Ecient Routing in Optical Networks. Alok Aggarwal Amotz Bar-Noy Don Coppersmith. Rajiv Ramaswami Baruch Schieber Madhu Sudan. IBM { Research Division

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1 Ecient Routing in Optical Networks Alok Aggarwal Amotz Bar-Noy Don Coppersmith Rajiv Ramaswami Baruch Schieber Madhu Sudan IBM { Research Division T. J. Watson Research Center Yorktown Heights, NY Abstract This paper studies the problems of dedicating routes to connections in optical networks. In optical networks, the vast bandwidth available in an optical ber is utilized by partitioning it into several channels, each at a dierent optical wavelength. A connection between two nodes is assigned a specic wavelength, with the constraint that no two connections sharing a link in the network can be assigned the same wavelength. This paper classies several models related to optical networks and presents optimal or near-optimal algorithms for permutation routing in many of these models. This work was supported in part by a grant No. MDA C-0075 from ARPA.

2 1. Introduction Fiber-optic networking technology using wavelength division multiplexing (WDM) oers the potential of building large wide-area networks capable of supporting thousands of nodes and providing capacities of the order of gigabits-per-second to each node in the network [Gre92, Ram93, CNW90]. In WDM optical networks, the vast bandwidth available in optical ber is utilized by partitioning it into several channels, each at a dierent optical wavelength. Each wavelength can carry data modulated at bit rates of several gigabits per second. In general, such a network consists of routing nodes interconnected by point-to-point ber-optic links (Figure 1). Each link can support a certain number of wavelengths. The routing nodes in the network are capable of routing a wavelength coming in on an input port to one or more output ports, independent of the other wavelengths. However, the same wavelength on two input ports cannot be routed to a common output port. The rst class of networks that we consider are non-recongurable, or switchless; i.e., the routing patterns at each of these routing nodes is xed. Such a routing node is shown in Figure 2. These networks are of practical importance because the entire network can be constructed out of passive (unpowered) optical components and hence made reliable as well as easy to operate, with all the control being done outside the network. Another motivation for studying such networks is that they form important components in the construction of recongurable networks (see Section 3). B λ 1 2 A 1 Fiber Link λ 2 3 λ 1 C E End Node 5 Routing Node λ 1 4 D Figure 1: A WDM network consisting of routing nodes interconnected by point-to-point ber-optic links. Some of the routing nodes have end-nodes attached to them that form the sources and destinations for network trac. The second class of networks, which we call recongurable networks, have optical switches at the routing nodes. By reconguring the switches, the routing pattern at a routing node can be changed. Optical switches will be required to build large networks because the switchless network requires a large number of wavelengths to support even simple trac patterns (as will be seen later in this 1

3 λ 1 λ 2 λ1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 λ 1 λ 2 λ 1 λ 1 λ 2 λ λ 1 λ 2 2 Wavelength Demux Wavelength Mux Figure 2: Structure of a non-recongurable (switchless) routing node. λ 1 λ 2 λ 1 λ 1 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 Optical Switch λ 2 λ 1 λ 1 λ 2 λ 2 λ 2 λ Optical 1 λ 1 Switch λ 1 λ 2 λ 2 λ 2 λ λ 1 λ 2 2 Wavelength Demux Wavelength Mux Figure 3: Structure of a recongurable routing node. The node can switch each wavelength at its input ports independent of the other wavelengths. The switch can be recongured to allow dierent interconnection patterns. paper). A recongurable routing node is shown in Figure 3. Optical switches can be classied into two categories: An elementary, or wavelength-independent switch does not distinguish between dierent wavelengths. A generalized, or wavelength-selective switch is capable of switching each wavelength independent of the other wavelengths and is functionally equivalent to w elementary switches, where w is the number of wavelengths. Some of the routing nodes in the network have end nodes attached to them via ber-optic links. We refer to end nodes as nodes. Each end node has a tunable optical transmitter and a tunable optical receiver. The transmitter can be tuned to transmit at any one of the available wavelengths and the receiver can be tuned to receive on any one of the available wavelengths. We are interested in the problem of setting up connections between dierent node pairs and would like to determine the number of wavelengths and switches required to support dierent trac patterns in these networks. Each connection is assigned a wavelength. The constraint imposed by the network is that there can be at most one connection using a given wavelength on any link. In switchless networks, once the routing pattern is set, the only choice remaining is in selecting the wavelength at which each node transmits and the wavelength at which it receives. In networks with 2

4 switches, additional degrees of freedom are obtained by changing the settings of the switches Preliminaries A permutation network is a network that can successfully route all sets of connections where each transmitter is connected to a single receiver and each receiver to a single transmitter. A non-blocking network (NBN) is a network that allows communication routes between pairs of transmitters and receivers to be connected and terminated dynamically. It handles two kinds of requests: connection requests and termination requests. A connection request species a pair of (transmitter, receiver) that has to be connected. It is assumed that both the transmitter and the receiver are idle when the request is initiated. A termination request species a pair (transmitter, receiver) that are currently connected and terminates this connection. Following Benes [Ben62], we distinguish between two types of non-blocking networks: rearrangeably NBNs, where existing connections can be rerouted to accommodate a new request, and wide-sense NBNs, where existing connections cannot be rerouted to accommodate a new request. In o-line routing, all the requests to be routed are known in advance. In on-line routing, future requests are not known in advance. Note that this distinction is relevant only for a wide-sense NBN (and also for a rearrangeably NBN if there is a cost associated with rearranging). An oblivious routing scheme always uses the same wavelength to satisfy a given connection request, regardless of the other connections in the network. Oblivious schemes are clearly on-line schemes. In partially oblivious routing, the wavelength that can be used to satisfy a connection request must be chosen from a subset of the available wavelengths in the network. The congestion of a routing algorithm is the maximum number of paths that go over a single edge in the network. The dilation of a routing algorithm is the maximum number of edges in a path used by the routing algorithm Previous Work The routing problem has been studied by Barry and Humblet [BH92, BH93a, BH93b], Pankaj [Pan92], and Pieris and Sasaki [PS93]. Barry and Humblet [BH92] derived an information-theoretic lower bound on the number of wavelengths required to support a given number of trac states in networks with and without switches. For example, permutation routing in a switchless network requires ( p n) wavelengths, where n is the number of nodes in the network. They also showed [BH92, BH93a] that oblivious permutation routing requires at least bn=2c + 1 wavelengths and could be done using dn=2e + 2 wavelengths. In a recongurable network with w available wavelengths, Barry and Humblet [BH93a] showed that the number of elementary 22 switches required to support permutation routing is (n log(n=w 2 )). All logarithms in this paper are to the base 2. 3

5 For the special case in which the transmitters are xed-tuned and the receivers are tunable, Pieris and Sasaki [PS93] showed that the number of elementary 2 2 switches required for permutation routing is (n log(n=w)), and constructed such a network using O(n log(n=w)) elementary switches. Pankaj [Pan92] obtained bounds on the number of wavelengths required for permutation routing in certain network topologies using generalized switches. His network model assumed a generalized switch at each routing node. For this model, Pankaj showed that (log n) wavelengths are required for permutation routing. He also showed that rearrangeably non-blocking permutation routing can be done with O(log 2 n) wavelengths and wide-sense non-blocking permutation routing with O(log 3 n) wavelengths in popular interconnection networks such as the shue exchange, de Bruijn, and hypercube networks Contributions of This Work We present almost tight bounds for most of the problems considered in the earlier papers. Our results are summarized in Table 1. Our rst set of results are for switchless networks. We prove that oblivious permutation routing in such networks can be done using dn=2e + 2 wavelengths, and prove that this is optimal; i.e., oblivious permutation routing in such networks requires dn=2e + 2 wavelengths. The upper bound has been obtained independently by Barry and Humblet [BH92, BH93a]. They have also obtained a lower bound that is lower than ours by at least 1. We demonstrate the existence of a switchless permutation network using O( p n log n) wavelengths. This result has been also obtained independently by Barry and Humblet [BH93b]. Both results give networks which are non-blocking in the wide-sense. Our result is accompanied by a polynomial time algorithm which dictates the tuning of the transmitters and receivers as the requests come online. This seems to be one of the strong contribution of our construction. The non-constructive aspect of the above result may be viewed as a drawback. We complement this result with a constructive version which is only slightly weaker. Dene (n) = 2 (log n)0:8+o(1). We show how to construct a switchless permutation network using O ( p n(n)) wavelengths. y We also provide non-trivial upper bounds for partially oblivious networks. For recongurable networks, we show the existence of a wide-sense non-blockingnetwork using O(n log nlog w ) switches and construct a wide-sense non-blocking network using O n log n(w) w 2 w 2 switches. Clearly all of these results apply also for rearrangeably non-blocking networks. For networks with generalized switches, we show that a permutation network using w wavelengths requires ( n log n ) generalized switches. Thus we can construct a permutation network with w n generalized switches using only O(log n) wavelengths, an improvement over Pankaj's results. We also derive an upper bound on the number of wavelengths required for any routing scheme in terms of congestion and dilation for the given routing and the given network. We show that p y Unfortunately, n(n) < n only for n >

6 Non-Recongurable (Switchless) Networks Number of Wavelengths Lower Bound Upper Bounds Previous Ours Oblivious Routing n d n e + 2 [BH92] d n e Non-oblivious (existence) ( p n) [BH92] O( p n log n) O( p n log n) [BH93b] Non-oblivious (constructive) ( p n) [BH92] O( p n(n) Partially oblivious (k 3) ( p n) [BH92] O n k+1 k 2k?1 k 2k?1 Recongurable Networks Number of Wavelengths Lower Bound Upper Bounds Previous Ours O(log 2 n) [Pan92] (rearrangeable) O(n) generalized switches (log n) [Pan92] O(log 3 n) [Pan92] O(log n) [c minfd; p mg] (wide-sense) O[c minfd; p mg] Number of Switches (w = number of wavelengths) Lower Bound Upper Bounds Previous Ours Elementary switches (existence)? n log n w [BH92] O n log n log w 2 w 2 Elementary switches (constructive)? n log n w [BH92] O n log n(w) 2 w 2 Generalized switches O n log n w Table 1: Summary of results. The results hold for both rearrangeable and wide-sense non-blocking cases unless specied otherwise. Notations used in the table: The number of nodes is n, c denotes congestion, d denotes dilation, m is the number of edges in the network, k denotes the number of wavelengths available for a connection, and (n) = 2 (log n)0:8+o(1). n log n w 5

7 there exists a class of networks for which this bound cannot be improved. The rest of the paper is organized as follows. Section 2 deals with non-recongurable (switchless) networks and Section 3 with recongurable networks (with switches). There are several open problems remaining to be solved; Section 4 gives a few such problems. 2. Non-Recongurable Optical Networks In this section we consider non-recongurable (or switchless) optical networks. We use = f 1 ; : : :; w g to denote the set of available wavelengths. The network is modeled as a bipartite multigraph z G(T; R; E) and a labelling function ` : E!, where T is the set of transmitters and R is the set of receivers and for an edge e from a transmitter t to a receiver r, the label `(e) denotes the wavelength which t can use to establish a connection to r. Since t may transmit to r using many possible wavelengths, there can be multiple edges between t and r. Thus two or more edges between a transmitter t and a receiver r will have dierent labels. A \tuning conguration" of the receivers and transmitters is formally described by a function W : T [ R!. The tuning con- guration is interpreted as follows. Every transmitter t transmits on the wavelength W (t). Every receiver r receives on the wavelength W (r). If e connects transmitter t 2 T to receiver r 2 R, then whenever t transmits using wavelength `(e), receiver r may receive this information only if it tunes to this wavelength. Moreover, since the network has no switches, all the receivers connected to t with edges labelled `(e) receive t's message if they tune to this wavelength. Consequently, if a receiver r is tuned to a wavelength, then only one transmitter that is connected to r by an edge labelled may use this wavelength. Note that once the graph G and the labeling are determined, the only choice remaining is in the tuning conguration of the transmitters and receivers. The assumption that G is bipartite is made only to make the presentation clearer. We can achieve the same results for networks where some of the nodes are both transmitters and receivers. In this paper we consider the special case of jt j = jrj = n. However, most of the results can be extended to the case where jt j 6= jrj. We consider the problem of constructing a non-blocking network using a minimum number of wavelengths Rearrangeably Non-Blocking Networks Barry and Humblet [BH92] proved that any non-recongurable rearrangeable NBN requires at least (1 + (n)) p n=e wavelengths, where e is the base of the natural logarithm, and (n) goes to zero faster than (ln p n)= p n. We show that there exists a rearrangeably NBN that uses O( p n log n) wavelengths. We show how to construct a network that uses p n(n) wavelengths (recall that (n) = 2 (log n)0:8+o(1) ). z A multigraph is a graph with multiple edges allowed between nodes. 6

8 In our upper bounds, the network has the following structure. The transmitter set T is partitioned into b blocks T 0 ; : : :; T b?1 each of cardinality either dn=be or bn=bc, where b is a parameter to be xed later. (For clarity of exposition we from now on omit the d:e b:c operators.) The receiver set R is partitioned k times, with k to be xed later. Each partition 1 i k partitions R into b blocks R i 0; : : :; R i b?1. (The cardinality of each such block may vary.) Our construction will use w = b k wavelengths, denoted i;j, for 1 i k and 0 j b? 1. The edges of the network are labelled as follows: for 1 i k, 0 j b? 1, and 0 a b? 1, all the transmitters in T a are connected to all the receivers in R i (a+j)mod b by edges labelled i;j. The construction above has the following two properties: G1 Transmitters in each block are identically connected to all of the receivers. G2 For any wavelength, if transmitters t 1 and t 2 belong to dierent blocks, then the set of receivers connected to t 1 by edges labelled is disjoint from the set of receivers connected to t 2 by edges labelled. To get a rearrangeably NBN, it is necessary and sucient to construct a network such that for any permutation = (1); : : :; (n), there is a way to tune the transmitters and receivers such that the connection requests (t; (t)), for 1 t n, are satised. A given tuning satises these connection requests if the following two properties are satised for all 1 t n: (i) Both t and (t) are tuned to the same wavelength, and there exists an edge e connecting t to (t) with `(e) =. (ii) For all transmitters t 0 6= t such that there exists an edge e 0 labelled connecting t 0 to (t), t 0 is not tuned to. Consider a permutation = (1); : : :; (n) that is to be routed. Property [G2] of our network implies that we can tune the transmitters of each block independently from the transmitters of other blocks. This is because transmitters from dierent blocks do not interfere. Property [G1] implies that in order to route, for any block of transmitters T i, we have to use n=b dierent wavelengths. For this, the n=b destination receivers of the transmitters in T i have to belong to n=b dierent blocks. Note that these blocks may belong to dierent partitions. Given a network G, dene the bipartite graph H(S; B; F ), where S, the input set, corresponds to the set of receivers, and B, the output set, corresponds to the set of blocks of receivers. A node r 2 S is connected by an edge in F to v i j 2 B if and only if the corresponding receiver r belongs to the corresponding block R i j. It is easy to verify that the graph H has the following two properties. H1 The degree of each node in S is at most k. H2 For 1 i k, each node in S is connected to exactly one node from the set fv i 0; : : :; v i b?1g. Theorem 1: The network G(T; R; E) is non-blocking if the corresponding graph H(S; B; F ) has the following matching property: 7

9 H3 The subgraph induced by any subset of n=b receivers and their neighbors in B contains a matching of cardinality n=b. Proof: Consider a permutation. Recall that Property [G2] of the construction implies that transmitters of each block can be tuned independently from the transmitters of other blocks. Fix a block of transmitters T i. The destination receivers of the transmitters in T i may be any subset of n=b receivers. Thus for any subset of n=b receivers, the receivers have to belong to dierent blocks. By the denition of H, this translates to the matching property [H3]. 2 In the rest of this section we prove the existence of a graph H with properties [H1], [H2], and [H3] for b = p n= log n and k = 4 log n. Then, we show how to construct such a graph with b = 2 p 2n and k = 2(n). The results of [FFP88] imply the existence of a graph H having all three properties in which b = p n and k = O(log n). The following theorem improves this result by a factor of p log n. Theorem 2: There exists a graph H(S; B; F ) with properties [H1], [H2], and [H3] in which b = p n= log n and k = 4 log n. Proof: We construct H(S; B; F ) probabilistically as follows. Let jsj = n, and let B be partitioned into k blocks B 1 ; : : :; B k of cardinality b each. We let each vertex in S pick k neighbors { one in each B i independently and at random. We now analyze the probability that this graph has the matching property: i.e., any subset of up to n=b vertices from S is contained in some matching. By Hall's Theorem [Hal35], such a matching does not exist if and only if there exists a set A of a vertices from S of cardinality at most n=b, such that jn(a)j < jaj, where N(A) denotes the set of neighbors of A. For a xed set A S and sets A i B i (where a = jaj, a i = ja i j) such that j [ i A i j < a, the probability that N(A) [ i A i is at most Q k i=1( a i b )a. Thus, the probability that there exist A and A i 's of cardinality a and a i respectively such that jn(a)j < jaj is at most! ky n a i=1! b a! ai ne2 a k?2 a : a i b k k?1 b k?1 Thus if kb cn=b ca for some constant c 2 and k 4 log c n, then this probability goes to zero as n?(a). For a xed a, there are at most a k = n 4 log c a choices for the a i s. Since there are at most n=b choices for a, it follows that the probability that there exist a and a i 's such that this happens is bounded by o(1). Thus under the conditions k = 4 log n and b = p n= log n, we get that with a positive probability, H(S; B; F ) has the required three properties. 2 Theorem 3: A graph H(S; B; F ) with properties [H1], [H2], and [H3] in which b = 2 p 2n and k = 2(n) can be constructed. Proof: First, we dene a concentrator. 8

10 Denition: An (x; y; `)-concentrator with expansion is a network with x inputs and y outputs such that every set of t ` inputs expands to at least t outputs. The size of the network is the number of edges and the depth of the network is the length of the longest path from an input to an output. We use the following result from Wigderson and Zuckerman [WZ93]. Theorem 4 (WZ93): For all x, there are explicitly constructible (x; 2 p x; p x)-concentrators with expansion, depth 1 and size x (x). We now show how to apply Theorem 4. For our case, we set x = 2n and = 1 and get that there exists a bipartite graph H 0 (S 0 ; B 0 ; F 0 ), where js 0 j = 2n, B 0 = 2 p 2n, and jf 0 j = 2n(2n) = 2n(n) with the desired matching property. However, graph H 0 does not satisfy Properties [H1] and [H2]. We modify H 0 so that it satises these two properties. First, we consider a subgraph of H 0 which excludes all input nodes whose degree is more than two times the average degree in H 0. Specically, the degree of each input node in this subgraph is at most 2(n). Clearly, this subgraph still has the desired matching property. Because the size of the original graph H 0 is 2n (n), there are at least n input nodes in this subgraph. Next, we duplicate each output node 2(n) times and split the neighbors of each output node among the copies as follows. We number the edges outgoing from each input node with the numbers 1 to 2(n). Now, the rst copy will have as edges the subset of the edges of the original node numbered 1, the second will have the subset numbered 2, and so forth. It is easy to see that the resulting graph has all the three properties Wide-Sense Non-Blocking Networks In this subsection we apply the above results to wide-sense NBNs. Denition: A connection request is one-sided if it species only an input. It is satised by connecting the input to any of its neighboring outputs. A network H is wide-sense one-sided NBN if it can satisfy any sequence of one-sided connection and termination requests without rerouting. Denition: A network H is a-limited wide-sense one-sided NBN if it can satisfy any sequence of requests in which at most a transmitters are connected simultaneously. We show that to get a wide-sense NBN G, it is sucient to make the corresponding graph H n=b-limited wide-sense one-sided NBN. Theorem 5: The network G(T; R; E) is wide-sense NBN if the corresponding graph H(S; B; F ) is n=b-limited wide-sense one-sided NBN. Proof: Recall that Property [G2] of our construction implies that transmitters of each block can be tuned independently from the transmitters of other blocks. Fix a block of transmitters T i. The destination receivers of the transmitters in T i at any given time, may be any subset of at most n=b receivers. Thus, in order to satisfy any sequence of requests for this block in G, at any given time, all the receivers connected to transmitters in T i have to belong to dierent blocks. By the 9

11 denition of H, this translates to the property that H is n=b-limited wide-sense one-sided NBN. 2 The following theorem is from [FFP88]. Theorem 6 (FFP88): A network H(S; B; F ) is a-limited wide-sense one-sided NBN if every set X of inputs of cardinality at most 2a has at least 2jXj neighbors. We remark that by following the proof in [FFP88] we can actually prove that for our special case we may weaken the property, and consider only sets of cardinality at most a. We conclude that the graph H(S; B; F ) is n=b-limited wide-sense one-sided NBN if it has the following property. H4 Every set X of inputs of cardinality at most 2n=b has at least 2jXj neighbors. Note that Property [H4] is stronger than Property [H3]. Theorem 7: There exists a graph H(S; B; F ) with properties [H1], [H2], and [H4] in which b = p n= log n and k = 10 log n. Proof: The proof is the same as the proof of Theorem 2. The only dierence is in the value of the constants. 2 Theorem 8: A graph H(S; B; F ) with properties [H1], [H2], and [H4] in which b = 4 p n and k = 2(n) can be constructed. Proof: We follow the construction given in the proof of Theorem 3. We use an explicit construction of (4n; 8 p n; 2 p n)-concentrators with expansion 2, depth 1, and size 8n (n), and extract from it a graph with properties [H1], [H2], and [H4] in which b = 8 p n and k = 2(n). 2 There is one problem with our construction. Any algorithm which decides how to tune the transmitters and receivers appears not to be polynomial. Borrowing terminology from [FFP88] we have to maintain the maximum critical set of inputs. For this, it appears that after each request, we have to check all subsets of idle inputs. There are two ways to alleviate this problem, one that works for the o-line case and the other for the on-line case. Suppose now that future requests are not known in advance. In this case we show how to get a polynomial decision algorithm by strengthening the properties H has to satisfy. Note that previous research on non-blocking networks did not address the problem of designing a network that can be operated by a polynomial time algorithm. Theorem 9: A network H(S; B; F ) is a-limited wide-sense one-sided NBN and has a polynomial time decision algorithm if for every subset X of inputs of cardinality at most a, even after we arbitrarily erase half of the edges adjacent to each input in X, X has at least 2jXj neighbors. Proof: The algorithm for making the assignment decision maintains the following invariant: 10

12 Let A be the set of inputs for which connections requests are currently active; and let the set T (A) denote the set of outputs to which these inputs are connected. Then, for any subset S of the remaining inputs of cardinality at most a, the cardinality of the neighborhood of S outside of T (A) is at least jsj. In other words, Hall's condition is satised by all potential sets of inputs in the subgraph given by deleting A and T (A). Clearly, if this invariant is maintained the network is indeed a-limited wide-sense one-sided NBN. We show how to maintain the invariant by induction on the size of the request sequence. Initially, the invariant follows from the property of H stated in the theorem. To satisfy the requests we maintain a maximal set C of \critical" inputs; that is, inputs that are not currently connected which pose bottlenecks to the invariant. A set C poses a bottleneck if there exists a set of outputs C 0 of the same cardinality as C, such that more than half of the neighbors of each input in C is in C 0 [ T (A). The reason such a set is critical is that the properties of H do not guarantee anything on the size of the neighbourhood of C outside T (A) [ C 0. We also maintain a matching that matches each of the inputs in C to an output in C 0. Initially, C (and C 0 ) is the empty set. Actually, we maintain the following modied invariant (which is stronger than the original invariant). Let A be the set of inputs for which connections requests are currently active; and let the set T (A) denote the set of outputs to which these inputs are connected. Let C be a maximal set of \critical" inputs. Then, (i) there exists a set of outputs C 0, where jcj = jc 0 j, such that there is a matching that matches the each of the inputs in C to an output in C 0 ; and (ii) for any subset S of the remaining inputs of cardinality at most a, the cardinality of the neighborhood of S outside of T (A) [ C 0 is at least 2jSj. Again, initially, the modied invariant follows from the property of H stated in the theorem. We now show how to satisfy a new request in polynomial time while maintaining the modied invariant. Consider a connection request. Let v be a new input for which a connection is requested. We distinguish between two cases. Case 1: Input v is in C. In this case we match v to its mate in C 0. We omit v from C (and add it to A), and omit its mate from C 0 (and add it to T (A)). Clearly, the modied invariant still holds. Case 2: Input v is not in C. We tentatively match v to one of its neighbors, denoted t(v), outside T (A) [ C 0. Such a neighbor must exist due to the invariant. Now, we compute the new inputs that has to be added to C. This is done incrementally. Let D be the current set of new inputs, and let D 0 be the set of outputs matched to these inputs. While there exists an input w outside C [ D [ A [ fvg such that more than half of its neighbors are in C 0 [ D 0 [ T (A) [ ft(v)g, 11

13 nd an output w 0 such that D [ fvg [ fwg can be matched to D 0 [ ft(v)g [ fw 0 g. This is done as follows. First, nd a matching M 1 of D [ fvg [ fwg to outputs outside T (A) [ C. Observe that as long as jd [ fvg [ fwgj a, it follows from the invariant that such a matching exists. (In the claim below we prove that jc [ D [ fvg [ fwgj a.) Consider the graph given by the union of M 1 and the matching M 0 of D to D 0. The connected component of this graph that includes v must be an odd path that starts with v and ends either with t(v) or with the additional output w 0. The links of the path alternate between M 0 and M 1. Similarly, the connected component that includes w must also be an alternating odd path that starts with w and ends either with t(v) or with the additional output w 0. The rest of the components must be either even paths or even cycles with alternating links. We construct the desired matching as follows: all the vertices from D that are in the \even" components are matched using the edges from M 0 ; v, w and all other the vertices from D that are in the \odd" component are matched using the edges from M 1. It is easy to see that the new matching matches D [ fvg [ fwg to D 0 [ ft(v)g [ fw 0 g. We update t(v) to be the mate of v in the new matching and repeat this step till no such vertex w is found. We connect v to the nal t(v), set the new set C to C [ D, and set the new set C 0 to C 0 [ D 0. Clearly, the modied invariant still holds. Now, consider a termination request. Let v be a new input for which a connection is terminated and let t(v) be its matched output. We delete v from A and t(v) from t(a). Next, we check if v has to be added to C, that is, if more than half of its neighbors are in C 0 [ T (A). If so, we add v to C and t(v) to C 0. In case v is does not have to be added to C, we check whether there are inputs that has to be deleted from C. While there exists an input w in C such that at least half of its neighbors are outside C 0 [ T (A), delete w from C and its mate from C 0. Repeat this step till no such vertex w is found. Clearly, the modied invariant still holds. 2 Claim: If jc [ Dj = jaj then there is no input w outside C [ D [ A [ fvg such that more than half of the neighbours of w are in C 0 [ D 0 [ T (A) [ t(v). Proof: To obtain a contradiction suppose that such a vertex w exists. We get that more than half of the neighbors of every input in C [ D [ fwg are in a subset of outputs of size 2jAj + 1. Observe that since we deal with a connection request jaj a? 1. Thus, jc [ D [ fwgj = jaj + 1 a and jaj + jcj + 1 < 2jAj + 2; a contradiction to the property of H stated in the theorem. 2 We conclude that the graph H(S; B; F ) is n=b-limited wide-sense one-sided NBN and has a polynomial time decision algorithm if it has the following property. H5 For every subset X of inputs of cardinality at most n=b, even after we arbitrarily erase half of the edges adjacent to each input in X, subset X has at least 2jXj neighbors. Note that this property does not seem to imply Property [H4]. Theorem 10: There exists a graph H(S; B; F ) with properties [H1], [H2], [H3], and [H5] in which b = p n= log n and k = 16 log n. Proof: We follow the construction given in the proof of Theorem 2. The construction would yield 12

14 a graph that does not have Property [H5] if there exists a set A of a vertices from S of cardinality at most n=b, and a set T of 2a vertices from B, such that for each input w in A half the neighbours of w are in T. For a xed set A S we estimate the probability that jt j 2a. This probability is at most!! a k=2 n k Y a k=2 i=1! a b 4a=k 4a=k b ne3 2 2k?4 a k=2?3 k k=2?2 b k=2?2! a : Thus if kb cn=b ca for some constant c 32 and k 2 log c n, then this probability goes to zero as n?(a), and the probability that there exist a such that this happens can now be bounded by o(1). Thus under the conditions k = 32 log n and b = p n= log n, we get that with a positive probability, H(S; B; F ) has the required four properties. 2 We remark that a similar construction is used in [BRSU93] to obtain ecient routing in \classical" networks Oblivious and Partially Oblivious Routing In this section we assume that whenever a transmitter has to communicate with a receiver it must use one out of a xed number k of wavelengths. In previous sections we considered the case k = w, where w denotes the total number of wavelengths available. Here, we consider the case k < w. The case k = 1 is called the oblivious routing problem since there is no freedom in choosing wavelengths. Note that this implies that G is a graph rather than multigraph. The case k 1 is called the partially oblivious routing problem. In this case, G is a multigraph with bounded multiplicity. An oblivious routing network can be described by an n n matrix M. The entry M(i; j) in the matrix is an integer in the range 1; : : :; w where w is the total number of wavelengths in the solution. The entry M(i; j) indicates that i transmits to j using wavelength M(i; j) in any permutation (:) for which (i) = j. Lemma 11: Let the matrix M be a solution to the oblivious routing. If = M(i; j) = M(i 0 ; j 0 ) for i 6= i 0 and j 6= j 0, then M(i; j 0 ) 6= and M(i 0 ; j) 6=. Proof: If either M(i; j 0 ) or M(i 0 ; j) is then any permutation (:) such that (i) = j and (i 0 ) = j 0 can not be satised. 2 Dene a legal coloring of an n n matrix M to be an assignment of colors to the entries of M with the following property: if is the color of M(i; j) and M(i 0 ; j 0 ) for i 6= i 0 and j 6= j 0, then M(i; j 0 ) and M(i 0 ; j) are not colored with. The above lemma reduces the oblivious routing problem to the problem of nding a legal coloring of an n n matrix with a minimum number of colors. We rst prove that dn=2e + 2 colors are needed and then construct an optimal solution with dn=2e + 2 colors. 13

15 Theorem 12: For n 6 and n = 4, any legal coloring of an n n requires at least dn=2e + 2 colors. Proof: To obtain a contradiction, assume that we are given a legal coloring with dn=2e + 1 colors. We mark each entry of the matrix with either R or C according to the following rule: An entry M(i; j) is an R-entry if its color appears more than once in row i; it is a C-entry if its color appears more than once in column j. In case its color does not appear again in both row i and column j it is marked arbitrarily. Since the coloring is legal it follows that an R-entry cannot match the color of any other entry in its column and a C-entry cannot match the color of any other entry in its row. For each line (row or column), let N(line) be the number of entries in the line marked compatibly with the line: N(row) counts the number of R-entries in that row, and N(column) counts the number of C-entries in that column. It follows that the sum of N(line) over all 2n lines is n 2 since each entry is counted once, either in its row or in its column. Thus the average value for C(line) is n=2. Assume now that n 4 is even. Since all the C-entries in a row (or all R-entries in a column) are colored with dierent colors, it follows that the number of colors in each line is at least 1 + n? N(line). Consequently, 1 + n? N(line) n=2 + 1 which implies that N(line) n=2. However, since the average value for N(line) is n=2, it must be that N(line) = n=2. Since the number of colors is n=2 + 1, all of the lines have the following structure: one color appears n=2 times and each of the other n=2 colors appears exactly once. We refer to the color that appears n=2 times as the dominating color of the line. For n 4: 2 (n=2 + 1) < 2n. Therefore, there are three lines with the same dominating color, say c. Without loss of generality, assume that two of these lines are rows. We claim that in this case c cannot appear in any entry outside these two rows { a contradiction. To see this, note that since the coloring is legal, the entries colored c in these two rows cannot share a column. Since there are n entries colored c in these two rows, for every column in the matrix, there is an entry in one of these rows colored c. However, this implies that c cannot appear anywhere else in all of these columns. Assume now that n 7 is odd. In this case we assume that we are given a legal coloring with (n + 3)=2 colors. Similar arguments to the even case show that 1 + n? N(line) (n + 3)=2. This implies that N(line) (n? 1)=2. However, since the average value for N(line) is n=2, it follows that there are at least n lines with N(line) = (n? 1)=2. Since the number of colors is (n + 3)=2, all these lines have the following structure: one color appears (n? 1)=2 times and each of the other (n + 1)=2 colors appears exactly once. We refer to the color that appears (n? 1)=2 times in such a line as the dominating color of the line, and to the line as a dominated line. For n 7, the number of colors (n + 3)=2 is strictly less than the number of dominated lines n. Therefore, there are two dominated lines with the same dominating color, say c. Suppose that these two lines are one row and one column. Consider the entry where this row and this column intersect. This entry cannot be colored by c. If this entry is an R-entry (respectively, an C-entry), then this row (respectively, column) has at least (n? 1)=2 + 1 entries marked R (respectively, C), contradicting the denition 14

16 of a dominated line. Thus, these two lines are either both rows or both columns. Without loss of generality assume that both are rows. The entries colored c in these two rows cannot share a column. Since there are n? 1 entries colored c in these two rows, there is only one column where color c may color entries not in these two rows. So, if we eliminate these two rows, and the one column, we are left with an (n? 2) (n? 1) matrix that is legally colored with (n + 1)=2 colors. We proceed to show that this is impossible. As before, we mark the (n?2)(n?1) entries with R and C. By similar arguments, we get that N(row) (n?1)=2 and N(column) (n?3)=2. We refer to the lines for which N(row) = (n?1)=2 or N(column) = (n? 3)=2 as dominated lines. In a dominated line there must be a dominating color appearing N(line) times while all the other colors appear exactly once. To lower bound the number of dominated lines note that if for all rows, N(Row) > (n? 1)=2 and for all columns N(column) > (n? 3)=2, the sum of N(line) over all lines is at least n+1 n?1 (n? 2) + (n? 1) = 2 2 2n 2?3n?1. However, there are only (n? 1)(n? 2) entries in the matrix. Therefore, there are at least 2 2n 2?3n?1? (n? 1)(n? 2) = 3n?5 dominated lines. 2 2 For n > 7: 2 (n + 1)=2 < (3n? 5)=2. Therefore, there exists a color which dominates at least three dominated lines. Following the same arguments as before it can be shown that these three lines cannot be either all rows or all columns. Thus, one of these lines is a row and one is a column. We get a contradiction by examining the entry where these lines intersect, as before. The remaining case is when n = 7, and the number of colors is four. Again, no color dominates three lines or one line and one column. Since 2 (7 + 1)=2 = (3 7? 5)=2, it follows that each of the four colors dominates exactly two lines. One of them must dominate two columns because there are only 7? 2 = 5 rows. Moreover, this color appears in these two dominated columns and in at most one row. If we omit these three lines, we are left with a 4 4 matrix that is colored legally with the remaining three colors. This is impossible by the even case proved earlier. 2 We note that for n = 2; 3; 4 we need n colors to cover the matrix. The case n = 5 is unique since we can color a 5 5 matrix with 4 < d5=2e + 2 colors as shown in Figure Figure 4: The solution for the oblivious routing for n = 5 with 4 wavelengths Now, we construct a solution using dn=2e + 2 wavelengths for n 6. We will construct a matrix M satisfying the conditions of Lemma 11. For an even n, the idea of the construction is well demonstrated by the routing matrix presented in Figure 5. In the example n = 12, and the wavelengths are denoted by 0; : : :; 7. In general, for an even n, we have the following matrix. The 15

17 entries of wavelength 0 are M n [0; i], for i = 0; : : :; n=2? 2, M n [j; n=2? 1], for j = 1; : : :; n=2? 1, M n [n=2; i], for i = n=2; : : :; n? 2, M n [j; n? 1], for j = n=2 + 1; :::; n? 1. It is easy to see that wavelength 0 obeys the conditions of Lemma 11. Now, for = 1; : : :; n=2? 1, the entries of wavelength are given by adding (modulo n) to the row index and subtracting (modulo n) from the column index of every entry of wavelength 0. Again, it is easy to see that these wavelengths also obey the conditions of Lemma 11. The rest of the entries are lled with the two wavelengths left. The entries of wavelength n=2 are M n [i; n? i? 1], for i = 0; : : :; n? 1, and the entries of wavelength n=2 + 1 are M n [i; n=2? i? 1], M n [n=2 + i; n? i? 1], for i = 0; : : :; n=2? j j j j j j { { { { { { j { { { { { { j j j j j j Figure 5: The solution for the oblivious routing for n = 12 with 8 wavelengths Next, assume that n is odd. The matrix M n is the matrix M n?1 with an additional top line and an additional last column. Since dn=2e + 2 = (n + 1)=2 + 2 = (n? 1)=2 + 3, we have an extra wavelength which is denoted by (n? 1)= The entries are as follows: (i) M n [0; j] = M n [i; n? 1] = (n? 1)=2 + 2, for j = 0 : : :n? 2 and i = 1 : : :n? 1, (ii) M[0; n? 1] = (n? 1)=2, and (iii) M n [i; j] = M n?1 [i? 1; j], for i = 1 : : :n? 1 and j = 0 : : :n? 2. The following theorem is implied by the above construction. Theorem 13: The matrix M n is a solution to the oblivious routing using dn=2e + 2 wavelengths Partially Oblivious Routing We now consider the case of partially oblivious routing. Let k be a bound on the number of wavelengths permitted to be used between any pair of transmitter receiver. In case k = O(log n) then w = ( p n) wavelengths are required, O( p n log n) are sucient (existentially), and p n(n) are sucient (constructively). (See Section 2.1.) 16

18 The existential upper bound for k = o(log n) can be achieved as follows. Assume as in the case of the k = O(log n) that we are looking at partitions of the receivers. If the degree is k this implies that we are looking at k dierent partitions. Assume that each such partition is to b blocks each of cardinality n=b. In this case the number of wavelengths is w = kb. To get the bound we have to nd the minimum w such that the failure probability is less than 1. Assume that n=b kb, otherwise the failure probability is clearly 1. Let = n=b. Borrowing the terminology of the proof of Theorem 2, we fail if and only if there exists a set A of vertices from S, such that jn(a)j <. For a xed set A S and sets A i B i (where a i = ja i j) such that j [ i A i j =? 1, the probability that N(A) [ i A i is at most Q k i=1( a i b ). Thus, the probability that there exist A and A i 's of cardinality and a i respectively such that jn(a)j < is at most n! k Y i=1 n= a i! ai n= n! n=! (? k) =k? 1 kn e 2?1 (2k?3)+2 n (k?2)+1 k (k?1)+1 : k?1 Y i=1 n= =k! 2 For k = 2, this expression is less than 1 if c log n= log log n, for some constant c. This gives w = kn= = O(n log log n= log n). Things look better for k > 2. Then, this expression is less than 1 if c log n (k?2)=(2k?3) k (k?1)=(2k?3), for some constant c. Thus, the number of wavelengths is O n k?1 k?1 2k?3 k 2k?3 : For example, for k = 3 it is O(n 2=3 ), for k = 4 it is O(n 3=5 ), and so on. As k increases the exponent of n tends from above to 1=2. Note that for such values it is always the case that kb = kn= = n=b. kn 3. Recongurable Optical Networks In this section we consider recongurable optical networks, i.e., networks with optical switches. The network is modeled by a layered multigraph G(T; R; M), where T is the set of transmitters, R is the set of receivers, and M is an undirected graph (where each edge is considered to be bidirectional) that connects T and R. We assume that M has nodes of degree four, corresponding to 22 switches. (This assumption is made for simplicity, as well as due to current technological limitations that only allow for construction of switches with constant degree.) An edge of G may be used several times each with a dierent wavelength. However, a routing with congestion c would require at least c wavelengths. Again, we assume that a connection is carried on the same wavelength on all links of the path. In this paper we consider the special case of jt j = jrj = n. However, most of the results can be extended to the case where jt j 6= jrj. We consider the problem of constructing a recongurable optical NBN. Our goal is to study the tradeos between the number of switches and the number of dierent wavelengths used in the network. As in Section 2 we dierentiate between rearrangeably 17

19 NBNs and wide-sense NBNs, and consider several variations of this problem. These variants arise because of dierent capabilities that can be attributed to the transmitters, or receivers, or the switches. Finally, we distinguish between the o-line and on-line cases. We consider two kinds of optical switches: generalized switches and elementary switches. Generalized switches, considered by Pankaj [Pan92], are fairly powerful in that they can change their state dierently for dierent wavelengths. Elementary switches are considered in [BH92, PS93]; these switches may not be set dierently for dierent wavelengths Non-Blocking Networks with Generalized Switches In his thesis, Pankaj [Pan92] considered networks of generalized switches of constant degree in which each receiver-cum-transmitter (i.e., each end-node) can be tuned to any wavelength. Pankaj showed that in order to permute n messages, any network that uses n switches must use (log n) wavelengths. He also described permutation routing algorithms for popular networks such as the shueexchange network, the DeBruijn network, and the hypercube. These algorithms use O(log 2 n) wavelengths to route messages in rearrangeable NBNs and O(log 3 n) wavelengths to route messages in wide sense NBNs. Theorem 15 creates an optical network for routing in rearrangeable NBNs and gives a routing algorithm which routes messages using (log n) wavelengths, using any rearrangeable NBN which uses n switches to route n= log n messages. Theorem 16 modies this algorithm to obtain a similar result for the wide-sense non-blocking network. Indeed, this algorithm is fairly general in that it will work on any O(log n) depth permutation network. Our constructions are based on the following proposition given in [Lei92]. Proposition 14: Given any permutation from k` elements to k` elements fx ij g k;` i=1;j=1, can be expressed as the product of three permutations 1, 2 and 3, where 1 and 3 preserve the row index of the elements and 2 preserves the column index (that is, if 1 (x ij ) = x then i 0 j 0 i = i0 and similarly for 2 and 3 ). Theorem 15: We can construct an optical recongurable rearrangeably NBN with w wavelengths and O( n log n ) generalized switches. w w Proof: We construct a network G with n inputs and n outputs labelled with a pair (i; j) for i = 1; : : :; n and j = 1; : : :; w. Our network uses a traditional rearrangeably non-blocking network w H for n inputs and n outputs, as a black box. It is well-known that such networks using O( n log n ) w w w w switches exist. (See, e.g. [Lei92].) The switches of G are just the switches of H replaced by generalized ones. In addition to the edges of H, for a xed i, and for j = 1; : : :; w, network G carries edges from every input (i; j) in G to the ith input in H, and similarly from the ith output of H to the outputs (i; j) of G. Each edge in G can carry all wavelengths 1 ; : : :; w. To route a permutation in this network, we decompose into using Proposition 14 above (with k = n and ` = w). The wavelength allotted to the message at input (i; j) is if and w only if 1 (x ij ) = x i. Similarly, the wavelength allotted to the message at output (i; j) is if and 18

20 only if 3 (x i ) = x ij. Observe that exactly one message in each wavelength arrives at any input of H, and similarly for the outputs. Notice further that the task of routing the messages within H, which is basically performing the permutation 2, decomposes into w dierent permutation routing tasks of size n=w each { one for each wavelength. We are done now since H can route any such permutation. 2 Theorem 16: We can construct an optical recongurable wide-sense NBN with 2w?1 wavelengths and O( n log n ) generalized switches. w w Proof: The idea here is similar to that of Theorem 15. The network here is the same as the one above, except for two dierences: (1) We now use a wide-sense NBN H for n=w terminals as our black box (rather than the rearrangeable network). Again, it is well-known that such networks using O( n log n ) switches exist [ALM90]. (2) Each edge can now carry 2w? 1 wavelengths (rather w w than exactly w as in the previous theorem). To route a request from input (i; j) to output (i 0 ; j 0 ) we look for a wavelength such that both the ith input of H and the i 0 th output of H are not currently using the wavelength. Such a wavelength must exist since the ith input of H is connected to at most w? 1 inputs of G (other than (i; j)), and hence must have at least w unused wavelengths. Similarly for the i 0 th output of H. Thus they must have one common unused wavelength. The message can now be routed on the wavelength since H is a wide-sense NBN. 2 Remark. When w = log n the above constructions can be adapted for the case where the terminals serve as the switches. In this case we identify terminal (i; j) with the ith switch in the jth layer of the NBN Tunable Non-Blocking Networks with Elementary Switches A dierent class of questions is posed by Barry and Humblet [BH92], who considered elementary switches. This scenario is also important since currently the estimated cost of making a generalized switch is much more than that of an elementary switch. For this case, Barry and Humblet showed a lower bound trade-o for the number of switches and the number of wavelengths used. More precisely, they showed that in a elementary switch network with w wavelengths, the number of switches must be (n log n ) for both rearrangeable and wide-sense NBNs. Pieris and Sasaki [PS93] w 2 construct such networks that use O(n log n ) switches. Here, we show tighter upper bounds on the w number of switches required in such networks by combining the arguments from Sections 2 and the previous subsection. Theorem 17: Given w wavelengths, there exists an optical rearrangeable NBN of size n that uses O(n log nlog w ) switches. Furthermore, we can construct an optical rearrangeable NBN that uses w 2 O(n log n(n) ) switches. w 2 Proof: Again, the idea is to use Proposition 14 about decompositions of permutations. The network G is constructed in three layers. The rst and third layer do the \column" permutations and the 19