d +)=. From this Corollary, if e = f, ffi(u) =(d +)= means we should choose the shortest edge. For the case of d =(e = f =), y reaking the request graph at the shortest edge, the difference etween our matching and a maximum matching is at most. For the case of d =(e = f =), the difference is at most. V. EXTENSIONS In the previous discussions we consider the case that all k output channels 0 ; ;::: ; k are availale at the time of scheduling. In this case, in the request graph there are always k vertices on the right side of a request graph. However, if the duration of connections is more than one time slot it may occur that not all of the output channels are availale at the time of scheduling, since some output channels may still e occupied y previously arrived connection requests. If the existing connections can e distured, i.e., e reassigned to a different output channel if needed, the algorithms discussed in the previous section can still e used. If it is not the case, which is true in optical urst switching, we can redraw the request graph y removing the right side vertices representing those occupied wavelength channels and all the edges incident to them. It is not difficult to see that we can still apply the algorithms introduced previously to these modified request graphs to otain the maximum matching. VI. CONCLUSIONS In this paper we have studied scheduling algorithms to resolve output contentions in WDM optical interconnects with wavelength conversion aility. We presented the solutions for the general case of limited range wavelength conversion, including the full range wavelength conversion. We have introduced the request graph and showed that the largest group of contention-free connection requests can e found y finding a maximum matching in the request graph. We studied two types of limited range wavelength conversions, circular symmetrical and non circular symmetrical. We showed that the request graphs under these types of wavelength conversions have some nice properties and presented fast, simple optimal algorithms which always find a maximum matching to achieve maximum network throughput. Compared with the general algorithm for finding a maximum matching in ipartite graphs with time complexity O(N k d) [], the distriuted algorithms we gave have time complexity of O(k) and O(dk) (independent of interconnect size N) for non-circular symmetrical and circular symmetrical wavelength conversion, respectively, where k is the numer of wavelengths and d is the conversion degree. In addition, our algorithms can e easily implemented in hardware, and used for time slotted WDM optical interconnects where connections hold for different numer of time slots. Interesting future work may include incorporating different QoS requirements, such as different priorities among connection requests, in the scheduling algorithm. REFERENCES [] J. Hopcroft and R. Karp An n algorithm for maximum matchings in ipartite graph, SIAM J. Comput., ():-, Dec. 97. [] F. Glover Maximum matching in convex ipartite graph, Naval Res. Logist. Quart.,, pp. -6, 967. [] W. Lipski Jr and F.P. Preparata Algorithms for maximum matchings in ipartite graphs, Naval Res. Logist. Quart.,, pp. -6, 98. [] B. Mukherjee, WDM optical communication networks:progress and challenges, JSAC, vol. 8, no. 0, pp. 80-8, 000. [] D. K. Hunter, M. C. Chia and I. Andonovic Buffering in optical packet switches, J. Lightwave Tech., vol. 6 no., pp , 998. [6] M. Kovacevic and A. Acampora, Benefits of wavelength translation in all-optical clear-channel networks, JSAC, vol., no., pp , 996. [7] N. McKeown, P. Varaiya, and J. Warland, Scheduling cells in an inputqueued switch, IEEE Eletron. Lett., pp. 7-7, 99. [8] N. McKeown, The islip scheduling algorithm input-queued switch, IEEE/ACM Trans. Networking, vol. 7, pp. 88-0, 999. [9] H. J. Chao, C. H. Lam and E. Oki, Broadand Packet Switching Technologies, Wiley-Interscience, 00. [0] Walter J. Goralski, Optical Networking and WDM, McGraw-Hill, 00. [] T. Tripathi and K. N. Sivarajan, Computing approximate locking proailities in wavelength routed all-optical networks with limited-range wavelength conversion, JSAC, vol. 8, pp. 9, 000. [] L.Xu, H. G. Perros and G. Rouskas, Techniques for optical packet switching and optical urst switching, IEEE communications Magazine, pp. 6-, Jan. 00. [] R. Ramaswami and G. Sasaki, Multiwavelength optical networks with limited wavelength conversion, IEEE/ACM Trans. Networking, vol. 6, pp. 7-7, 998. [] X. Qin and Y. Yang, Nonlocking WDM switching networks with full and limited wavelength conversion, IEEE Trans. Comm., vol 0, no., 00. [] Y. Yang and J. Wang, WDM optical interconnect architecture under two connection models, Proc. of IEEE Hot Interconnects 0, Symposium on High Performance Interconnects, pp. 6-, Palo Alto, CA, 00. [6] Y. Yang, J. Wang and C. Qiao Nonlocking WDM multicast switching networks, IEEE Trans. Parallel and Distriuted Systems, vol., no., pp. 7-87, 000. [7] M.S. Borella and B. Mukherjee Efficient scheduling of nonuniform packet traffic in a WDM/TDM local lightwave network with aritrary transceiver tuning latencies, JSAC, vol., no., pp. 9-9, 996. [8] G.N. Rouskas and V. Sivaraman Packet scheduling in roadcast WDM networks with aritrary transceiver tuning latencies, IEEE/ACM Trans. Networking, vol., no., pp. 9-70, /0/$7.00 (C) 00 IEEE Proceedings of the International Parallel and Distriuted Processing Symposium (IPDPS 0)