# Generalized DCell Structure for Load-Balanced Data Center Networks

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2 of servers in a DCell k. Then, to construct a DCell k,wetake t k 1 +1 DCell k 1 s and connect them in such a way that (a) there is exactly one edge between every pair of distinct DCell k 1 s, and (b) we have added exactly one edge to each vertex. Requirement (a) means that, if we contract each DCell k 1 to a single point, then the DCell k is a complete graph on t k 1 +1 vertices. This imitation of the complete graph is what we believe gives the DCell structure many of its desirable properties. Requirement (b) is the reason why we must have exactly t k 1 +1 DCell k 1 sinadcell k. It ensures that every server has the same number of links and is the reason why DCell scales doubly exponentially in the number of servers. This is precisely the point of divergence from the original DCell proposal. There, one specific way of meeting requirements (a) and (b) was proposed, which we name the α connection rule later on. But there are many other possibilities. Before we can make this idea more precise, we need to discuss how we label the vertices. Each server is labeled by a vector id [a k,a k 1,..., a 0 ].Here a k specifies which DCell k 1 the server is in; a k 1 specifies which DCell k inside that DCell k 1 the server is in; and so on. So 0 a 0 <n, and for i 1, wehave0 a i <t i We can convert a vector id to a scalar uid (unique identifier) as follows: u = a 0 + a 1 t 0 + a t a k t k 1. (1) Note that we have 0 u t k 1. Most often, we will label servers just by [a, b] where a a k is the number of the DCell k 1, and b is the uid corresponding to [a k 1,..., a 0 ]. Using these notions, we can define mathematically what a connection rule is. Namely, it is a perfect matching ρ L of the vertices {0,..., t L 1 } {0,..., t L 1 1} that must satisfy the following two properties: 1) ρ L must be the identity, so that the graph is undirected. (This is also implicit in the term perfect matching.) ) For all a c, there exist b and d such that ρ L ([a, b]) = [c, d]. This ensures that there is a L-level link between each pair of distinct DCell L 1 s. This encapsulates precisely the requirements (a) and (b) above. We summarize the construction in the following definition. Definition 1: A generalized DCell with parameters n, k 0, and R =(ρ 1,..., ρ k ) is constructed as follows: ADCell 0 is a complete graph on n vertices. From here we proceed recursively until we have constructed a DCell k : A DCell L consists of t L 1 +1 DCell L 1 s, where t L 1 is the number of vertices in a DCell L 1. Edges are added according to the connection rule ρ L. B. Connection Rules In this section, we give four examples of connection rules. For n =, k =, the graphs are shown in Fig. 1. α. The connection rule for the original DCell is { [b +1,a] if a b, α L :[a, b] [b, a 1] if a>b. β. A mathematically simple connection rule is () β L :[a, b] [a+b+1 (mod t L 1 +1),t L 1 1 b]. (3) γ. For t L 1 even, we can leave b unchanged by the switch, except for a change inside the DCell 0. [a + b (mod t L 1 +1),b 1] if b is odd, γ L :[a, b] [a (b +1)(mod t L 1 +1),b+1] if b is even. (4) δ. For t L 1 even: [a + b +1(mod t L 1 +1),b+ tl 1 ] if b< tl 1 δ L :[a, b], [a b + tl 1 1 (mod t L 1 +1),b tl 1 otherwise. (5) In the rest of this paper, when the specific connection rule used is not important, we will speak of just DCells. If we need to make reference to a specific connection rule, we will speak e.g. of α-dcells, meaning DCells with R =(α 1,..., α k ).In this context, we should clarify why the requirement that t k 1 be even is not a practical problem for the γ and δ connection rules. It turns out that t k is even for k 1. Thus, for even n, there is no problem, while for odd n, only a different rule for the 1-level links is needed. Since almost all real-world switches have an even number of ports, we will restrict ourselves to even n whenever convenient. III. GRAPH PROPERTIES In this section, we give expressions and bounds for the number of servers and the diameter. We also investigate the symmetries of the different connection rules. Due to the page limitation, we omit the proofs of the theorems in Sec. III and Sec. IV, which can be found in [6]. A. Number of Servers No closed-form expression for the exact number of servers t k in a DCell k is known. However, it is clear from the DCell construction that t k satisfies the following recurrence relation: t k+1 = t k (t k +1) (6) t 0 = n. This permits t k to be easily and quickly computed for small n and k. Refer to Table I for values of t k. Following a hint by D. E. Knuth in [8], we use the methods of [1] to solve Equation (6), leading to the following theorem. Theorem 1: We have t k = c k, (7) ]

3 3 (a) α-dcell (b) β-dcell (c) γ-dcell (d) δ-dcell Fig. 1: Generalized DCells with different connection rules for n =,k =.DCell 1 groupings are shown in different colors. where the constant c is well approximated by the first few terms of the infinite product ( c = n + 1 ) ( ) 1/ i i=0 4 ( ) t i + 1. (8) B. Diameter It is desirable for a data center network to have as small a diameter as possible. For if the diameter is large, then communication between some pairs of servers will necessarily be slow, regardless of the routing algorithm used. 1) An Upper Bound: In [3], it is shown that the diameter of α-dcells satisfies D k+1 1. (9) In fact, the proof carries over easily to all generalized DCells since it uses only the DCellRouting which is independent of the connection rule. ) A Lower Bound: A well-known (e.g. [4, p.38]) lower bound on the diameter of a graph G with N vertices and maximum degree Δ is D log N log Δ. (10) Using Theorem 1, this inequality leads to the following theorem. Theorem : The diameter D is bounded below by D k log c log(n + k 1). (11) Since c is only slightly larger than n + 1, Theorem can be used to show that D k 1 (1) when k n +1. Since n, this includes the realistic cases k =3and k =4. Together with inequality (9), this narrows the diameter down to within a factor of 4. 3) Dependence on Connection Rule: Table I compares the diameter for different connection rules for some small values of n and k. The diameters of structures constructed by the new connection rules are significantly smaller than that of the original DCell connection rule. For example, for n =and k =4, the diameter of the original DCell is 7, whereas it is n k t k α β γ δ , , ,63, TABLE I: Comparison of the diameters of different connection rules. at most 17 in the new structures. Low diameter leads to higher capacity and also leads to better load-balancing in our case. C. Symmetry Symmetry is of importance to data center networks primarily because it facilitates the initial wiring. 1) α-dcell: It turns out that, at least for n 3, every graph automorphism of a generalized DCell respects its leveled structure; that is, a DCell L will be mapped to another DCell L for all L, and all link levels are preserved. Depending on the connection rule, however, there can be much stronger requirements on graph automorphisms. For α-dcell, it appears that there is only one nontrivial symmetry. Theorem 3: For k and 3 n 6, the automorphism group of an α-dcell k is isomorphic to C, the Cycle group of order. ) Other connection rules: It is straightforward to prove the following theorem. Theorem 4: Suppose the k-level connection rule of a DCell is of the form: ρ k :[a, b] [a + b +1(mod t k 1 +1),g(b)], (13) where g is any permutation on {0,..., t k 1 1}. Then the map τ :[a, b] [a +1(mod t k 1 +1),b] (14) is a graph automorphism. τ generates a cyclic subgroup of the automorphism group of order t k Proof: We have τ([a, b]) = [a +1(mod t k 1 +1),b] (15) [(a +1)+b +1(mod t k 1 +1),g(b)] (16) =[(a + b +1)+1(mod t k 1 +1),g(b)] (17) = τ([a + b +1(mod t k 1 +1),g(b)]). (18)

4 4 As for the second assertion, clearly τ is of order t k 1 +1, since for no smaller number c do we have a+c a (mod t k 1 +1). Note that β, γ, and δ are all of this form. Hence, these connection rules lead to significantly more symmetric graphs than the α rule. This group of symmetries consists of exactly the rotational symmetries that are apparent in Fig. 1. IV. ROUTING Since link-state routing is feasible only for small networks [7], it is important to have an efficient, locally computable routing algorithm. In [3], a recursive routing algorithm called DCellRouting is presented for the α-dcell. A similar algorithm can be devised for any generalized DCell. In this section, we state a number of results concerning the path length distribution and flow distribution when using DCellRouting. A. Path-Length Distribution As shown in [3], the longest path using DCellRouting is k+1 1. Fix a vertex v in a DCell k and let Ni k denote the number of servers that are exactly i hops away from v in DCellRouting. It turns out that Ni k is independent of the choice of v, asthe following theorem shows. Theorem 5: Ni k satisfies N0 k =1, (19) Ni 0 = δ i0 +(n 1)δ i1, (0) N k i i 1 i + = N k 1 j=0 N k 1 j N k 1 i 1 j, for k, i 1. (1) Here δ ij is the Kronecker delta, which is 1 if i = j and 0 otherwise. B. Flow Distribution Theorem 6: In all-to-all communication using DCellRouting, the number of flows F L carried by a L-level link is t k 1 for L = k, k 1 t L 1 (1 + t j ) for 1 L k 1, F L = j=l () k 1 (n 1) (1 + t j ) for L =0. j=0 Using Theorem 6 and Theorem 1, we can derive from the exact expression for F L a fairly tight upper bound that is more readily compared to the previously known bound k L t k [3]. Corollary 1: We have F 0 < n 1 n + 1 For 1 L<k,wehave F L < t L t L 1 t L + 1 k (t k +0.6) = n 1 n + 1 k t k (1 + o(1)). (3) k L (t k +0.6) = t L t L 1 t L + 1 k L t k (1+o(1)). (4) n k DCR SP-α SP-β SP-γ SP-δ (a) Mean n k DCR SP-α SP-β SP-γ SP-δ (b) Standard deviation TABLE II: Expected value and standard deviation of path length distribution. DCR and SP stand for DCellRouting and shortest path routing, respectively. Finally, we point out that a simple double counting argument shows that the expected value of the path-length distribution is related to the flow distribution as follows. Theorem 7: The expected value of the path-length distribution is given by k L=0 E = F L t k 1. (5) V. EXPERIMENTAL RESULTS In this section, we compare empirically the performance of DCellRouting and shortest path routing for the various connection rules. The simulations were necessarily restricted to small n and k; due to the the doubly exponential growth of DCells, these are the only realistic values of n and k. A. Path-Length Distribution Table II compares, for some small n and k, the mean and standard deviation of the path length distribution when using DCellRouting or shortest path routing. Shortest path routing for the γ connection rule has the lowest expected value and standard deviation, making it the rule of choice. Fig. shows the different path length distributions for n =4and k =3. The other cases look similar. B. Flow Distribution The flow distributions by link level using shortest path routing and DCellRouting are shown in Fig. 3 for n =4,k =3. Again, this figure is representative of the other cases as well. We observe that DCellRouting does a poor job of load-balancing. Shortest path routing for α-dcell does better than DCellRouting on average, but has significant bottlenecks that exceed even those of DCellRouting. Shortest path routing for the β, γ, and δ connection rules does better on average and also exhibits very good load-balancing: there are no significant bottleneck links. We believe the asymmetry of α-dcell leads to bottlenecks in the flow distribution and the symmetry in β,δ,γ-dcell leads to

5 5 Fig. : Path length distribution for n =4and k =3. (a) α-dcell balanced flow distribution. It appears that γ is again the rule of choice for all-to-all communication using shortest path routing. VI. CONCLUSION AND FUTURE WORK In this paper, we present a generalized DCell framework and several structures derived from this framework. We have studied the common graph properties such as scaling, diameter, and symmetry. We also show that the newly introduced structures have much smaller diameter and better load-balancing properties than the original DCell. The generalized DCell structure introduces a new and huge degree of freedom: the choice of connection rule. We explored only a few of the many possible connection rules chosen largely for their mathematical simplicity and symmetry and we are far from having a full understanding of how to design an optimal connection rule. We think the further exploration of this new degree of freedom has the potential to lead to great improvements of the DCell structure. The improved loadbalancing and smaller diameters we found are strong evidence for this. Our future work is to design a practical and scalable shortestpath routing algorithm for the generalized DCell framework. We cannot directly use link-state based routing protocols such as OSPF, since they can only scale to at most a thousand routers. In our future work, we plan to design a shortest-path based routing protocol by taking advantage of the fact that the network topology is known in advance in data centers. We also plan to program servers to handle network failures. ACKNOWLEDGMENT This work was performed when Markus Kliegl, Jason Lee, Jun Li, and Xinchao Zhang were visiting students and David Rincón was a visiting academic mentor for the MSRA and UCLA IPAM RIPS-Beijing program at Microsoft Research Asia. The four students are equal contributors. Funding was provided by the NSF and MSRA. REFERENCES [1] A. V. Aho and N. J. A. Sloane. Some Doubly Exponential Sequences, Fibonacci Quarterly, Vol. 11, pp , (b) γ-dcell Fig. 3: Distribution of flows by link level using all-to-all communication for n =4and k =3. The error bars indicate the maximum and minimum values. The performance of β-dcell and δ-dcell is similar to that of γ-dcell. [] C. Guo, G. Lu, D. Li, H. Wu, X. Zhang, Y. Shi, C. Tian, Y. Zhang, and S. Lu. BCube: A High Performance, Server-centric Network Architecture for Modular Data Centers, in Proc. of ACM SIGCOMM, 009. [3] C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu. DCell: A Scalable and Fault-Tolerant Network Structure for Data Centers, in Proc. of ACM SIGCOMM, 008. [4] F. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, 199. [5] D. Li, C. Guo, H. Wu, K. Tan, Y. Zhang, and S. Lu. FiConn: Using Backup Port for Server Interconnection in Data Centers, in Proc. of IEEE INFOCOM, 009. [6] M. Kliegl, J. Lee, J. Li, X. Zhang, C. Guo, D. Rincon. The Generalized DCell Structures and Their Graph Properties, Microsoft Research Technical Report, MSR-TR [Online]. Available: microsoft.com/apps/pubs/?id=10319 [7] J.T.Moy.OSPF: Anatomy of an Internet Routing Protocol. Addison- Wesley Professional, [8] N. J. A. Sloane, Ed. Sequence A The On-Line Encyclopedia of Integer Sequences, 009. [9] J. Snyder. Microsoft: Datacenter Growth Defies Moore s Law, PC- World, 007. [Online]. Available: microsoft datacenter growth defies moores law.html [10] M. Al-Fares, A. Loukissas, and A. Vahdat, A Scalable, Commodity Data Center Network Architecture, in Proc. of ACM SIGCOMM, 008.

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