Implementing Load-Balanced Switches With Fat-Tree Networks

Transcription

1 Implementing Load-Balaned Swithes With Fat-Tree Networks Hung-Shih Chueh, Ching-Min Lien, Cheng-Shang Chang, Jay Cheng, and Duan-Shin Lee Department of Eletrial Engineering & Institute of Communiations Engineering National Tsing Hua University, Hsinhu, Taiwan, R.O.C. Abstrat Load-balaned swithes have reeived a lot of attention lately as they are muh more salable than other existing swith arhitetures in the literature. One of the most salient features of load-balaned swithes is the simpliity of implementing deterministi and periodi onnetion patterns for its swith fabris. In this paper, we propose to use fat-tree networks as the swith fabris in load balaned swithes. Fat-tree networks have been widely used for interonneting omputers in data enters and for other appliations in Network-on-Chip (NoC). One of the main problems in fat-tree networks is that the link apaity has to be inreased rapidly from the leaves to the root of the tree. This poses a serious salability problem as the omplexity of implementing the swithes near the root of the tree ould be very high, espeially when a fat-free network is required to be nonbloking. As we only require an N N fat-tree network to realize a set of N permutations needed for the implementation of N N load-balaned swithes, in this paper we show that the implementation omplexity an be greatly redued. For this, we first derive a lower bound on the link apaity for eah swith in a fat-tree network. By using the uniform mapping property of the bit-reversal permutation, we show that there exists a set of N permutations that ahieves the lower bound. To further redue the implementation omplexity, we propose a fully meshed fattree network that replaes the upper half of the tree by a simple mesh. We then show a fully meshed fat-tree network an be implemented by folding a banyan-type network that realizes this set of N permutations. I. INTRODUCTION Load-balaned swithes [] [] have reeived a lot of attention reently as they are muh more salable than other existing swith arhitetures in the literature. In Figure, we show a typial two-stage load-balaned swith: the first stage is for load-balaning that onverts inoming traffi into the uniform traffi, and the seond stage is for swithing of the uniform traffi. It was shown in [] that load-balaned swithes ahieve % throughput and have delay performane omparable to ideal output-buffered swithes (when the traffi is heavy and bursty). In this paper, we onsider the disrete-time setting and assume that time is slotted and synhronized so that a fixedsize paket an be transmitted over a link within a time slot. One of the most salient features of load-balaned swithes is that the onnetion patterns for the swith fabris in both stages in Figure are deterministi and periodi. As suh, N Fig.. Load-balaning Swithing The generi two-stage load-balaned swith. there is no need to find mathings as required in most inputbuffered swithes, and there are no omputational overheads in load-balaned swithes. Speifially, for an N N loadbalaned swith, the swith fabris in both stages in Figure only need to realize in every period of N time slots any N N N permutation matries P, P,..., P N that satisfy P + P + + P N = e, () where e is the N N matrix with all its elements being. Clearly, there are many possible hoies of the N permutation matries P, P,..., P N for implementing N N loadbalaned swithes. In the literature, there are two well-known onditionally nonbloking swithes, i.e., rotators (whih realize all of the powers of the irular shift matrix) and symmetri TDM swithes [] [], that an be used for realizing the needed onnetion patterns for the swith fabris in load balaned swithes. In [] and [], respetively, it was shown that twister networks (whih are speial types of multistage interonnetion networks) and degenerate banyan networks (whih only use half of the inputs/outputs in the lassial banyan networks) an be used as rotators and symmetri TDM swithes, and hene they an be used as the swith fabris in load balaned swithes. In this paper, we propose to use fat-tree networks [] as the swith fabris in load balaned swithes. Fat-tree networks (and many variants of them) are ommonly used for the onstrutions of data enter interonnetion networks []. They have also been widely deployed in Network-on-Chip (NoC) []. As suggested by its name, a fat-tree network is a swithing network onstruted from a omplete binary tree, N

2 where eah leaf is an input/ouput port and eah non-leaf node is a swith. To aommodate the traffi multiplexed into the tree, the link apaity of a swith has to be inreased rapidly from the leaves to the root of the tree (this is why it is alled a fat-tree network). In partiular, if a fat-tree network with N input/output ports is required to be nonbloking, i.e., the fatfree network an realize all of the N! permutations between its input and output ports, then eah node has to be a nonbloking swith and the link apaity of a swith has to be inreased exponentially from the bottom to the top of the tree. This poses a serious salability issue in designing the swithes near the root of the tree when N is very large. Fortunately, as our purpose in this paper is to use fat-tree networks as the swith fabris in load balaned swithes, we only require a fat-tree network with N input/output ports to realize N permutation matries P, P,..., P N that satisfy the ondition in () in every period of N time slots. We will show that the salability problem in a nonbloking fat-tree network an be solved in this senario by appropriate hoies of the N permutation matries P, P,..., P N and by replaing the upper half of the fat-tree network by a simple mesh. In this paper, we first show a lower bound on the link apaity for eah swith in a fat-tree network that is apable of realizing any N permutation matries P, P,..., P N satisfying the ondition in (). The lower bound is derived based on averaging the traffi flows that need to go through a swith in the fat-tree network. Unfortunately, both rotators and symmetri TDM swithes require link apaities that are substantially higher than those given by the lower bound. Then we propose new permutation matries P, P,..., P N that not only satisfy the ondition in () but also ahieve the lower bound, i.e., they an be realized by a fat-tree network with link apaities speified by the lower bound. The idea is based on the bit-reversal permutation previously proposed in []. One key property of the bit-reversal permutation is the uniform mapping property that maps the inputs in a subtree uniformly to the outputs in other subtrees. We show that any permutation that has the uniform mapping property an be realized by a fat-tree network with link apaities speified by the lower bound. We also show that the bitreversal permutation and its variants obtained by irular shifts all have the uniform mapping property, and hene they an be realized by a fat-tree network with link apaities speified by the lower bound. To further redue the implementation omplexity, we propose a fully meshed fat-tree network by replaing the upper half of a fat-tree network by a simple mesh. As suh, there is no need to implement the swithes in the upper half of the usual fat-tree network, and hene we do not have the salability problem for the swithes near the root in the usual nonbloking fat-tree network. We show that any permutation that has the uniform mapping property an also be realized by a fully meshed fat-tree network. To implement the swithes in a fully meshed fat-tree network, i.e., in the lower half of a fat-tree network, we onsider a speifi banyan-type network. Suh a banyan-type network is onstruted by onneting a olletion of reversed baseline networks (with a muh smaller size) and another olletion of baseline networks (with a muh smaller size). We show that a permutation an be realized by suh a banyantype network if and only if the permutation has the uniform mapping property. By folding suh a banyan-type network, we are able to map suh a folded banyan-type network to the orresponding fully meshed fat-tree network with the same number of input/output ports. We further show that eah swith in the orresponding fully meshed fat-tree network an be onstruted by a olletion of swithes in the folded banyan-type network. As any banyan-type network possesses the self-routing property, the orresponding fully meshed fattree network also inherits suh a self-routing property. This solves both the swith implementation problem and the routing problem in a fully meshed fat-tree network. This rest of this report is organized as follows. In Setion II, we introdue fat-tree networks and show a lower bound on the link apaity for eah swith in a fat-tree network that an realize the permutations needed for load-balaned swithes. In Setion III, we introdue the bit-reversal permutation and show that any permutation that has the uniform mapping property, inluding the bit-reversal permutation and its variants obtained by irular shifts, an be realized by a fat-tree network with the link apaities speified by the lower bound. We then propose fully meshed fat-tree networks in Setion IV and show that a fully meshed fat-tree network an be implemented by folding a speifi banyan-type network. We also show that any permutation that has the uniform mapping property an be realized by a fully meshed fat-tree network with the link apaities speified by the lower bound. Finally, the report is onluded in Setion VI, where we address possible extensions of our work. II. FAT-TREE NETWORKS A fat-tree network, first proposed in [], is a swithing network onstruted from a omplete binary tree. To explain how a fat-tree network works, we first onsider a omplete binary tree with n leaves, indexed from to n (see Figure for a omplete binary tree with leaves). In suh a omplete binary tree, there are n + levels, indexed from to n. The root is the only node at level and a node is at level j+ if it is a hild of a node at level j for j n. Index the root as node (, ), and reursively index the two hildren of node (j, k) as node (j +, k) and node (j +, k + ) for j n and k j. Clearly, there are j nodes at level j and in this report we also all node (j, k) as the k th node at level j for j n and k j. Note that the n leaves are nodes (n, ), (n, ),..., (n, n ), and we also all node (n, x) as leaf x for x n in this report. For j n and k j, let T (j, k) be the subtree rooted at node (j, k), and let S(j, k) be the set of all of the leaves in the subtree T (j, k), namely, S(j, k) = {x : k n j x (k + ) n j } ()

3 Fig.. A omplete binary tree with leaves. Note that the total number of leaves in the subtree T (j, k) is S(j, k) = n j. For example, for the omplete binary tree with leaves in Figure, we have S(, ) = {, }, S(, ) = {, }, and S(, ) = S(, ) S(, ) = {,,, }. For x n, let the n-tuple (I n (x), I n (x),..., I (x)) be the binary representation of x, i.e., x = n m= I m(x) m, where I m (x) is the (n m+) th most signifiant bit of x for m n. Then the set S(j, k) an be alternatively expressed as S(j, k) = {x : x n, and I n j+m (x) = I m (k) for m j}. () In other words, S(j, k) ontains all of the leaves of whih the first j most signifiant bits are the same as the last j most signifiant bits of k (note that the first n j most signifiant bits of k are ). Fig.. Level A nonbloking fat-tree network with input/output ports. Now we show how one onstruts a n n nonbloking fat-tree network (with n input/output ports) by using the omplete binary tree with n leaves. For this, we view every leaf of the tree as both an input port and an output port of a n n swithing network and view every non-leaf node in the tree as a nonbloking swith (see Figure for a nonbloking fat-tree network with input/ouptut ports). For j n and k j, let the upward apaity C u (j, k) of node (j, k) be the number of parallel links from node (j, k) to its parent. For j n and k j, let the downward apaity C d (j, k) of node (j, k) be the number of parallel links from node (j, k) to its two hildren. In this report, we assume that the downward apaity of a node is evenly split between its two hildren. For suh a n n fat-tree network to be nonbloking, it has to realize all of the ( n )! n n permutations between its input and output ports. In other words, for eah n n permutation there is a non-onfliting path for every pair of input/output ports speified by the permutation. Clearly, this requires that the upward apaity and downward apaity of a node in the tree must not be less than the total number of leaves in the subtree rooted at this node. As the total number of leaves in the subtree T (j, k) is S(j, k) = n j for j n and k j, we have C u (j, k) n j, j n, k j, () C d (j, k) n j, j n, k j. () Conversely, if the upward apaities and the downward apaities of the nodes in a fat-tree network satisfy the onditions in () and (), respetively, then there is always a non-onfliting path from an input to the root and there is always a nononfliting path from the root to an output, and it follows that the fat-tree network is nonbloking. Therefore, the onditions in () and () are the neessary and suffiient onditions for a fat-tree network to be nonbloking. In this report, we define a n n nonbloking fat-tree network to be the fat-tree network with C u (j, k) = n j for j n and k j, and C d (j, k) = n j for j n and k j. Note that the link apaity of a swith in a nonbloking fattree network grows exponentially from the leaves to the root, and this poses a serious salability problem in designing the swithes near the root. As mentioned in Setion I, our purpose in this report is to use fat-tree networks as the swith fabris in load balaned swithes, and hene we only require a n n fat-tree network to realize n permutation matries P, P,..., P n that satisfy the ondition in () in every period of n time slots. As suh, it seems that the upward apaities and the downward apaities of the nodes in suh a fat-tree network ould be greatly redued. In the following theorem, we first show lower bounds for the upward apaities and the downward apaities of the nodes in suh a fat-tree network. Theorem Suppose that a n n fat-tree network is apable of realizing n permutation matries P, P,..., P n that satisfy the ondition in () in every period of n time slots. Then for eah k j, we have C u (j, k) n j n j { n j n j, if j n/, = n j, if n/ + j n, C d (j, k) n j n j { n j n j, if j n/, = n j, if n/ j n. Proof. Consider node (j, k), where j n and k j. Our idea for the proof of () and () is by averaging. Consider a frame of n time slots, indexed from to n, alled the tagged frame. Assume that there is always a paket () ()

4 at every input port in every time slot and assume that fat-tree network realizes permutation matrix P i in the i th time slot for i =,,..., n. Sine the sum of the n permutation matries P, P,..., P n is a n n matrix with all its elements being, there is exatly one paket transmitted from every input to every output in the tagged frame. For j n, every leaf in the subtree T (j, k) sends a paket to every leaf that is not in the subtree T (j, k) in the tagged frame, and eah of these pakets has to go through an upward link of node (j, k). As there are n j leaves in the subtree T (j, k) and there are n n j leaves that are not in the subtree T (j, k), the total number of pakets that have to go through the upward links of node (j, k) in the tagged frame is n j ( n n j ). On the average, there are n j n j pakets that need to go through the upward links of node (j, k) per time slot. Clearly, the upward apaity C u (j, k) of node (j, k) is not less than the maximum number of pakets that need to go through its upward links in a time slot, and hene is also not less than the average number of pakets that need to go through its upward links per time slot. Therefore, it follows that C u (j, k) n j n j for j n, whih is the desired result in (). For j n, every leaf that is not in the subtree T (j +, k) (resp., subtree T (j +, k + )) sends a paket to every leaf in the subtree T (j +, k) (resp., subtree T (j +, k + )) in the tagged frame, and eah of these pakets has to go through a downward link of node (j, k) that is direted to node (j +, k) (resp., node (j +, k + )). On the average, there are n j n j pakets that need to go through the downward links of node (j, k) per time slot that are direted to node (j +, k) (resp., node (j +, k + )). As we assume that the downward apaity of a node is evenly split between its two hildren, we then see that C d (j, k) n j n j for j n, whih is the desired result in (). Observe that lower bounds for the upward apaities and the downward apaities in () and () are the same as those of nonbloking fat-tree networks in () and () for the lower half of the tree, but are smaller for the upper half of the tree. As the salability problem is mainly due to the design of the swithes in the upper half of the tree, it is of highly importane and interest to see if there exist n permutation matries P, P,..., P n that satisfy the ondition in () and ahieve the lower bounds in () and (). It an be seen that the n permutation matries realized by rotators and symmetri TDM swithes do not ahieve the lower bounds in () and (). In the next setion, we will find n permutation matries P, P,..., P n that satisfy the ondition in () and ahieve the lower bounds in () and (). III. BIT-REVERSAL PERMUTATION In the previous setion, we derive lower bounds for the upward apaities and the downward apaities of a fat-tree network to realize the needed permutations for a load-balaned swith. In this setion, we will show that these lower bounds are indeed ahievable by using the bit-reverse permutation introdued in [] and its variants obtained by irular shifts. We first introdue some notations that will be used for desribing the bit-reversal permutation. Let Z N = {,,..., N }. For a permutation σ on Z N, we denote P σ as the N N permutation matrix orresponding to σ. For a set S Z N, let σ(s) be the range of S under σ, i.e., σ(s) = {σ(x) : x S}. Let σ be the irular shift permutation on Z N, i.e., σ (x) = (x + ) mod N for all x N. Also, let σ i be the identity permutation on Z N for i = and let σ i = σ i σ for i. Clearly, σ i is the permutation that performs irular shift permutation i times for i, i.e., σ(x) i = (x + i) mod N for x N. As it is easy to see that P σ + P σ + + P σ N = e, () we have that P σ, P σ,..., P σ N satisfy the ondition in (). Furthermore, for a permutation σ on Z N, we denote σ i = σ i σ for i, i.e., σ i (x) = σ(σ(x)) i = (σ(x) + i) mod N for x N. Sine P σi = P σ i P σ for i, we see from () that P σ + P σ + + P σn = (P σ + P σ + + P σ N )P σ = ep σ = e. () Note that as σ is the identity permutation on Z N, we have σ = σ σ = σ. Therefore, it follows from () that P σ, P σ, P σ,..., P σn satisfy the ondition in (). Definition (Bit-Reversal Permutation) Let N = n and let (I n (x), I n (x),..., I (x)) be the binary representation of x Z N, where I m (x) is the (n m + ) th most signifiant bit of x for m n. The bit-reversal permutation π on Z N is the permutation suh that I m (π(x)) = I n+ m (x), for m n, () namely, π(x) = n m= I n+ m(x) m, for x N. Note that P π, P π, P π,..., P πn satisfy the ondition in (), where π i = σ i π for i N. In Figure, we show the permutations π, π,..., π on Z. The olumn marked with on the top row is π, and the olumn marked with i on the top row is π i for i. As π, π,..., π satisfy the ondition in (), the matrix in Figure is a Latin square, where every symbol in Z = {,,,..., } appears exatly one in every row and every olumn. One prominent property of the bit-reversal permutation is the uniform mapping property as defined below. Definition (Uniform Mapping Property) Let N = n. A permutation σ on Z N is said to have the uniform mapping property if σ(s(j, k)) S(n j, l) = () for all j n, k j, and l n j, where S(j, k) and S(n j, l) are given by ().

5 Fig.. The permutations π, π,..., π on Z = {,,,..., }. When a permutation σ is realized in a fat-tree network, σ(s(j, k)) is the set of outputs with their inputs in the subtree T (j, k). As there are n j leaves in the subtree T (j, k) and there are n j subtrees at level n j, the uniform mapping property implies that all of the n j leaves in the subtree T (j, k) are mapped uniformly to the n j subtrees at level n j. In the following theorem, we show that the bit-reversal permutation and its variants obtained by irular shifts all have the uniform mapping property. Theorem Let N = n. The permutations π, π,..., π N all have the uniform mapping property. Proof. Sine we know that π = π, it suffies to show that π i has the uniform mapping property for i N. Let i N, j n, k j, and l n j. From (), we see that it suffies to show that π i (S(j, k)) S(n j, l) ontains exatly one element. Let x S(j, k). Then we have from I m (π(x)) = I n+ m (x) for m n in () and I n+ m (x) = I (n j)+(j+ m) (x) = I j+ m (k) for m j in () that π(x) = = = n I m (π(x)) m = m= n m=j+ n j m= n I n+ m (x) m m= I n+ m (x) m + I n+ m j (x) m+j + j I n+ m (x) m m= j I j+ m (k) m m= = q (x) j + r (j, k), () where q (x) = n j m= I n+ m j(x) m and r (j, k) = j m= I j+ m(k) m. Let r (j, k) + i = q (i, j, k) j + r (i, j, k), where q (i, j, k) and r (i, j, k), respetively, are the quotient and the remainder of r (j, k) + i divided by j. Then we have from () that (π(x) + i) mod n = (q (x) j + r (j, k) + i) mod n = ((q (x) + q (i, j, k)) j ) mod n + r (i, j, k). () Note that sine x S(j, k), we have from () that k n j x (k + ) n j. As x goes from k n j, k n j +,..., (k + ) n j, the last n j most signifiant bits (I n j (x), I n j (x),..., I (x)) goes from (,,..., ), (,,..., ),..., (,,..., ). As q (x) = n j m= I n+ m j(x) m, it then follows that {q (x) : x S(j, k)} = {,,..., n j }. () It is easy to see from () that {(q (x) + q (i, j, k)) j mod n : x S(j, k)} = {q j : q n j }. () As suh, we have from (), (), and r (i, j, k) j that π i (S(j, k)) = {π i (x) : x S(j, k)} = {(π(x) + i) mod n : x S(j, k)} = {((q (x) + q (i, j, k)) j ) mod n + r (i, j, k) : = {q j + r (i, j, k) : q n j } = {y : y n, x S(j, k)} and I m (y) = I m (r (i, j, k)) for m j}.() Furthermore, we have from () that S(n j, l) = {y : y n, and I j+m (x) = I m (l) for m n j}.() Therefore, we see from () and () that there is exatly one element in π i (S(j, k)) S(n j, l), and it is uniquely determined by the binary representation (I n j (l),..., I (l), I j (r (i, j, k)),..., I (r (i, j, k))). In the following theorem, we show that any permutation that satisfies the uniform mapping property an be realized by a fat-tree network with link apaities speified by the lower bounds in Theorem. Theorem Let N = n. Suppose that an N N fat-tree network has link apaities given by the lower bounds in () and (), i.e., the inequalities in () and () hold with equality. Then suh an N N fat-tree network an realize any permutation on Z N that satisfies the uniform mapping property. Proof. Let σ be a permutation on Z N that satisfies the uniform mapping property. To show that the permutation σ an be realized by the fat-tree network, we need to show that there is a non-onfliting path for every pair of input/output ports speified by the permutation σ. In fat, we will show that the

6 shortest paths for all of the pairs of input/output ports speified by σ are non-onfliting paths. The shortest path from an input port x to its output port σ(x) is given by first going up the tree from x to the first ommon anestor of x and σ(x), and then going down the tree to σ(x). (i) We first show that there is no onflit in the upward links of node (j, k) for j n and k j by proving that C u (j, k) is not less than the total number of shortest paths that go through its upward links. We onsider the two ases j n/ and n/ + j n separately. Case : j n/. Note that a shortest path needs to go through an upward link of node (j, k) if its input x is a leaf of the subtree T (j, k) and its output σ(x) is a leaf outside the subtree T (j, k). The set of the leaves that are outside the subtree T (j, k) an be written as k ks(j, k ). Sine in this ase we have j n/, the set of the leaves in the subtree T (j, k ) an be expressed as the union of the sets of the leaves in the subtrees T (n j, l), l = n j k, n j k +,..., n j (k + ), and hene we have S(j, k ) = n j (k +) l= n j k S(n j, l). Therefore, it follows from the uniform mapping property in () and () that the total number of shortest paths that go through the upward links of node (j, k) is given by σ(s(j, k)) ( k ks(j, k )) = σ(s(j, k)) ( k k n j (k +) l= n j k S(n j, l)) = k k n j (k +) l= n j k (σ(s(j, k)) S(n j, l)) = k k n j (k +) l= n j k σ(s(j, k)) S(n j, l) n j (k +) = k k = C u (j, k). () l= n j k = ( j ) n j Case : n/ + j n. Clearly, the total number of shortest paths that go through the upward links of node (j, k) is bounded above by S(j, k), i.e., the total number of leaves in the subtree T (j, k). As in this ase we have from () that C u (j, k) = n j = S(j, k), the proof is ompleted. (ii) Now we show that there is no onflit in the downward links of node (j, k) for j n and k j by proving that C d (j, k) is not less than the total number of shortest paths that go through its downward links. We onsider the two ases j n/ and n/ j n separately. Case : j n/. Note that a shortest path needs to go through a downward link of node (j, k) that is direted to node (j +, k) (resp., node (j +, k + )) if its input x is a leaf outside the subtree T (j +, k) (resp., subtree T (j +, k + )) and its output σ(x) is a leaf of the subtree T (j +, k) (resp., subtree T (j +, k + )). The set of the leaves that are outside the subtree T (j +, k) (resp., subtree T (j +, k + )) an be written as k ks(j +, k ) (resp., k k+s(j +, k )). Sine in this ase we have j + n/, the set of the leaves in the subtree T (j +, k ) an be expressed as the union of the sets of the leaves in the subtrees T (n j, l), l = n j k, n j k +,..., n j (k + ), and hene we have S(j +, k ) = n j (k +) l= n j k S(n j, l). As suh, it follows from the uniform mapping property in () and () that the total number of shortest paths that go through the downward links of node (j, k) that are direted to node (j +, k) (resp., node (j +, k + )) is given by σ( k ks(j +, k )) S(j +, k)) = σ( k k n j (k +) l= n j k S(n j, l)) S(j +, k)) = k k n j (k +) l= n j k σ(s(n j, l)) S(j +, k)) = k k = k k = C d(j, k). n j (k +) l= n j k σ(s(n j, l)) n j (k +) S(j +, k)) l= n j k = ( j+ ) n j Similarly, the total number of shortest paths that go through the downward links of node (j, k) that are direted to node (j+, k+)) is given by σ( k ks(j+, k )) S(j+, k+ )) = C d(j, k). Therefore, the total number of shortest paths that go through the downward links of node (j, k) is exatly C d (j, k). Case : n/ j n. Clearly, the total number of shortest paths that go through the downward links of node (j, k) is bounded above by S(j, k), i.e., the total number of leaves in the subtree T (j, k). As in this ase we have from () that C d (j, k) = n j = S(j, k), the proof is ompleted. From Theorem, Theorem, and the fat that the N permutations P π, P π, P π,..., P πn, where N = n, satisfy the ondition in (), we obtain the following theorem. Theorem Let N = n. Suppose that an N N fat-tree network has link apaities given by the lower bounds in () and (). Then suh an N N fat-tree network an realize the N permutations P π, P π, P π,..., P πn, and hene an be used as the swith fabri for an N N load-balaned swith. IV. FULLY MESHED FAT-TREE NETWORKS There is a very important observation from the proof of Theorem. Note that the total number of paths that go through the subtree T (j, k) to another subtree T (j, k ) at the same level is σ(s(j, k)) S(j, k ) under the shortest path routing for realizing a permutation σ in a fat-tree network. If σ satisfies the uniform mapping property, then it an be seen from () that this number is n j for j n/.

7 In partiular, for j = n/, we have n j = when n is even and we have n j = when n is odd. Therefore, the onstrution omplexity an be greatly redued if we simply provide diret links among the subtrees at level n/ and route pakets diretly through these links. In other word, we replae the upper half of the tree by a simple mesh, and this leads to a muh more simplified onstrution, alled a fully meshed fat-tree network. Speifially, a n n fully meshed fat-tree network is onstruted by n/ n/ n/ nonbloking fat-tree networks. There are n n/ links from eah root of a n/ n/ fat-tree network to the root of another n/ n/ fat-tree network. In Figure, we show a fully-meshed fat-tree network. To see why the n/ n/ n/ fat-tree networks in a n n fully meshed fat-tree network are nonbloking, observe that level j in the n/ n/ n/ fat-tree networks is the level j + n/ in the original n n fat-tree network with link apaities given by the lower bounds in () and (). Let C u (j, k) (resp., C d (j, k)) be the upward (resp., downward) apaity of node (j, k) in the n/ n/ n/ fat-tree networks for j n/ (resp., j n/ ) and k j. Then we have from () and () that C u (j, k) = C u (j + n/, k) = n j n/ = n/ j, for j n/, () C d (j, k) = C d (j + n/, k) = n j n/ = n/ j, for j n/. () The link apaities in () and () are exatly the same as those required for a nonbloking n/ n/ fat-tree network. Fig.. A fully meshed fat-tree network. Level By following the same argument as in the proof of Theorem, we also have the following theorem. Theorem Let N = n. Consider the N N fully meshed fat-tree network as desribed in this setion. Suh an N N fully meshed fat-tree network an realize any permutation on Z N that satisfies the uniform mapping property. In partiular, it an realize the N permutations P π, P π, P π,..., P πn, and hene an be used as the swith fabri for an N N load-balaned swith. V. IMPLEMENTATION OF THE SWITCHES IN A FULLY MESHED FAT-TREE NETWORK It is shown in the previous setion that a n n fully meshed fat-tree network is apable of realizing the needed n permutations for a n n load-balaned swith. Although the onstrution omplexity of a n n fully meshed fattree network is muh smaller than that of a n n nonbloking fat-tree network, it still needs to implement n/ n/ n/ nonbloking fat-tree networks. As eah node in these n/ n/ nonbloking fat-tree networks is itself a nonbloking swith with many input/output ports, the onstrution omplexity is still very high. To further redue the onstrution omplexity, we will show that one does not need to implement nonbloking swithes for the nodes in these n/ n/ n/ fat-tree networks. Speifially, for a node with upward apaity j and downward apaity j, it an be implemented by a olletion of j swithes. Our idea of ahieving this is folding a speifi banyan-type network. A. A Banyan-Type Network In this setion, we onsider a n n banyan-type network (see e.g., []), where n is an even number. A n n banyantype network is a multistage interonnetion network with n stages, indexed from to n. Eah stage onsists of n swithes, indexed from to n. As there are two inputs and two outputs in a swith, there are n inputs and n outputs in eah stage. For eah stage, index the upper input (resp., output) and the lower input (resp., output) of swith k as input (resp., output) k and input k +, respetively. To ompletely speify the banyan-type network onsidered in this report, we need to desribe how the n outputs from one stage are onneted to the n inputs of the next stage. For x n, let (I n (x), I n (x),..., I (x)) be the binary representation of x, where I m (x) is the (n m + ) th most signifiant bit of x for m n. For j n/, output x of the j th stage is onneted to input y of the (j +) th stage, where y has the following binary representation: (I n (y), I n (y),..., I (y)) = (I n (x), I n (x),..., I j+ (x), I j (x), I j (x),..., I (x), I j+ (x)). () In the middle of the banyan-type network, output x of the (n/) th stage is onneted to input y of the (n/ + ) th stage, where y has the following binary representation: (I n (y), I n (y),..., I (y)) = (I n/ (x), I n/ (x),..., I (x), I n (x), I n (x),..., I n/+ (x)). () Finally, for n/ + j n, output x of the j th stage is onneted to input y of the (j + ) th stage, where y has the following binary representation: (I n (y), I n (y),..., I (y)) = (I n (x), I n (x),..., I n j+ (x), I (x), I n j+ (x), I n j (x),..., I (x)). () In Figure, we show a banyan-type network with suh onnetions. Readers who are familiar with the onstrutions

8 of banyan-type networks might observe that the first n/ stages are n/ n/ n/ reversed baseline networks and the last n/ stages are n/ n/ n/ baseline networks. They are joined by a perfet shuffle in the middle. With suh an observation, we will show how one an fold this banyan-type network from the middle to onstrut a fully meshed fat-tree network in Setion V-B. Stage Fig.. A banyan-type network. It is well-known that for every input/output pair in a banyantype network, there is a unique routing path from the input to the output, and it an be used for the self-routing of a paket from the input to the output []. The unique routing path for an input/output pair is speified by setting the j th swith (in the j th stage) on the path aording to the j th most signifiant bit of the output. Speifially, onsider an input/output pair (i, o) and let u j (resp., v j ) be the input (resp., output) of the j th swith on its routing path for j n. The unique routing path for the input/output pair (i, o) is speified by starting the path from input i, i.e., u = i, and setting the j th swith on the path in suh a way that its input u j is onneted to the upper output link (resp., lower output link) if I n j+ (o) = (resp., I n j+ (o) = ) for j n, i.e., (I n (v j ), I n (v j ),..., I (v j ), I (v j )) = (I n (u j ), I n (u j ),..., I (u j ), I n j+ (o)). () In Appendix A, we show that for j n/, the binary representations of u j and v j are given as follows: (I n (u j ), I n (u j ),..., I (u j )) = (I n (i),..., I n/+ (i), I n/ (i),..., I j+ (i), I n (o), I n (o),..., I n j+ (o), I j (i)) (I n (v j ), I n (v j ),..., I (v j )) = (I n (i),..., I n/+ (i), I n/ (i),..., I j+ (i), I n (o), I n (o),..., I n j+ (o), I n j+ (o)). () Furthermore, for n/ + j n, the binary representations of u j and v j are given as follows: (I n (u j ), I n (u j ),..., I (u j )) = (I n (o), I n (o),..., I n/+ (o),..., I n j+ (o), I n (i), I n (i),..., I j+ (i), I j (i)), (I n (v j ), I n (v j ),..., I (v j )) = (I n (o), I n (o),..., I n/+ (o),..., I n j+ (o) I n (i), I n (i),..., I j+ (i), I n j+ (o)). () Therefore, we have from () (with j = n) that (I n (v n ), I n (v n ),..., I (v n )) =(I n (o), I n (o),..., I (o)), i.e., v n = o, and hene the end v n of the unique routing path is indeed output o. Theorem A permutation on Z n an be realized by the n n banyan-type network as desribed in this setion if and only if the permutation has the uniform mapping property. Proof. Let σ be a permutation on Z n. Suppose that σ has the uniform mapping property. We show that σ an be realized by the n n banyan-type network by ontradition. Assume that the routing paths for two distint input/output pairs (i, σ(i )) and (i, σ(i )), where i i, share a ommon link between stages j and j + for some j n. It follows that the two routing paths traverse the same output of a swith in the j th stage, and we have from () (in the ase that j n/) and () (in the ase that n/ + j n ) that I m (i ) = I m (i ), for j + m n, () I m (σ(i )) = I m (σ(i )), for n j + m n. () From () and (), we see that the i, i S(n j, l), where l = n j m= I j+m(i ) m. From () and (), we also see that σ(i ), σ(i ) S(j, k), where k = j m= I n j+m(σ(i )) m. It follows that {σ(i ), σ(i )} σ(s(n j, l)) S(j, k) and hene σ(s(n j, l)) S(j, k), ontraditing to σ(s(n j, l)) S(j, k) = in (). Conversely, suppose that σ an be realized by the banyantype network. To show that σ has the uniform mapping property, it suffies to show that σ(s(n j, l)) S(j, k) = for all j n, k j, and l n j. We first prove that σ(s(n j, l)) S(j, k) for all j n, k j, and l n j by ontradition. Assume that σ(s(n j, l)) S(j, k) for some j n, k j, and l n j, then there exist i i and i, i S(n j, l), suh that σ(i ), σ(i ) S(j, k). It follows from () that () and () hold. Therefore, in the ase that j n/ (resp., n/ + j n), we see from (), (), and () (resp., (), (), and ()) that the routing paths for the two distint input/output pairs (i, σ(i )) and (i, σ(i )) share a ommon link between stages j and j +, and we have reahed a ontradition. Let j n and l n j. Sine σ is a

9 permutation, we have j k= σ(s(n j, l)) S(j, k) = σ(s(n j, l)) j k= S(j, k) = σ(s(n j, l) = S(n j, l) = j. () As we have already proved that σ(s(n j, l)) S(j, k) for all k j, it is lear that () holds if and only if σ(s(n j, l)) S(j, k) = for all k j. We note that a theorem similar to Theorem was previously shown in Theorem in [] for the reverse-exhange network. Furthermore, the speifi banyan-type network in this setion an be shown to be equivalent to the baseline network and the reverse-exhange network with fixed input/output ports by using the trae and guide in Li s book []. B. Folding the Banyan-Type Network In the following, we desribe how to onstrut a n n fully meshed fat-tree network by folding the n n banyantype network speified in Setion V-A (when n is an even number). (i) For i n, the i th input and the i th output of the banyan-type network is merged as the i th leaf of the fully meshed fat-tree network. (ii) Eah link from an input port to the first stage (resp., from the last stage to an output port) and eah direted link from stage j to stage j + for j n/ (resp., n/ + j n ) in the banyan-type network is viewed as an upward link (resp., a downward link) in the fully meshed fat-tree network. (iii) For j n/ and k n j, let F j (k) (resp., F n j+ (k)) be the olletion of the swithes in the j th stage (resp., (n j + ) th stage) with indies in S(n j +, k) = {x : k j x (k + ) j } in the banyan-type network. It is easy to see that for j n/, the set of all of the n swithes in the j th stage (resp., the (n j + ) th stage) is partitioned into n j sets of swithes F j (), F j (),..., F j ( n j ) (resp., F n j+ (), F n j+ (),..., F n j+ ( n j )), eah ontaining j swithes. For j n/ and k n j, onstrut the swith at node (n j, k) of the fully meshed fat-tree network by the olletion of the j swithes in F j (k) F n j+ (k). Note that for j n/ and k n j, eah of the j swithes in F j (k) has two upward links, and hene the upward apaity of the swith at node (n j, k) is j, whih is the same as that in (). Also, for j n/ and k n j, eah of the j swithes in F n j+ (k) has two downward links, and hene the downward apaity of the swith at node (n j, k) is j, whih is the same as that in () Furthermore, it an be seen from the link onnetions in () for the first n/ stages in the banyan-type network that the two output links of eah swith in F j (k) are onneted to two different swithes in F j+ ( k/ ) for j n/ and k n j. Therefore, the upward links of the swith at node (n j, k) are all onneted to the swith at node (n j, k/ ) in the fully meshed fat-tree network for j n/ and k n j. Similarly, it an be seen from the link onnetions in () for the last n/ stages in the banyan-type network that the two output links of eah swith in F n j+ (k) are onneted to one swith in F n j+ (k) and another swith in F n j+ (k + ) for j n/ and k n j. Therefore, half of the downward links of the swith at node (n j, k) are onneted to the swith at node (n j +, k) and the other half are onneted to the swith at node (n j +, k +) in the fully meshed fat-tree network for j n/ and k n j. Finally, the link onnetions in () for the perfet shuffle in the middle of the banyan-type network guarantee that exatly one of the n/ output links of the n/ swithes in F n/ (k) is onneted to a swith in F n/+ (k ) for all k, k n/. Therefore, there is exatly one link from the swith at node (n/, k) to the swith at node (n/, k ) in the fully-meshed fat-tree network for all k, k n/ (note that this implies that there is an internal link inside the swith at node (n/, k) for k n/ ). The proof of the above laims is given in Appendix B. For example, we show in Figure the folded banyantype network. The swithes represented by solid (resp., dotted) squares are from the first (resp., seond) half of the banyan-type network. In partiular, the swith at node (, ) in the fully meshed fat-tree network ontains the swith with index in the st stage and the swith with index in the th stage of the banyan-type network. Level Fig.. Constrution of a fully meshed fat-tree network by folding the banyan-type network in Setion V-A. VI. CONCLUSION In this report, we proposed to use fat-tree networks as the swith fabris in load balaned swithes. One of the main problems in fat-tree networks is that the link apaity has to be inreased rapidly from the leaves to the root of the tree, and hene the omplexity of implementing the swithes near the root of the tree ould be very high. As we only require a fat-tree network to realize a set of N permutations needed for the implementation of N N load-balaned swithes, we showed that the implementation omplexity an be greatly redued. We first derived a lower bound on the link apaity for eah swith in a fat-tree network, and then we found a

10 set of N permutations that ahieves the lower bound by using the uniform mapping property of the bit-reversal permutation. To further redue the implementation omplexity, we also proposed a fully meshed fat-tree network that replaes the upper half of the tree by a simple mesh, and showed a fully meshed fat-tree network an be implemented by olleting swithes in a folded banyan-type network that realizes this set of N permutations. There are some researh topis that require further study. (i) Inremental update of the number of lineards: In this report, the number of input/output ports is assumed to be a power of. For the purpose of inremental update of the number of lineards, there are solutions by using twister networks [] and degenerated banyan networks []. It seems that the approahes used there might also be appliable to our setting in this report. (ii) Uniform mapping property: It is our belief that the uniform mapping property should be equivalent to the ondition previously stated in Theorem in []. As shown in Theorems in [], there are other sets of permutations that have the uniform mapping property. In partiular, if a permutation σ has the uniform mapping property, then pσ defined by (pσ)(x) = pσ(x) mod N for x Z N, also has the uniform mapping property when p is an odd number. This implies that one an ombine the bit-reversal permutation and the n permutations in a n n symmetri TDM swith to form another set of permutations that an also be used for implementing load-balaned swithes with fat-tree networks. Further development along this diretion will be reported separately. APPENDIX A PROOF OF () AND () We first note from u = i and () (with j = ) that the binary representations of u and v given by (I n (u ), I n (u ),..., I (u )) = (I n (i),..., I n/+ (i), I n/ (i),..., I (i), I (i)), (I n (v ), I n (v ),..., I (v )) = (I n (i),..., I n/+ (i), I n/ (i),..., I (i), I n (o)). By using () with j =,,..., n/ (in that order) and () with j =,,..., n/ (in that order), we an obtain the binary representations of u j and v j for j n/ as follows: (I n (u j ), I n (u j ),..., I (u j )) = (I n (i),..., I n/+ (i), I n/ (i),..., I j+ (i), I n (o), I n (o),..., I n j+ (o), I j (i)) (I n (v j ), I n (v j ),..., I (v j )) = (I n (i),..., I n/+ (i), I n/ (i),..., I j+ (i), Thus, () is proved. I n (o), I n (o),..., I n j+ (o), I n j+ (o)). After the perfet shuffle in the middle, we have from () (with j = n/), (), and () (with j = n/ + ) that the binary representations of u n/+ and v n/+ are given by (I n (u n/+ ), I n (u n/+ ),..., I (u n/+ )) = (I n (o), I n (o),..., I n/+ (o), I n (i), I n (i),..., I n/+ (i), I n/+ (i)), (I n (v n/+ ), I n (v n/+ ),..., I (v n/+ )) = (I n (o), I n (o),..., I n/+ (o), I n (i), I n (i),..., I n/+ (i), I n/ (o)). Finally, by using () with j = n/+, n/+,..., n (in that order) and () with j = n/ +, n/ +,..., n (in that order), we an obtain the binary representations of u j and v j for n/ + j n as follows: (I n (u j ), I n (u j ),..., I (u j )) = (I n (o), I n (o),..., I n/+ (o),..., I n j+ (o), I n (i), I n (i),..., I j+ (i), I j (i)), (I n (v j ), I n (v j ),..., I (v j )) = (I n (o), I n (o),..., I n/+ (o),..., I n j+ (o) I n (i), I n (i),..., I j+ (i), I n j+ (o)). Therefore, () is proved. APPENDIX B PROOF OF THE CLAIMS IN SECTION V-B In this appendix, we show the following laims in Setion V-B: (i) The two output links of eah swith in F j (k) are onneted to two different swithes in F j+ ( k/ ) for j n/ and k n j. (ii) The two output links of eah swith in F n j+ (k) are onneted to one swith in F n j+ (k) and another swith in F n j+ (k + ) for j n/ and k n j. (iii) Exatly one of the n/ output links of the n/ swithes in F n/ (k) is onneted to a swith in F n/+ (k ) for all k, k n/. (i) First we show that the two output links of eah swith in F j (k) are onneted to two different swithes in F j+ ( k/ ) for j n/ and k n j. To see this, let j n/ and k n j, and onsider a swith in F j (k) with index x S(n j +, k). From () and I n j+ (k) = (as k n j ), we see that the binary representation of x is given by (I n (x), I n (x),..., I (x)) = (, I n j (k), I n j (k),..., I (k), I (k), I j (x), I j (x),..., I (x)). () Clearly, the binary representations of the two outputs x and

11 x + of swith x in the j th stage are given by (I n (x), I n (x),..., I (x)) = (I n j (k), I n j (k),..., I (k), I (k), I j (x), I j (x),..., I (x), ), () (I n (x + ), I n (x + ),..., I (x + )) = (I n j (k), I n j (k),..., I (k), I (k), I j (x), I j (x),..., I (x), ). () Let output x (resp., x + ) of swith x in the j th stage be onneted to input y (resp., z) of swith y/ (resp., z/ ) in the (j + ) th stage. Then we have from (), (), and () that the binary representations of y and z are given by (I n (y), I n (y),..., I (y)) = (I n j (k), I n j (k),..., I (k), I j (x), I j (x),..., I (x),, I (k)), () (I n (z), I n (z),..., I (z)) = (I n j (k), I n j (k),..., I (k), I j (x), I j (x),..., I (x),, I (k)). () As we have from () and () that y/ = n j m= n j I m (k) j+m + j m= j I m (x) m, z/ = I m (k) j+m + I m (x) m +, m= m= it follows from n j m= I m(k) j+m = (k I (k)) j = k/ j and j m= I m(x) m j that k/ j y/ ( k/ + ) j, () k/ j z/ ( k/ + ) j. () As suh, we see from (), (), and () that y/ = z/ and y/, z/ S(n j, k/ ), and hene the two output links of swith x in F j (k) are onneted to two different swithes in F j+ ( k/ ). (ii) Now we show that the two output links of eah swith in F n j+ (k) are onneted to one swith in F n j+ (k) and another swith in F n j+ (k + ) for j n/ and k n j. To see this, let j n/ and k n j, and onsider a swith in F n j+ (k) with index x S(n j +, k). As before, the binary representations of the two outputs x and x + of swith x in the (n j + ) th stage are given by () and (), respetively. Let output x (resp., x + ) of swith x in the (n j + ) th stage be onneted to input y (resp., z) of swith y/ (resp., z/ ) in the (n j + ) th stage. Then we have from (), (), and () that the binary representations of y and z are given by (I n (y), I n (y),..., I (y)) = (I n j (k), I n j (k),..., I (k),, I j (x), I j (x),..., I (x), I (x)), () (I n (z), I n (z),..., I (z)) = (I n j (k), I n j (k),..., I (k),, I j (x), I j (x),..., I (x), I (x)). () As we have from () and () that y/ = n j m= n j I m (k) j+m + j m= z/ = I m (k) j+m + j + m= I m (x) m, j m= I m (x) m, it follows from n j m= I m(k) j+m = k j = k j and j m= I m(x) m j that k j y/ (k + ) j, () (k + ) j z/ (k + ) j. () As suh, we see from (), (), and () that y/ = z/, y/ S(n j +, k), and z/ S(n j +, k + ), and hene the two output links of swith x in F n j+ (k) are onneted to one swith in F n j+ (k) and another swith in F n j+ (k + ). (iii) Finally, we show that exatly one of the n/ output links of the n/ swithes in F n/ (k) is onneted to a swith in F n/+ (k ) for all k, k n/. To see this, let k n/, and onsider a swith in F n/ (k) with index x S(n/ +, k). It an be seen that () () still hold for j = n/, and hene the binary representations of x and its two outputs x and x + are given by (I n (x), I n (x),..., I (x)) = (, I n/ (k), I n/ (k),..., I (k), I n/ (x), I n/ (x),..., I (x)), () (I n (x), I n (x),..., I (x)) = (I n/ (k), I n/ (k),..., I (k), I n/ (x), I n/ (x),..., I (x), ), () (I n (x + ), I n (x + ),..., I (x + )) = (I n/ (k), I n/ (k),..., I (k), I n/ (x), I n/ (x),..., I (x), ). () Let output x (resp., x+) of swith x in the (n/) th stage be onneted to input y (resp., z) of swith y/ (resp., z/ ) in the (n/ + ) th stage. Then we have from (), (), and

12 () that the binary representations of y and z are given by (I n (y), I n (y),..., I (y)) = (I n/ (x), I n/ (x),..., I (x),, I n/ (k), I n/ (k),..., I (k), I (k)), () (I n (z), I n (z),..., I (z)) = (I n/ (x), I n/ (x),..., I (x),, I n/ (k), I n/ (k),..., I (k), I (k)). () From () and (), we have y/ = z/ = n/ m= n/ m= I m (x) n/+m + n/ m= I m (x) n/+m + n/ + I m (k) m, n/ m= I m (k) m. Let k = n/ m= I m (x) m (note that k ( n/ ) = n/ ). It then follows from n/ m= I m(k) m n/ that k n/ y/ (k + ) n/, () (k + ) n/ z/ (k + ) n/. () Sine k, k + n/, we see from (), (), and () that y/ S(n/ +, k ) and z/ S(n/ +, k + ), and hene the two output links of swith x in F n/ (k) are onneted to one swith in F n/+ (k ) and another swith in F n/+ (k +). As I n/ (x), I n/ (x),..., I (x) goes from (,,..., ), (,,..., ),..., (,,..., ) (note that there are n/ swith in F n/ (k)), we see from k (, ), (, ),..., ( n/, n/ ). As suh, exatly one of the n/ output links of the n/ swithes in F n/ (k) is onneted to a swith in F n/+ (k ) for all k n/. = n/ m= I m (x) m that (k, k + ) goes from [] C.-M. Lien, C.-S. Chang, J. Cheng, D.-S. Lee, and J.-T. Liao, Using banyan networks for load-balaned swithes with inremental update, in Proeedings IEEE International Conferene on Communiations (ICC ), Cape Town, South Afria, May,. [] C. E. Leiserson, Fat-trees: Universal networks for hardware-effiient superomputing, IEEE Transations on Computers, vol., pp., Otober. [] M. Al-Fares, A. Loukissas, and A. Vahdat, A salable, ommodity data enter network arhiteture, in Proeedings ACM Speial Interest Group on Data Communiation (SIGCOMM ), Seattle, WA, USA, August,. [] H. Hossain, M. Akbar, and M. Islam, Extended-butterfly fat tree interonnetion (EFTI) arhiteture for network on hip, in Proeedings IEEE Paifi Rim Conferene on Communiations, Computers and Signal Proessing (PaRim ), Vitoria, B.C., Canada, August,. [] C. -L. Wu and S. -Y. Feng, The Reverse-Exhange Interonnetion Network, IEEE Transations on Computers, vol. -, pp., September. [] S.-Y. R. Li, Algebrai Swithing Theory and Broadband Appliations, San Diego, CA: Aademi Press,. REFERENCES [] C.-S. Chang, D.-S. Lee, and Y.-S. Jou, Load balaned Birkhoff-von Neumann swithes Part I: One-stage buffering, Computer Communiations, vol., pp.,. [] I. Keslassy, S.-T. Chung, K. Yu, D. Miller, M. Horowitz, O. Sloggard, and N. MKeown, Saling Internet routers using optis, in Proeedings ACM Speial Interest Group on Data Communiation (SIGCOMM ), Karlsruhe, Germany, August,. [] I. Keslassy, S.-T. Chung, and N. MKeown, A load-balaned swith with an arbitrary number of lineards, in Proeedings IEEE International Conferene on Computer Communiations (INFOCOM ), Hong Kong, China, Marh,. [] Y. Shen, S. Jiang, S. S. Panwar, and H. J. Chao, Byte-foal: a pratial load-balaned swith, in Proeedings IEEE Workshop on High Performane Swithing and Routing (HPSR ), Hong Kong, China, May,. [] J.-J. Jaramillo, F. Milan, and R. Srikant, Padded frames: a novel algorithm for stable sheduling in load-balaned swithes, IEEE/ACM Transations on Networking, vol., pp., Otober. [] C.-L. Yu, C.-S. Chang, and D.-S. Lee, CR swith: A load-balaned swith with ontention and reservation, IEEE Transations on Communiations, vol., pp., Otober. [] C.-M. Lien, C.-S. Chang, J. Cheng, D.-S. Lee, and J.-T. Liao, Twister networks and their appliations to load-balaned swithes, in Proeedings IEEE International Conferene on Computer Communiations (INFOCOM ), San Diego, CA, USA, Marh,.