Multilayer Structure of Data Center Based on Steiner Triple System


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1 Journal of Computational Information Systems 9: 11 (2013) Available at Multilayer Structure of Data Center Based on Steiner Triple System Jianfei ZHANG 1, Zhiyi FANG 1, S. Q. ZHENG 2, Guannan QU 1, 1 College of Computer Science and Technology, Jilin University, Changchun , China 2 Department of Computer Science, University of Texas at Dallas, Richardson TX 75083, USA Abstract Topology structure is considered as a primary factor which largely influences the performance of data center (DC). There are lots of results which are deployed graph theories or block design theories. In this paper, we develop an innovate DC topology structure named Multilayer Steiner Triple System, abbr. MST S. We will prove that MSTS is able to be scaled in faster than double exponential speed and with short diameter. This significant property makes MSTS be built with large number of commercial connection equipments without sacrificing communication performance. Keywords: Data Center; Network Topology; Steiner Triple System; Scalable 1 Introduction In recent years, as Internet getting popular, online application services, such as online games, ebusinesses, online videos, SNS, are becoming a hot spot. All of these services are running around a core of data. Enterprises build their own data center (DC) for storing and processing huge amount of data. With data increasing, data centers have to be extended. It is no doubt that the topology structure of data center is precisely the key point to meet this demands. A well designed topology structure is able to connect as many computation units as possible, with high scalability and short diameter. These requirements raise challenges of designing DC topology structure. Numbers of works were proposed. Bus interconnection network [1] is a networks that the buses provide communication medium between computation units. The solutions of Bus networks are reduce to solving hypergraph. Since it is a pointtopoint networks, when it is scaled, its diameter increases rapidly. Hypercube [2] structure C n is a multidimension structure which includes 2 n computation units where n is the number of the dimensions and the diameter of the interconnection networks. But in a C n, each node has to connect with n other nodes. Every node is required to deploy n communication ports, such that it is not suitable for a large scale Corresponding author. address: (Guannan QU) / Copyright 2013 Binary Information Press DOI: /jcis6123 June 1, 2013
2 4372 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) networks. Complete graph structure K n includes n computation units, and these units link to each other. In this way, the distance between any pair of computation units is exactly one. However, every computation unit has to deploy (n 1) communication port. For K n with large size n, n! connection wires are hard to be implemented. Nstar structure also faces the scalable issue. In Nstar structure S n [3, 4], n! computation units by n!(n 1)/2 switch equipments. DCell structure [5, 6] is a novel data center topology structure which is constructed with miniswitches. It can be scaled in doubly exponential speed as the node degree increase and keep small diameter. In this paper, we present an innovate topology structure called Multilayer Steiner Triple System (MST S). We will prove that, the number of computation units connected by MST S increases even faster than doubly exponentially, however, the diameter of a MST S l is only 2 l+1 1. Comparing to DCell structure, with the same diameter, MST S can connect times number of computation units. It improves the performance of the communication significantly. And computation units can be connected by lowend switch equipments. It largely reduces the implementation cost. 2 Preliminaries 2.1 Block design Combinatorial Design [7] is a way of selecting subsets from a finite set to meet specific conditions, and the conditions may involve set members, set intersections and so on. Block Design is included in Combinatorial Design as one of important parts. A Design is defined on a finite set X = {x 1, x 2,, x v }, where element x i is called Treatment or Variety. And all treatments are partition into a nonempty collection of subsets of X. The collection of subsets is like B = {B 1, B 2,, B b }, where B i is called a Block. The number of blocks in which a treatment appears is called the valence of that treatment. The number of blocks in which any pair of treatments appears is called the covalence of that pair of treatments. A Regular Design is a Block Design which satisfies two conditions : I: all blocks have the same size of k, k 2; II: all treatments have the same valence, such that every treatment appears in the same number r of blocks, where r is called replication number. From above definitions, we have bk = vr, (1) where v is the number treatments, b is the number of blocks, r is the replication number and k is the size of block. If all v treatments appear in the same block of a design, such block is called Complete. But if there is at least one incomplete block, we call the design is Incomplete. In an incomplete block design, assuming x and y are two different treatments, their covalence is represented as λ xy. We say a block design is balanced, if each pair of treatments has the same covalence, and they appear in the same number λ of blocks. λ is called the index of the design. The block design which has been studied more frequently is Balanced Incomplete Block Design, abbr. BIBD. It is represented as a fivetuple (v, b, r, k, λ). And in a (v, b, r, k, λ)bibd, we have
3 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) r(k 1) = λ(v 1). (2) 2.2 Steiner triple system Steiner Systems is a type of block design with λ = 1. A steiner systems, denoted by S(t, k, v), is based on a set V with cardinality v. and V is partitioned into a subsets collection B who has size k. Each telement subset of V is contained exactly in one member of B. In terms of hypergraph, each element in V is a vertex, and each ksubset in B is a hyperedge, S(t, k, v) is a uniform hypergraph who has v nodes, and each edge contains k nodes. S(2, 3, v), is a special steiner systems with t = 2 and k = 3. We use ST S(v) to present a steiner triple system of cardinality v. A ST S(v) is built on a elements set of v. The v elements are partition into b blocks and each block contains 3 elements. Each two elements appears in exactly one of b blocks. A S(2, 3, v) has C(v, 2) 2subsets and C(v, 3) 3subsets. Every 3subset contains C(3, 2) = 3 2subsets. Thus a solution to S(2, 3, v) has b = C(v, 2)/C(3, 2) 3subsets. Resolving the equation, we get: b = v(v 1). (3) 6 r is the number of 3subsets that an element appears. Apparently, according to (1), we have rv = 3b, i.e. Replacing the b in Equation (4) by Equation (3), we have r = 3b v. (4) and r = v 1 2, (5) v = 2r + 1. (6) The following is an example of ST S(9) on set V = {1, 2, 3, 4, 5, 6, 7, 8, 9}, ST S(9) = {{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {1, 8, 9}, {2, 4, 6}, {2, 5, 8}, {2, 7, 9}, {3, 4, 9}, {3, 5, 7}, {3, 6, 8}, {4, 7, 8}, {5, 6, 9}}, where ST S(9) is based on a elements set of 9. The elements are partition into b = 12 blocks, and each elements appears in r = 4 blocks. Any pair of elements presents in one block. The necessary and sufficient condition [8] for the existence of ST S(n) is n 1 or 3 ( mod 6 ). (7)
4 4374 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) And we say that an positive integer n is admissible if and only if n is congruent to 1 or 3 mod 6. Apparently, the hypergraph representation of ST S(v) is a regular, uniform and linear hypergraph. A ST S(v) hypergraph can connect v vertices with degree of v 1 v(v 1). It has hyperedges with 2 6 rank 3, and any two different nodes in hypergraph are connected by exactly one common edge. 3 Multilayers Structure Based on STS In this article, we consider a hyper networks modeled by ST S(n) as a basic interconnection networks, and construct a multilayers interconnection hypernetworks recursively by STShyper networks, denoted by MST S. The MST S solution is an excellent hypernetworks with advantages of high scalability and short communicating path. 3.1 M ST S structure The computation units in MST S have multiple network ports and are connected by lowend connection equipments. In M ST S computation units communicate with each other via commercial connection equipments, such as small scale switches. The whole hypernetworks is built recursively, such that highlevel MST S l is constructed by lowlevel MST S l 1. The MST S 0 is constructed by n computation units and a nports switch, all computation units connect to the switch directly. MST S 0 is the basic block to build larger MST S. Considering a MST S 0 as a single unit, we construct level1 MST S 1 using v MST S 0 s. And in a MST S 1, the rules of connections among all MST S 0 are refer to the solution of ST S(v). Since one MST S 0 is as an element of a ST S(v), each element presents in r = n 3subsets. By the Equation (6), MST S 0 consists of 2r + 1 MST S 0 s. Certainly, the value of 2r + 1 should satisfy the condition of Equation (7). We will prove this later. According to the Equation (3), all 2r + 1 MST S 0 s are connected via v(v 1) 3ports switches. We label MST S 6 0 with i where i [1, v]. For every 3subset {i, j, k} in solution of ST S(v), we pick one computation unit which has not connect to any level1 switches from i, j and k MST S 0, and connect them by one 3port switch. (1,2,4) (1,3,7) (1,5,6) (2,3,5) (2,6,7) (3,4,6) (4,5,7) Fig. 1: A MST S 1 hypernetworks with n=3.
5 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) An example of MST S 1 with n = 3 is shown in Fig 1. In MST S 1, we treat MST S 0 as a virtual unit, and all MST S 0 s form hypernetworks according to the solution of ST S(v). The level2 and higher MST S l are constructed in the same way. If the MST S l 1 has been constructed and each MST S l 1 has r l 1 computation units, we can build a MST S l with v l = 2r l MST S l 1 s. As construction of MST S 1, we treat MST S l 1 as a virtual unit, and all MST S l 1 s form a solution of ST S(v l ). Let r l denote the total amount of computation units of MST S l, we have and v l = 2r l 1 + 1, (8) r l = v l r l 1. (9) for l 1. The basic block MST S 0 is different with v 0 = 1 and r 0 = n, where n is the number of computation units in MST S 0. Since in different level i, v i, the number of virtual node MST S i 1, are different. In order to make sure that for i 1, v i always satisfies the condition (7), we have follow theorem. Theorem 1 If n 0 or 1 (mod 3), then ST S(V l ) exists for l 1. Proof According to the necessary and sufficient condition for existence of a ST S(v), there exists a solution of ST S(v), if and only if n 1or3 ( mod 6 ), where n is the number of connection equipment ports in MST S 0. If n 0 or 1 ( mod 3 ), then r 0 = n 0 or 1 ( mod 3 ). We consider following two cases: Case I: suppose r 0 = 3m, m = 1, 2,, n,. According to (6), we have v 1 = 2r 0 + 1, thus v 1 = 6m + 1, m = 1, 2,, n,. Obviously, v 1 satisfies the condition (7). According to Equation (9), r 1 = v 1 r 0, such that r 1 = 3(6m 2 + m). Apparently, r 1 0 (mod 3). Let m 1 = 6m 2 + m, we have r 1 = 3m 1. And v 2 = 6m satisfies the condition (7). In the same way, for i 1, v i = 6m i satisfies the condition (7), where m i 1 = 6m 2 i 2 + m i 2. Case II: suppose r 0 = 3m + 1, m = 1, 2,, n,. According to Equation (6), we have v 1 = 2r 0 +1, thus we have v 1 = 6m+3, m = 1, 2,, n,. Therefore v 1 satisfies the condition (7). From Equation (9), we have r 1 = v 1 r 0, where r 1 = 3(6m 2 + 5m + 1). Let m 1 = 6m 2 + 5m + 1, we have r 1 = 3m 1. v 2 = 6m satisfies the condition (7). As same as case I, for i 1, v i satisfies the condition (7). Concluding above two cases, theorem 1 is proved. 3.2 Properties of MSTS Firstly, we investigate how many computation units a MST S l can be included. Theorem 2 2 l (n )2l 1 4 < r l < 2 l (n + 1) 2l 1.
6 4376 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) Proof From Equation (9) and (8), we have r l = r l 1 (2r l 1 + 1). Thus, we get that is r l = 2(r l )2 1 8 > 2(r l )2 1 4, Solving above formulation iteratively, we get r l > 2(r l )2. and r l > 2 l (n )2l 1 4, (10) Equation (11) can be simplified as r l = 2(r l 1 + 1) 2 3r l 1 2 < 2(r l 1 + 1) 2 1. (11) Solve it iteratively, we get r l + 1 < 2(r l 1 + 1) 2. From Equation (10) and Equation (12), Theorem 2 is proved. r l < 2 l (n + 1) 2l 1. (12) From theorem 2, we find out that the number of computation units included in a MST S l increases even more rapidly than doubly exponentially. Table 1 shows some examples. The first column of Table 1 lists the number of computation units in MST S 0 n. The other three columns are value of l, which is the number of layers of whole MST S structure from 1 to 3 respectively. And the numbers in other cells are the number of computation units connected by a MST S with corresponding n and l connects. For instance, when n = 6 and l = 3, the number of computation units in a MST S 3 is more than 299million. With the same n, as layer number increasing, the amount of computation units scales hundreds even thousands times. Not only the amount is increasing, but the increasing speed is accelerating. In columns, a small addition to n leads to big expanding of amount of computation units in any layer. Although every computation unit has to deploy more than one communication port, compared to the size of the whole network being extended to, the number of links that one computation unit has is still small. The diameter of a networks is defined as the length of longest shortest path between every pair of nodes. The length of communication path between any pair of computation units in a
7 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) Table 1: Computation Unit Number in MST S l n l = 1 l = 2 l = , ,941, ,653 6,880,407,471 data center is not longer than the diameter. It means that the diameter is a upper bound of the path length. Small diameter provides efficient communication. The following theorem proves the diameter of the MST S l networks. Theorem 3 The diameter of a llevel MST S l is 2 l+1 1. Proof As we know, for 1 i < l, any two different MST S i s are connected by one connection equipment. Without loss of generality, let s and d be source and destination computation units respectively, where s and d belong to different MST S l 1. We use n 1 and n 2 to denote the medium computation units which are in the same MST S l 1 as s and d respectively. In the case of longest path, s n 1 and d n 2. The problem is reduced to to find the subpaths between s to n 1 and n 2 to d. The length is Solving Equation (13) recursively, we get Theorem 3 is proved. Len l = 2Len l (13) Len l = 2 l+1 1. Diameter is one of important factors for designing a data center topology structure. From theorem 2, we have r l > 2 l n 2l. Such that 2 l < log r l n logn 2l < log r l n. The diameter of MST S is less than 2 log r l n 1. As a comparison, a fattree structure has a diameter of 2 log N 2, where N is the number of nodes. Let D ft and D msts be the diameter of fattrees and MST S respectively. With same number of nodes, r l = N, we have D ft /D msts log n 2. According to theorem 1, n = 3m or 3m + 1 for m = 1, 2, 3,, n,, thus [D ft /D msts ] log 3m 2 or [D ft /D msts ] log 3m+1 2. A hypercube structure has a diameter of log N 2, with N is the number of nodes. And in the same way, we have a similar result where D hc is the diameter of a hypercube. [D hc /D msts ] log 3m 2 /2 or [D hc /D msts ] log 3m+1 2 /2. From Table 2, no matter what the value of n is, to connect same number of nodes, the diameter of Fattrees structure is multiple times diameter of MST S. And the same result is held with
8 4378 J. Zhang et al. /Journal of Computational Information Systems 9: 11 (2013) Table 2: Diameter Ratio m 3m [D ft /D msts ] [D hc /D msts ] 3m + 1 [D ft /D msts ] [D hc /D msts ] 1 3 log 3 2 (log 3 2)/2 4 log 4 2 (log 4 2)/2 2 6 log 6 2 (log 6 2)/2 7 log 7 2 (log 7 2)/2 3 9 log 9 2 (log 9 2)/2 10 log 10 2 (log 10 2 )/2 hypercube structure, when n > 4. Comparing to a DCell structure whose diameter is 2 k+1 1, k is the number of levels of a DCell, MST S can connect 2 l times computation units than DCell with the same diameter. In another words, diameter of a MST S is shorter than DCell with the same scale. 4 Concluding Remarks In this paper, we presented an innovative topology networks of data center. called Multilayer Steiner Triple System (MST S). The construction of MST S structure is based on the theory of Steiner Triple System which is a type of block design. In M ST S structure, computation units are connected by lowend commercial connection equipments. The level0 consists of one nports connection equipments and n computation units. And in each higher level, MST S l consists of MST S l 1 s and organizes them according to solution of a ST S. Each computation unit has one leveli links, which connects with two other computation units in different MST S i 1 s. We proved that M ST S has outstanding properties on scalability and communication path. By selecting n carefully, an available solution to construct the M ST S for any layer can be guaranteed. The amount of computation units scales by speed of times double exponent, and the diameter of a MST S is also competitive. References [1] JC. Bermond, F.O. Ergincan, Bus Interconnection Networks, Discrete Applied Mathematics, issue 68, pp. 115, [2] Y. Saad, M.H. Schultz, Topological Properties of Hypercubes, IEEE Trans. on Computers, issue 7, vol. 37, pp , [3] S.G. Akl, T. Wolff, Efficient sorting on the star graph interconnection network, Telecommunication System, issue 10, pp. 320, [4] D.K. Saikia, R.K. Sen, Two ranking schemes for efficient computation on the star interconnection network, IEEE Trans. on Parallel and Distributed Systems, vol 7, no.4, pp , [5] C.Guo, H.Wu, K.Tan, L.Shi, Y.Zhang, S.Liu, DCell: A Scalable and FaultTolerant Network Structure for Data Centers, Proc. of the ACM SIGCOMM 2008 conference on Data communication, August 1722, [6] M. Kliegl, J.Lee, J. Li, X. Zhang, C. Guo, D. Rincon, Generalized DCell Structure for Load Balanced data center networks, proc. of Infocom 2010, [7] W.D. Wallis, Combinatorial Designs, CRC Press1988. [8] Stephen J. Hartley, Aaron H. Konstam, Using Genetic Algorithms to Generate Steiner Triple Systems, Proc. of the 21st Annual ACM Computer Science Conference, 1993.
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