Wavelength Assignment Problem on All-Optical Networks with k Fibres per Link Luciano Margara1, Janos Simon2 1
Computer Science Department, University of Bologna. Email:
2
Computer Science Department, University of Chicago. Email:
[email protected]
[email protected]
Abstract. Given a (possibly directed) network, the wavelength assignment problem is to minimize the number of wavelengths that must be assigned to communication paths so that paths sharing an edge are assigned di erent wavelengths. Our generalization to multigraphs with k parallel edges for each link (k bres per link, with switches at nodes) may be of practical interest. While the wavelength assignment problem is NP-hard, even for a single bre, and even in the case of simple network topologies such as rings and trees, the new model suggests many nice combinatorial problems, some of which we solve. For example, we show that for many network topologies, such as rings, stars, and speci c trees, the number of wavelengths needed in the k- bre model is less than 1=k fraction of the number required for a single bre. We also study the existence and behavior of a gap between the minimum number of wavelengths and the natural lower bound of network congestion, the maximum number of communication paths sharing an edge. For optical stars (any size) while there is a 3/2 gap in the single bre model, we show that with 2 bres the gap is 0, and present a polynomial time algorithm that nds an optimal assignment. In contrast, we show that there is no xed constant k such that for every ring and every set of communication paths the gap can be eliminated. A similar statement holds for trees. However, for rings, the gap can be made arbitrarily small, given enough bres. The gap can even be eliminated, if the length of communication paths is bounded by a constant. We show the existence of anomalies: increasing the number of bres may increase the gap. 1 Introduction We study a collection of interesting combinatorial questions, motivated by optimization problems in the context of optical interconnection networks. For the purposes of this paper, an all-optical network consists of routing nodes interconnected by point-to-point bre-optic links, which can support a certain number of wavelengths. Links are bidirectional. Each message travels through the network on a speci c wavelength that, in this model, cannot be changed during the transmission. Two variants of this model of optical networks have been studied intensively (see [2] for an up-to-date survey): the directed model, in which two messages traveling on the same bre-optic link in the same direction must have
di erent wavelengths, and the undirected model, in which two messages passing through the same bre-optic link must have di erent wavelengths no matter in which direction they are traveling 3 In what follows, a message will be called a request and a set of requests will be called an instance. Given a network G and an instance I on G, the wavelength-routing problem consists of nding a routing scheme R for I and an assignment of a wavelength to each request of I, such that no two paths of R sharing an edge have the same wavelength, and such that the total number of wavelengths is minimized. In this paper we do not address the problem of obtaining a good routing scheme for a set of requests on a given network. We will assume that the routing scheme is given as part of the input, or that it is uniquely determined by the topology of the network we are considering (as in the case of optical trees). We will focus on the problem of assigning wavelengths to a given set of communication paths. The problem of nding an optimal wavelength assignment is NP-hard even if we restrict our attention to very simple network families such as rings or trees [4, 5, 6]. In some cases, there exist polynomial time greedy algorithms (see for example [16, 9]) which provide approximate solutions for this problem in terms of the network congestion (the maximum number of connection paths which share a bre-optic link) which, in turn, is a lower bound on the optimal number of wavelengths. We de ne and analyze a new optical network model: Each point-to-point bre-optic link consists of k distinct optical bres (the same k for each link). This assumption is very natural 4 , and suggests many interesting algorithmic and combinatorial questions. In this new model, each time we send a message through a link we need to specify which bre we want to use. Two paths sharing a link can be given the same wavelength if they pass through distinct bres. We ask the following basic question: can we reduce the number of wavelengths by a factor strictly larger than k using k bres per link ? We prove that for a number of network topologies of practical interest, this question has an armative answer. We also identify many challenging and (in our view) interesting problems, and provide solutions to some of them. The main results of this paper are: { We show (Thm. 4) that for any k; m 1 there exists a network G, an instance I on G, and a routing R for I, such that the minimal number of wavelengths needed using k bres per link is at least m, while the number of wavelengths needed using k +1 bres per link is 1. Note that this gives an armative answer our basic question for instance I. { For optical star networks we are able to show signi cant improvement by using multiple bres. In the undirected single bre model every instance can be routed using a number of wavelengths equal to 3=2 times the congestion 3
Brief justi cation for the models: physical links ( bres) are undirected. Current repeaters and routers aren't. 4 We do not discuss practicality: it is easy to justify wiring that uses multiple bres, and it is possible to build appropriate switches. Whether such switches can be made economical is not clear. This may also depend on the bene ts of multiple bres. We hope that papers like ours will eventually determine the amount of these bene ts.
of the network [15] and this is the best ratio achievable. In contrast using 2 bres per link it is possible to route any set of requests on an undirected star network optimally (i.e. using a number of wavelengths equal to the congestion of the network, where the congestion of the network using k bres is de ned as the largest number of paths through any bre{equivalently, this is the value of the congestion of the single bre version of the same network, divided by k). Moreover, we give a polynomial time algorithm (Thm. 5) that assigns paths to bres. { In the case of optical tree networks we prove that there is no single constant k so that for every tree all instances can be routed using a number of wavelengths equal to the congestion of the busiest link. This is true both for undirected (Thm. 9) and directed (Thm. 10) networks. The theorem holds even if we restrict the underlying graph to be the family of undirected trees of height 2. Note that this does not mean that it is impossible to eliminate the gap for a xed graph. In fact, we prove that for binary trees of height 2, 4 bres are enough for closing the gap between number of wavelengths and network congestion (Thm. 7). { For ring networks we give a polynomial time algorithm (Thm. 12) which takes as input an optical ring G with n nodes (either directed or undirected), an instance I on G, and a routing scheme R for I and computes a wavelength assignment for R whose cardinality is at most 1 + 1=k times larger than the congestion caused by R on G, where k is the number of bres per link. Note that using one bre per link the best ratio achievable between number of wavelengths and network congestion is 2 [16]. We also prove (Thm. 13) that for every k 1 there exist an optical ring G with n nodes and k bres per link, an instance I on G, and a routing scheme R for I such that the cardinality of any wavelength assignment for R is 2k=(2k ? 1) = 1 + 1=(2k ? 1) times larger than the network congestion caused by R on G. Finally, we show (Thm. 14) that if all the requests have a length uniformly bounded by an arbitrarily chosen constant c, then there exists a value kc (which depends only on c) such that using kc bres per link it is possible to close the gap between the number of wavelengths used and the network congestion for ring of any size. { We show that, perhaps surprisingly, adding bres may increase the gap between load and number of wavelengths (Thm. 7). The following question remains open: Is it true that given a xed network G there exists a number k 1 such that using k bres per link it is possible to route all the possible instances on G using a number of wavelengths equal to the congestion of the network ? Due to limited space some of the proofs are omitted, or only sketched.
2 Basic De nitions
The standard optical model. We represent an all-optical network as a graph G = (V; E), where V = f1; : : :; ng represents the set of nodes of the network and E f(i; j) j i; j 2 V g represents the set of node-to-node connections available in
the network. A request is an ordered pair of vertices (s; d); s; d 2 V; corresponding to a message to be sent from node s to node d. An instance I is a collection of requests. Note that a given request can appear more than once in the same instance. Let I = f(s1 ; d1); : : :; (sm ; dm )g be an instance on G. A routing scheme R = fp1; : : :; pmg for I is a set of simple directed paths on G. Each path pi is a sequence (v1 ; : : :; vk ) of distinct vertices of V such that v1 = si and vk = di . We say that a path p = (v1 ; : : :; vk ) contains the edge (i; j) if there exists h; 1 h k ? 1, such that vh = i and vh+1 = j. A legal wavelength assignment W of cardinality m for a routing scheme R is a map from R to [1; : : :; m] such that if two elements p; q 2 R share an edge then W(p) 6= W(q), i.e., they are given distinct wavelengths. This de nes two variant models, depending on the interpretation of \.. .sharing an edge. ..": { Directed model. Two paths p and q share the edge (i; j) if both p and q contain the edge (i; j). { Unirected model. Two paths p and q share the edge (i; j) if both p and q contain at least one edge in the set f(i; j); (j; i)g. The wavelength-routing problem can be formulated as follows: Given a graph G and an instance I on G, nd a routing scheme R for I and a legal wavelength assignment W for R such that the cardinality of W is the minimum among all possible routing schemes R for I and legal wavelength assignments W for R. The new optical model. A legal k-wavelength assignment W of cardinality m for a routing scheme R is a map from R to [1; : : :; m] such that if k + 1 elements p1 ; : : :; pk+1 2 R share an edge then there exist 1 i; j k + 1 such that W(pi ) 6= W(pj ). Our de nition is equivalent to considering a multigraph obtained from G by replacing every edge by a set of k parallel edges. Note that we consider both directed and undirected legal k-wavelength assignments. In the directed model k paths p1 ; : : :; pk 2 R share the edge (i; j) i every pi ; 1 i k, contains the edge (i; j), while in the undirected model k paths p1 ; : : :; pk 2 R share the edge (i; j) if every pi ; 1 i k, contains at least one edge in the set f(i; j); (j; i)g. Number of wavelengths and network congestion. Let I be an instance on a graph G = (V; E). Let R = fp1; : : :; pm g be a routing scheme for I. We de ne the con ict graph Gc = (Vc ; Ec) for R, and G as having vertices Vc = R and edges Ec = f(pi; pj ) j pi and pj share an edge of Gg. We denote by W(R; G; k) the cardinality of the best possible legal k-wavelength assignment for R. It is easy to verify that W(R; G; 1) is equal to the chromatic number of Gc . Let L(R; G; ), the load of be the maximum number of paths of R sharing the edge . Let L(R; G) be the maximum of L(R; G; ) over all the edges of G. It is easy to verify that W(R; G; k) d k1 L(R; G)e. In the 1- bre case (k=1) L(R; G) is called the congestion of the network. Similarly, we will call the quantity d k1 L(R; G)e the k-congestion (or, when k is clear from the context) simply, congestion. Fix a graph G. Let T = fIi gi2N be a sequence of instances on G, and let S = fRigi2N be a sequence of routing schemes. We say that S produces a k-gap r 1 on G if and only if W(Ri; G; k) r: for every i 1 : d k1 L(Ri ; G)e
We denote by Gap(S; G; k) the maximumk-gap that S produces on G. We de ne the k-gap of G, denoted by Gap(G; k), as the supremum of Gap(S; G; k) taken over all possible sequences S. Again, we will omit k when its value is clear from the context. We de ne N(G) as the minimum k such that Gap(G; k) = 1, if such a k exists. Acyclic networks. Acyclic networks are modeled by acyclic graphs. In acyclic graphs an instance I uniquely determines its routing scheme, so we omit R: I will denote both a set of requests and the associated set of paths.
3 More bres can help In this section we prove that there exist a network G, an instance I on G, and a routing scheme R for I such that the ratio between the number of wavelengths needed for R using k bres and the number of wavelengths needed for R using k + 1 bres can be made arbitrarily large. We start with two observations.
Observation 1 Let I be an instance on some graph G = (V; E), and let R be a routing scheme for I . If the con ict graph Gc associated to R and G contains no triangles (cliques with 3 nodes) then W(R; G; 2) = 1.
Observation 2 Let Gc be an arbitrary graph. Then we can construct, in time
polynomial in the size of Gc a graph G, an instance I on G, and a routing scheme R for I on G, such that Gc is the con ict graph of (R; G).
We use these observations to prove the following result. Theorem3. Given any m 1 it is possible to construct a graph G = (V; E), an instance I on G, and a routing scheme R for I such that: W(R; G; 1) m and W(R; G; 2) = 1. Sketch of proof. Let Gc be any graph with chromatic number at least m and with maximum clique at most 2 (for the construction of such a graph see for example [11]) From Gc , we construct a graph G, an instance I on G, and a routing scheme R for I on G such that Gc is the con ict graph of (R; G). Then we conclude that W(R; G; 1) m and W(R; G; 2) = 1.
We generalize Thm. 3 as follows. Theorem4. Given any m 1 and k 2 it is possible to construct a graph G = (V; E), an instance I on G, and a routing scheme R for I such that W(R; G; k ? 1) m and W(R; G; k) = 1. Sketch of proof. It is possible to construct a network G an instance I on G, and a routing scheme R for I such that: for every subset S R of k+1 paths there is an edge e in G such that all the paths of S share the edge e, and L(R; G) = k+1. As a consequence we have that W(R; G; k) jRj=k, while W(R; G; k + 1) = 1.
4 Star Networks An n-star is a graph G = (V; E) such that V = fc; x1; : : :; xng and E = f(c; xi); i = 1; : : :; ng. The node c is the center of the star, while the nodes xi are the leaves of the star. In the case of star networks using the single bre directed model it is possible to eciently (in polynomial time) route all instances with a number of wavelengths equal to the network congestion (this problem is equivalent to computing the chromatic index of bipartite graphs [18]). This is no longer true in the undirected model. The best ratio achievable in the undirected model between number of wavelengths and network congestion is 3=2 [15]. Also, computing the optimal wavelength assignment in this model is an NP-hard problem (it is equivalent to the edge-coloring problem of multigraphs, which is an NP-hard problem [8]). In the next theorem we show a rather surprising result: using 2 bres it is always (independently of the size of the star) possible to nd a wavelength assignment whose cardinality is equal to the network congestion. Moreover, this can be done in polynomial time. Theorem 5. Let G be any n-star. Then, in the undirected model, N(G) = 2. Proof. Let G be an n-star with vertex set V = fc; x1; : : :; xng. Let I be any set
of paths on G. We have L(I; G) = maxfdeg(xi ) j i = 1; : : :; ng, where deg(xi ) is the number of paths touching the node xi . Without loss of generality we assume L(I; G) even. We prove that there exists a legal 2-wavelength assignment for G of cardinality L(I; G)=2. We rst add to I as many new paths as we can without increasing L(I; G). At the end of this procedure each node has degree L(I; G) except for at most one node. For assume this is not the case. Then there are two distinct nodes xi and xj of less than maximum degree. Adding the path (xi ; xj ) to I does not increase L(I; G), contradicting the maximality of I. Assume that xi is the only node with degree d < L(I; G). Since n ? 1 nodes have degree L(I; G) and xi has degree d and since each path has 2 endpoints we know that (n ? 1)L(I; G) + d is even. Since L(I; G) is even we conclude that also d is even. We now add two new nodes xn+1 and xn+2 to G. Then we add to I: { (L(I; G) ? d)=2 paths from xi to xn+1, { (L(I; G) ? d)=2 paths from xi to xn+2, and { (L(I; G) + d)=2 paths from xn+1 to xn+2 . Now, all the nodes of G have even degree L(I; G). Consider a new graph G0 = (V 0 ; E 0) where V 0 = V n fcg and E 0 = f(xi ; xj ) j the path from xi to xj belongs to I g : G0 is L(I; G)-regular (each node has degree L(I; G)) with L(I; G) even, so E 0 can be partitioned [13] into L(I; G)=2 subsets E10 ; : : :; EL0 (I;G)=2 such that each graph G0 = (V 0 ; Ei0) is 2-regular. Let Ii I be the set of paths corresponding to Ei0 . It is easy to verify that L(Ii ; G) = 2 and therefore W(Ii ; G; 2) = 1. This implies that there exists a legal 2-wavelength assignment W for (I; G) with jW j = L(I; G)=2 as claimed.
Since there is a polynomial time algorithm to partition a graph into 2-factors (2-regular spanning subgraphs) [1], and the other procedures used in the constructive proof above are clearly in polynomial time we have: Corollary6. Let G be any n-star and I be any instance on G. An optimal wavelength assignment for (I; G) in the 2- bres undirected model can be computed in polynomial time.
5 Tree Networks We rst consider the well studied H-graph network [10], the complete binary tree of height 2 in the undirected model. We show that the H-graph is the simplest network that needs more than 2 bres per link to close the gap between number of wavelengths and network congestion: 4 bres are necessary and sucient. We also show that the H-graph is a simple example of a network that exhibits a monotonicity anomaly: using more bres may increase the gap. Theorem7. Let G be an H -graph. In the undirected model we have: N(G) = 4, and Gap(G; 5) > 1. The proof, a small explicit instance, will be presented in the full journal version. If only leaf-to-leaf communication paths are allowed then it is possible to prove that N(G) = 2 in the undirected model and N(G) = 1 in the directed model. We now prove that, in a directed H-graph, 2 bres per link are not enough for closing the gap between number of wavelengths and network congestion. The problem of determining the minimum number of bres necessary for closing the above mentioned gap in a directed H-graph is still open. Theorem8. Let G be a directed H -graph. Then N(G) 3. Proof. We construct a sequence S of instances such that Gap(S; G; 2) 89 . Let x be the root of G. Let y1 and y2 be the left and the right child of x. Let z1 ; z2 ; z3; and z4 be the leaves of G listed from left to right. Let I be de ned as follows: 3 copies of path (z1 ; z2), 2 copies of path (z2 ; z1), 3 copies of path (z3 ; z4), 2 copies of path (z4 ; z3 ), 2 copies of path (z2 ; z3), 1 copy of path (z3 ; z2 ), 1 copy of path (z1 ; y2), 1 copy of path (y1 ; z4), 1 copy of path (y2 ; z1), 1 copy of path (z4 ; y1 ), and 1 copy of path (z4 ; z1). We have jI j = 18 and L(I; G) = 4. Let Ij be a set of paths consisting of j copies of I. Let S = fIj gj 2N . It is easy to verify that jIj j = 92 L(Ij ; G). Since the largest subset Q of Ij such that W(Q; G; 2) = 1 has cardinality 8 (this can be proven by exhaustive analysis), we have 9 L(I ; G) or, equivalently, W(Ij ; G; 2) 9 : W(Ij ; G; 2) jI8j j = 16 j 1 L(I ; G) 8 2 j By de nition of Gap we conclude that Gap(S; G; 2) 98 , so Gap(G; 2) 98 . For general optical trees of height 2 we have the following result.
Theorem 9. For every k 1 1there exists an undirected optical tree T of height 2 such that Gap(T; k) 1 + 2k2 . Proof. Let k 1 be an integer. We de ne an undirected optical tree T of height
2 as follows. X is the root of T with left child L and right child R; L has 2k children l0 ; : : :; l2k?1 and R has 2k children r0 ; : : :; r2k?1. We de ne on T a set P of undirected paths as follows. P consists of 2k ? 1 copies of left ? short ? paths, 2k ? 1 copies of right ? short ? paths, and 1 copy of long ? paths, where left ? short ? paths = f(l0 ; l1); (l2 ; l3); : : :; (l2k?2; l2k?1)g right ? short ? paths = f(r0 ; r1); (r2; r3); : : :; (r2k?2; r2k?1)g long ? paths = f(li ; ri) j i = 2h; h = 0; 1; : : :; k ? 1g [ (li ; r(i+2) mod 2k ) j i = 2h + 1; h = 0; 1; : : :; k ? 1 The cardinality of P is 4k2 and L(P; T) = 2k. Let Ij be the set of paths on T obtained by taking j copies of P. Trivially, the cardinality of Ij is 4jk2 and L(Ij ; T) = 2jk. Let S = fIi gi2N . Let P 0 be any subset of Ij such that W(P 0 ; T; k) = 1. We claim that the cardinality of P 0 is at most 2k2 and that if P 0 contains at least one long path, then the cardinality of P 0 is at most 2k2 ? 1. To prove our claim we proceed as follows. Let P 0 be the union of k copies of the paths in the set left ? short ? paths and k copies of the paths in the set right ? short ? paths. It is not dicult to prove that W(P 0; T; k) = 1 and that the cardinality of P 0 is 2k2. If we insert a long path in P 0 , in order to maintain W(P 0 ; T; k) = 1, we are forced to remove 2 short paths from P 0 decreasing its cardinality by one. We can insert in P 0 at most k ? 1 other long paths without violating the property W(P 0; T; k) = 1. Each insertion of a long path in P 0 forces us to remove at least one short path. This completes the proof of our claim. Ij contains 2jk long paths. Since W(P 0; T; k) = 1, P 0 contains at most k long paths. This means that for assigning wavelengths to all long paths we need at least 2j distinct new wavelengths. We call them long wavelengths. Each long wavelength can be given to at most 2k2 ? 1 paths in Ij . This means that to complete the wavelength assignment of Ij we still have to assign2 wavelengths to 2?1) 4 jk ? 2 j (2 k 2 2 at least 4jk ? 2j(2k ? 1) paths and then we need at least = kj2 2k2 j 1 new wavelengths. So W(Ij ; T; k) 2j + k2 . Since k L(Ij ; T) = 2j, we conclude that Gap(S; T; k) 1 + 21k2 as claimed. A similar result can be proven in the directed model. Theorem 10. For every k 1 there exists a directed optical tree T of height 2 such that Gap(T; k) 1 + 4k12 . We omit the proof of this theorem since it is similar to the proof of Theorem 9.
6 Ring Networks An n-ring is a graph G = (V; E) with V = fx0; : : :; xn?1g and E = f(xi; xi+1); i = 0; : : :; n ? 2g [ f(xn?1; x0)g.
For ring networks, if we are not interested in routing schemes, there is no di erence between the directed and the undirected model. In fact, once we are given a routing scheme R for an instance I in a directed optical ring, the set of paths of R can be partitioned into two disjoint sets of paths, C, paths routed in the clockwise direction and CC, routed in the opposite direction. Since there are no con icts among paths belonging to C and CC (they use di erent directions on every edge), the original problem is equivalent to the two undirected wavelength assignment problems given by the set of requests in C and CC. For this reason, we will consider only the problem of assigning wavelengths to a set I of undirected paths on a ring. In the proof of next theorem we give an algorithm which takes as input an optical ring G of any size and an instance I on G and produces a 2-legal wavelength assignment whose cardinality is at most 3=2 larger than the network congestion caused by I (plus 3).
Theorem11. There exists a polynomial time algorithm which, given any set I of paths on any n-ring G, produces a legal 2-wavelength assignment W such that ) jW j 23 L(I;G 2 + 3. Sketch of proof. Let G = (V; E) be an n-ring, with vertex set V = fx0; : : :; xn?1g
and edge set E = f(xi; xi+1); i = 0; : : :; n ? 2g [ f(xn?1; x0)g. Let I be any set of paths on G. Without loss of generality, we assume that: { each path of I starting at xi and ending at xj (denoted by (xi ; xj ), passes through nodes xi+1 ; xi+2; : : :; xj ?1, where sums are taken modulo n, { each edge e 2 E has full load L = L(I; G), i.e., exactly L(I; G) paths pass through e, and { there are no paths starting from or ending at x0. We we say that a path (xi ; xj ) is a regular path if i < j and is a crossing path if i > j. Let cp1; : : :; cpL be the crossing paths for (I; G). Our algorithm computes a 2-legal wavelength assignment W as follows. We rst assume that every crossing path cpi = (si ; di) can be given two di erent wavelengths at the same time. The rst wavelength is associated to the segment si ; : : :; x0 and the second wavelength is associated to the segment x0; : : :; di. Taking advantage of this (illegal) assumption, it is possible to nd a legal 1-wavelength assignment W 0 such that jW 0 j = L (this problem is equivalent to the wavelength assignment problem on the line). W 0 applied to a regular path returns a single wavelength, while W 0 applied to a crossing path returns a pair of wavelengths associated to the rst and to the second segment of the path. We say that a subset S = fcpi1 ; : : :; cpih g of crossing paths is cyclic according to W 0 i : { for every j = 1; : : :; h ? 1, the wavelength associated to the second segment of cpij is equal to the wavelength associated to the rst segment of cpij+1 and { the wavelength associated to the second segment of cpih is equal to the wavelength associated to the rst segment of cpi1 . We now partition the set of crossing paths into cyclic subsets S1 ; : : :; Sm . Note that this decomposition is unique up to a relabeling of the cyclic subsets. Consider now a generic cyclic set Si having cardinality 4h for some h 1. Let Wi0
be the set consisting of the 4h distinct wavelengths associated to the crossing paths belonging to Si . Let Ii I be the set of paths having a wavelength belonging to Wi0 . It is easy to verify that L(Ii ; G) = 4h and that L(I n Ii ; G) = L ? 4h. We claim that there exists a wavelength assignment W for the paths in Ii with cardinality 3h. To prove this fact we proceed as follows. Step 1. To each crossing path cpi2j+1 ; j = 0; : : :; 2h ? 1 of Si we assign a new wavelength w. We assign w also to each path in Ii whose wavelength according to W 0 is one of the two wavelengths associated to the two segments of cpi2j+1 . Globally, in Step 1, we use 2h new wavelengths. Step 2. To each pair of crossing paths cpi4j+2 and cpi4j+4 ; j = 0; : : :; h ? 1 of Si we assign a new wavelength. Globally, in Step 2, we use h new wavelengths. The wavelength assignment W de ned in Step 1 and 2 has the following properties: jW j = 3h, W is a 2-legal assignment for (Ii ; G), L(I n Ii ; G) = L ? 4h, and jW j = (3=2)(L(Ii ; G)=2). Assume for a moment that all Si s have cardinalities multiple of 4. In this easy case we just have to repeat Step 1 and 2 until all cyclic sets have been considered. Unfortunately, in general not all Si s have cardinality 4h for some h 1. If the cardinality of jSi j = 4h + d; d = 1; 2; 3, we need to construct a wavelength assignment W in a slightly more complicated way. We now describe the basic idea for dealing with the case jSi j = 4h + 1. The other 2 cases can be treated analogously and the complete proof will be given in the full paper. If jSi j = 4h + 1, we distinguish two cases. If there exists another Sj such that jSj j = 4h+1, we construct W for Si [ Sj using 6h wavelengths using a technique which is similar to that used in Step 1 and 2. If Si is the unique cyclic set with cardinality 4h+1 then we construct W for Si alone using 3h+1 wavelengths. This is one of the three cases in which we are forced to use one additional wavelength. This completes the sketch of the proof. The result given in Thm. 11 for the 2 bres model can be generalized to the k bres model using a similar proof technique.
Theorem 12. There exists a polynomial time algorithm which, given any in-
stance I on? any n-ring G, produces a legal k-wavelength assignment W such ) that jW j 1 + k1 L(I;G k + ck , where ck depends on k but is independent of I and n (and then L(I; G)).
In the next theorem we prove that if we consider optical rings of any size with k bres per link, the ratio between wavelengths and network congestion can be made arbitrarily close to 2k2?k 1 . As a consequence, we have that, no matter how many bres we use, it is impossible to close the gap between wavelengths and network congestion (as we did in the case of optical stars) for the entire family of optical rings at the same time. Theorem 13. Let Ak be any algorithm such that for every n-ring G and every instance I on G nds a legal k-wavelength assignment W for (I; G) such that ) 2k jW j L(I;G k . Then 2k?1 .
Sketch of proof. Let Gn = (V; E) be an n-ring with n even, where
V = fx0 ; : : :; xn?1g and E = f(xi ; xi+1); i = 0; : : :; n ? 2g [ f(xn?1; x0)g. Let In = P1 [ P2 where P1 = (xi ; xi+n=2) j 0 i n=2 ? 1 ; P2 = (xi+n=2 ; xi+1) j 0 i n=2 ? 2 : It is easy to see that L(In ; Gn) = n=2 and that jInj = n ? 1, while it takes a little e ort to prove that the largest subset Q of In such that W(Q; Gn; k) = 1 has cardinality 2k ? 1. As a consequence of this consideration we have W(In ; Gn; k) n?1 2k?1 and then W(In ; Gn; k) 2k ? 2k : (1) 1 2k ? 1 n(2k ? 1) k L(I; G) If < 2k2?k 1 using Equation 1 with n large enough we get a contradiction. We end this section by considering a slightly di erent version of the standard wavelength assignment problem for optical rings in which the maximum length of input requests is at most c < n. In this framework it is possible to prove the following result. Theorem14. If we consider requests of length at most c, then there exists kc (which depends only on c) such that for every n-ring G we have Gap(G; kc) = 1. Sketch of proof. Let I be any instance on any n-ring G. As usual we assume that for every edge e of G we have L(I; G; e) = L(I; G). First we note that the number of distinct crossing paths (de ned in the proof of Thm. 11) of I is at most c. As a consequence of this fact, we claim that it is possible to nd a set of requests I 0 I such that L(I 0 ; G) c and L(I n I 0 ; G) = L(I; G) ? L(I 0 ; G): (2) The set I 0 can be constructed as follows. We start from I 0 = fcpg where cp is any crossing path. Then we move clockwise on the ring adding to I 0 a sequence of adjacent paths until we reach a new crossing path cp0 whose rst endpoint is equal to the rst endpoint of another crossing path cp00 2 I 0 . Note that this must happen within at most c complete rounds since the number of distinct crossing paths is at most c. At this point we do not add cp0 to I 0 but we remove from I 0 all the paths (possibly none) inserted before cp00. It takes a little e ort to prove that I 0 satis es properties (2). We repeat this procedure until all paths of I have been considered getting a partition of I into m subsets Ii0 ; 1 i m; such that L(Ii0 ; G) c and
m X i=1
L(Ii0 ; G) = L(I; G):
(3)
It remains to be proven that there exists kc such that using kc bres per link Gap(S; G; kc) = 1 for every sequence of instances S. It can be proven (due to limited space we omit this part of the proof) that chosing kc = c! no sequence of instances can produce a gap bigger than 1. Acknowledgments. We would like to thank Amit Kumar and Bruno Codenotti for many useful discussions and Jon Kleinberg for having suggested the proof of Thm. 4.
References 1. R. P. Anstee. An algorithmic proof of Tutte's f-factor theorem. Journal of Algorithms, 6:112{131, 1985. 2. B. Beauquier, J.-C. Bermond, L. Gargano, P. Hell, S. Perennes, and U. Vaccaro. Graph problems arising from wavelength-routing in all-optical networks. Proc. of Workshop on Optics in Computer Science WOCS'97. 3. N. K. Cheung, N. K., and G. Winzer. Special issue on dense WDM networks. Journal on Selected Areas in Communications, 8, 1990. 4. T. Erlebach and K. Jansen. Scheduling of virtual connections in fast networks. In Proc. of Parallel Systems and Algorithms (PASA), 13{32, 1996. 5. T. Erlebach and K. Jansen. Call scheduling in trees, rings and meshes. In Proc. of HICSS, 1997. 6. M. C. Golumbic and R. E. Jamison. The edge intersection graphs of paths in a tree. Journal of Combinatorial Theory, Series B, 38:8{22, 1985. 7. P. E. Green. Fibre{Optic Communication Networks. Prentice{Hall, 1993. 8. I. Holyer. The NP{completeness of edge coloring. SIAM Journal of Computing, 10(4):718{720, 1981. 9. C. Kaklamanis, G. Persiano, T. Erlebach,and K. Jansen. Constrained bipartite edge coloring with applications to wavelength routing. Proc. of ICALP'97, Lecture notes in Computer Science vol. 1256:493{504, 1997. 10. J. Kleinberg and A. Kumar. Wavelength conversion in optical networks In Proc. 10th ACM-SIAM Symposium on Discrete Algorithms, 1999. 11. L. Lovasz. On chromatic number of nite set-systems. Acta Math. Acad. Sci. Hungar, 19:59{67, 1968. 12. A. D. McAulay. Optical Computer Architectures. John Wiley, 1991. 13. J. Petersen. Die Theorie der Regularen Graphen. Acta Math. 15, 193{220, 1891. 14. R. Ramaswami. Multiwavelength lightwave networks for computer communication. IEEE Communications Magazine, 31(2):78{88, Feb. 1993. 15. R. E. Tarjan. Decomposition by clique separators. Discrete Mathematics, 55(2):221{232, 1985. 16. A. Tucker. Coloring a family of circular arcs. SIAM Journal of Applied Mathematics, 29(3):493{502, 1975. 17. R. J. Vetter and D. H. C. Du. Distributed computing with high-speed optical networks. IEEE Computer, 26(2):8{18, Feb. 1993. 18. G. Wilfong and P. Winkler. Ring routing and wavelength translation. In Proc. of the 9th Annual ACM-SIAM Symposium on on Discrete Algorithms (SODA), 333-341, 1998.
This article was processed using the LATEX macro package with LLNCS style
