Many-to-Many Disjoint Path Covers in DCell by Shirley Wang

www.seipub.org/ie

Information Engineering (IE) Volume 4, 2015 doi: 10.14355/ie.2015.03.003

Many-to-Many Disjoint Path Covers in DCell Xi Wang*1, Jianxi Fan1, Lantao You2 School of Computer Science and Technology, Soochow University, Suzhou 215006, China

Suzhou Industrial Park Institute Of Services Outsourcing, Suzhou 215006, China

wangxi0414@163.com

Abstract Data center networks (DCNs for short) become more and more important with the development of cloud computing. For any integers k≥0 and n≥2, the k-dimensional DCell with n-port switches, Dk,n, has been proposed for one of the most important DCNs as a server centric DCN structure. In this paper, we give a construction scheme for many-to-many r-disjoint path covers in Dk , n with 1≤r≤⌊(n+k-1)/2 ⌋. Keywords DCell; Disjoint Path Cover; Hamiltonian Path

Introduction The performance improvement of cloud computing faces great challenges of how to connect a large number of servers in building a data center network with promising performance [1]. Guo et al. have recently proposed a novel data center network called DCell [2], which addresses the needs of a data center. Its desirable properties include doubly exponential scaling, high network capacity, large bisection width, small diameter and high faulttolerance. The many-to-many node-disjoint paths covering (DPC for short) problem in DCNs have been an important subject because DPC routing would offer many benefits such as fault tolerance, software testing, etc [3,4,5]. A DCN can be represented by a simple graph G= (V ( G ) , E ( G ) ) , where V ( G ) represents the node set and E ( G ) represents the edge set, and each node represents a server and each edge represents a link between servers (switches can be regarded as transparent network devices [2]). For a set S= {s1 , s2 , , sr } of r source nodes (sources for short) and a set T= {t1 , t2 , , tr } of r sink nodes (sinks for short) in V (G ) , the many-to-many r-DPC problem is to determine whether there exist r-node disjoint paths, each path joining a source and a sink, these r paths cover V (G ) . There are paired and unpaired types of many-to-many r-DPC problems. In the paired type, each source should be joined to a specific sink, that is, s j should be joined to t j for 1 ≤ j ≤ r . In this paper, we discuss about some interesting properties of paired many-to-many DPC in DCell networks. Then, we give a construction scheme for paired many-to-many r-DPCs in DCell networks with 1 ≤ r ≤ (n + k − 1) / 2  . This work is organized as follows. Section 2 provides the preliminary knowledge. Some basic properties are given in Section 3. We make a conclusion in Section 4. Preliminaries In this section, we provide some graph theoretical terminology and give the definition of DCell. Let G1 and G2 be two subgraphs of G. We use G1 ∪ G2 to denote the subgraph induced by V (G1 ) ∪ V (G2 ) of G. E ′ {(u, v) ∈ E (G ) For U ⊆ V (G ) , we use G[U ] = (U , E ′) to denote the subgraph induced by U in G where= | u , v ∈ U } . A path in a graph is a sequence of nodes, P = 〈u0 , u1 , …, u j , …un −1 , un 〉 , in which no nodes are repeated

and u j , u j +1 are adjacent for any integer 0 ≤ j < n . The reverse of P is 〈un , un −1 , …, u1 , u0 〉 , denoted by P −1 . We use V ( P ) to denote the set of all nodes appearing in P and E ( P) to denote the set of all edges appearing in P. The

Information Engineering (IE) Volume 4, 2015

www.seipub.org/ie

path, starting from ui and ending with u j in a path Q, can be denoted by P(Q, ui , u j ) . The length of a path P, l ( P) , is the number of edges in P. We also write the path 〈u1 , u2 , , uk 〉 as 〈u1 , Q1 , ui , ui +1 , , uk 〉 where Q1 is the path 〈u2 , u3 , , ui −1 〉 . Hence, it is possible to write a path as 〈u1 , Q, u2 , u3 , , uk 〉 if l (Q) = 0 .

A path in a graph G containing every nodeof G is called a Hamiltonian path (HP). HP(u, v, G ) can denote a Hamiltonian path beginning with a node u and ending with another node v in graph G. A graph G is called paired (resp. unpaired) many-to-many r-disjoint path coverable (DPC-able for short) if 2r ≤| V (G ) | , G has a paired (resp. unpaired) r-DPC for any set S of r sources and any set T of r sinks in G such that S ∩ T = ∅ . A paired many-tomany r-DPC-able graph is unpaired many-to-many r-DPC-able. DCell uses a recursively defined structure to interconnect servers. Each server connects to different levels of DCell through multiple links. For any integers k ≥ 0 and n ≥ 2 , we use Dk , n to denote a k-dimensional DCell with n-port switches. D0,n is a complete graph on n nodes for any integer n ≥ 2 . We use N 0 to denote the number of nodes in

D0,n with N 0 = n . Furthermore, we use N k to denote the number of nodes in Dk , n for k ≥ 1 , where = N k N k −1 ( N k −1 + 1) . The node u of Dk , n can be labeled by a tuple [uk , , ui , , u0 ] such that u0 ∈ {0,1, , n − 1} and ui ∈ {0,1, , N i −1} for all i = 1, 2, , k .

The Dk , n is defined recursively as follows. (1) D0,n is a complete graph consisting of n nodes. (2) For any integer k ≥ 0, Dk , n is built from N k −1 + 1 disjoint copies Dk −1, n , according to the following steps. (2.1) We use Dki −1, n to denote the graph obtained by prefixing the label of each node of one copy of Dk −1, n with i for i = 0,1, , N k −1 . (2.2) Node u = [uk , uk −1 , , u0 ] in Dku−1, n k

u= k

v0 + ∑ kj =−11 (v j N j −1 )

is adjacent to node v = [vk , vk −1 , , v0 ] in

Dkv−1, n k

if and only

and vk = u0 + ∑ kj =−11 (u j N j −1 ) + 1 .

By definition of DCell, we have the following Lemma. Lemma 1. There exists an unique external edge, denoted by e( Dkα−1, n , Dkβ−1, n ) ,

which connects Dkα−1, n to Dkβ−1, n

with α ≠ β . For any integer d ≥ 1 , when two adjacent nodes u and v have a leftmost differing element at the position d, we say that v is the d-neighbor of u or the edge (u , v) is an edge of dimension d. We use (u ) d to denote the d-neighbor of u if d ≥ 1 . For all integer 0 ≤ i ≤ n − 1 , let zi = [uk , , u1 , i ] , we say that zi is a 0-neighbor of u or the edge (u , zi ) is an edge of dimension 0 if zi ≠ u . Let (u= )0 {zi | 0 ≤ i ≤ n − 1 and zi ≠ u} , we use (u )0 to denote the 0-neighbors of u. For any node u = [uk , uk −1 , , u0 ] in Dk , n , we use (u )i to denote the i-element ui of u . Main Results Some properties of the DCell networks have received considerable attentions in the literature [2,7]. Theorem 2. For any two integers k ≥ 0 and n ≥ 2, the number of servers in Dk , n satisfies N k ≥ (n + 1 2) 2 − 1 2 . k

Theorem 3. For any two integers k ≥ 0 and n ≥ 2 , Dk , n is Hamiltonian-connected except for D1,2 . By Theorem 3, we have the following three Lemmas. Lemma 4. D0,n is a paired many-to-many r-disjoint path coverable with 1 ≤ r ≤ (n − 1) / 2  . Lemma 5. For any two integers k ≥ 0 and n ≥ 2 , Dk , n is paired many-to-many 1-disjoint path coverable, except for D1,2 . Lemma 6. For any two integers k > 2 and n = 2 , or k > 0 and n > 2 .Let G = Dk , n [V (θm=1 Dkω−θ 1, n )] be a sub graph of

Dk , n with 2 ≤ m ≤ N k −1 . For any two distinct nodes u and v in G with (u ) k , (v) k ∉ V (G ) , (u ) k ≠ (v) k , (u ) k = ω1 , and (v) k = ωm , there exists a Hamiltonian path of G joining u and v.

www.seipub.org/ie

Information Engineering (IE) Volume 4, 2015

Lemma 7. D1,4 is a paired many-to-many r-disjoint path coverable with 1 ≤ r ≤ 2 . Proof. The Lemma is verified by a computer program. Lemma 8. For any integer n ≥ 4 , D1,n is a paired many-to-many r -disjoint path coverable with 1 ≤ r ≤  n / 2  . Proof. By Lemma 7, the lemma holds for D1,4 . For any integer n > 4 , the lemma is proved by induction on r and we omitted for limitation of length. Lemma 9.

For any integers k ≥ 0 and n ≥ 2 , Dk , n is paired many-to-many r-disjoint path coverable with

1 ≤ r ≤ (n + k − 1) / 2  and n + k is even, except for D1,2 .

Proof. By Lemma 4, the lemma holds with n = 0 . By Lemma 5, the lemma holds for r = 1 with k ≥ 0 and n ≥ 2 , except for D1,2 . We will prove this lemma by induction on the dimension k of Dk , n for other cases. We assume that Dk , n is paired many-to-many r-DPC-able for τ ≥ 1 and n ≥ 3 or τ ≥ 2 and n = 2 with 1 ≤ r ≤ (n + τ − 1) / 2  and n + τ

is even. In the following, we will prove the Dτ +1,n is paired many-to-many r-DPC-able. That is, given a set of r + 1 sources S = {s1 , s2 , , sr +1} and a set of r + 1 sinks T = {t1 , t2 , , tr +1} in Dτ +1,n , we will prove that there exist r + 1 disjoint

paths joining S and T in Dτ +1,n with 1 ≤ r ≤ (n + τ ) / 2 and S ∩ T = ∅ , such that the union of these r paths covers V ( Dτ +1, n ) .

For

W0 = θr +11 G θr +11 Gti = = si , W1 =

any

integer 1 ≤ i ≤ r + 1 ,

let Gsi = Dτ( ,sni )1

and Gti = Dτ(t,in)1 .

What's

more,

let

, W2 = WS ∪ WT , W3 = S ∪ T , G0 = Dτ +1, n [V (W2 )] ,and G1 = Dτ +1, n [ Dτ +1, n G0 ] . Furthermore, for

any integer 0 ≤ j ≤ Nτ , let S j= S ∩ V ( Dτj, n ) , T j= T ∩ V ( Dτj, n ) , and R= S j ∪ T j . Then, let p=max {| R0 |,| R1 |, j  ,| RNτ |} .We can claim the following three cases with respect to m. τ

Case 1. 1 ≤ p ≤ r . By Theorem 2, we have Nτ + 1 ≥ 1 2 + (n + 1 2) 2 ≥ 2(n + τ ) ≥ 2(r + 1) . For any integer 1 ≤ i ≤ r+1, that ui , vi ∉ {si , ti }, xi = (ui )τ +1 , yi = (vi )τ +1 , and ( x1 )1 ≠ ( x2 )1

find ui ∈ V (Gs ), vi ∈ V (Gt ), and xi , yi ∈ V (G1 ) such i

≠  ≠ ( xτ +1 )1 ≠ ( y1 )1 ≠ ( y2 )1 ≠  ≠ ( yτ +1 )1 . What's more, for any integer 1 ≤ i ≤ r + 1 , by Lemma 6, there exists a

 Dτ( ,xn) ∪ Dτ( ,yn ) if i ≤ r , Hamiltonian path Qi joining (ui )τ +1 and (vi )τ +1 in G =  (x ) (y ) r r + 1. G0  i =1 ( Dτ , n ∪ Dτ , n ) if i = i 1

i 1

Case 1.1. p = 1 . We can confirm that P1 = 〈 HP( s1 , u1 , Gs ) 1

, Q1 , HP(v1 , t1 , Gt )〉, P2 = 〈 HP( s2 , u2 , Gs ), Q2 , HP(v2 , t2 , Gt )〉 ,  , and Pτ +1 = 〈 HP( sτ +1 , sτ +1 , Gsτ ) + ( si′ , si′′ ), Qτ +1 , HP (tτ +1 , 1

tτ +1 , Gtτ )〉 are r + 1 disjoint paths such that V ( ir=+11 Pi ) = V ( Dτ +1, n ) . +1

Case 1.2.2 ≤ p ≤ r. For any integer 0 ≤ i ≤ Nτ with Ri >0, we use {sα , sα , , sα , tβ , tβ , , tβ } to denote R j with 1

|Si |

|Ti |

α1 , , α|S | , β1 , , β|T | ∈ {1, 2, ,τ + 1} and α1 ≠  ≠ α|S | ≠ β1 ≠  ≠ β|T | . According to the induction hypothesis, these i

exist | Ri | disjoint paths Pα , Pα , , Pα , Qβ , Qβ , , Qβ 1

such that that

joining {sα , , sα , vβ , , vβ } and {uα , , uα , tβ , , tβ } 1

|Si |

|Ti |

|Si |

|Ti |

the union of these | Ri | paths covers V ( Dτ +1, n ) for any integer 0 ≤ i ≤ Nτ . We can confirm

P1 = 〈 P1′ , Q1 , Q1′ 〉 , P2 = 〈 P2′ , Q2 , Q2′ 〉 , ,

and

Pτ +1 = 〈 Pτ +1′ , Qτ +1 , Qτ +1′ 〉

are

r +1

disjoint

paths

such

that V ( ir=+11 Pi ) = V ( Dτ +1, n ) . Case 2. r + 1 ≤ p < 2r .The case is similar to the Case 1, so we skip it. Case 3. p ≥ 2r .Choose Rλ such that | Rλ |= p with 0 ≤ λ ≤ Nτ . Let q = 2r + 2 − p . Then, we can claim the following three cases with respect to q . Case 3.1. q = 0 . According to the induction hypothesis, there exist r disjoint paths P1 , P2 , , Pr joining {s1 , s2 , , sr }

Information Engineering (IE) Volume 4, 2015

www.seipub.org/ie

and {t1 , t2 , , tr } in Dτλ,n such that V ( ir=+11 Pi ) = V ( Dτ +1, n ) . Choose Pφ and Pϕ such that and tr +1 ∈ V ( Pϕ ) with φ , ϕ ∈ {1, 2, , r} . Then, we can identify the following two cases with respect to φ and ϕ . Case 3.1.1. φ = ϕ . Let Q0 = P( Pφ , sr +1 , tr +1 ) . It is easy to verify that l ( Pφ ) ≥ l (Q0 ) + 2 . Choose four distinct nodes sr0+1 , s1r +1 , tr0+1 , and t1r +1 such that ( sr0+1 , sr +1 ) , ( sr +1 , s1r +1 ) , (tr0+1 , tr +1 ) , (tr +1 , t1r +1 ) ∈ E ( Pφ ) . What's more, let x = sr +1 , Q1 = P( Pφ , tr +1 , s1r +1 ) , Q2 = P( Pφ , tr +1 , sr0+1 ) , and tr +1 if tr +1 ∈ {sr0+1 , s1r +1},  = y  s1r +1 if l (Q= l (Q1 ) + 1 ≥ 2, 0)  0 l (Q2 ) + 1 ≥ 2. 0)  sr +1 if l (Q=

Then, let Q3 P= = ( Pφ , sφ , sr0+1 ), Q4 P= ( Pφ , sφ , s1r +1 ), Q5 P ( Pφ= , sφ , tr0+1 ), Q6 P= ( Pφ , sφ , t1r +1 ), Q7 P= ( Pφ , tφ , sr0+1 ), Q8 P( Pφ , tφ , , s1r +1 ), Q9 = P ( Pφ , tφ , tr0+1 ) , and Q10 = P( Pφ , tφ , t1r +1 ) .Then, let  sφ  0  sr +1  u =  s1r +1 0 tr +1 t1  r +1

if sφ ∈ {sr0+1 , s1r +1 , tr0+1 , tr1+1},

tφ  0  sr +1  v =  s1r +1 0 tr +1 1 tr +1

if tφ ∈ {sr0+1 , s1r +1 , tr0+1 , tr1+1},

if l (Q3 ) < l (Q4 ) ≤ l (Q5 ), if l (Q4 ) < l (Q3 ) ≤ l (Q5 ), if l (Q5 ) < l (Q6 ) ≤ l (Q3 ), if l (Q6 ) < l (Q5 ) ≤ l (Q3 ).

and

if l (Q7 ) < l (Q8 ) ≤ l (Q9 ), if l (Q8 ) < l (Q7 ) ≤ l (Q9 ), if l (Q9 ) < l (Q10 ) ≤ l (Q7 ), if l (Q10 ) < l (Q9 ) ≤ l (Q7 ).

Let u ′ (= = u )τ +1 , v′ (= v)τ +1 , x′ ( x= )τ +1 , y ′ ( y= )τ +1 , G2 Dτ(u, n′) ∪ Dτ( v, n′) , and G3 = G1 G2 . By Lemma 6, there exists a 1

Hamiltonian path Pφ′ joining u ′ and v′ in G2 and exists a Hamiltonian path Pr′+1 joining x′ and y ′ in G3 . We can confirm that P1 , , Pφ = 〈 P( Pφ , sφ , u ), Pφ ′ , P( Pφ , v, tφ )〉, , and Pτ +1 = 〈 x, Pr +1′ , P ( Pτ +1 , y, tr +1 )〉 are r+1 disjoint paths such that V ( ir=+11 Pi ) = V ( Dτ +1, n ) . Case 3.1.2. φ ≠ ϕ . The case is similar to the Case 3.1.1, so we skip it. Case 3.2. q=1. Choose x, y, sφ , and tφ such that x ∈ V (G0 ), y ∈ V (G1 ), sφ , tφ ∈ W3 , and φ ∈ {1, 2, ,τ + 1} with tφ x=  sφ

if sφ ∈ V (G1 ),  sφ and y =  if tφ ∈ V (G1 ). tφ

if sφ ∈ V (G1 ), if tφ ∈ V (G1 ).

Let S ′ = S \ {sφ } and T ′ = T\ {tφ } . According to the induction hypothesis, these exist r disjoint paths P1 , P2 , , Pi , , Pr +1 joining S ′ and T ′ such that V ( ir=+11 Pi ) = V ( Dτ +1, n ) with i ≠ φ . Choose Pϕ with x ∈ V ( Pϕ )

and ϕ ∈ {1, 2, , r + 1} . Q0

Select

two

distinct

nodes

and

x1 such

that ( x 0 , x) , ( x, x1 ) ∈ E ( Pϕ ) .

Let

P= ( Pϕ , sϕ , x ), Q1 P= ( Pϕ , sϕ , x ), Q2 P( Pϕ , tϕ , x ), and Q3 = P( Pϕ , tϕ , x ) . Then, let 0

 sϕ  = u  x0  1  x

if sϕ ∈ {x 0 , x1}, if l (Q0 ) < l (Q1 ), if l (Q1 ) < l (Q0 ).

www.seipub.org/ie

Information Engineering (IE) Volume 4, 2015

and tϕ  = v  x0  1  x

if tϕ ∈ {x 0 , x1}, if l (Q2 ) < l (Q3 ), if l (Q3 ) < l (Q2 ).

Let u ′ = (u )τ +1 , v′ = (v)τ +1 , and x′ = ( x)τ +1 .Then, we can claim the following cases with respect to u ′ , v′ , x′ , and y . Case 3.2.1. (u ′)τ +1 ≠ (v′)τ +1 ≠ (x ′)τ +1 ≠ ( y )τ +1 . Choose δ , ω Let

G3 =

Dτ(x, n′)τ +1

∪ Dτ( ,yn)τ +1

∪ Dτδ, n

∪ Dτω, n

and G4 = G0

(

vDτλ, n

∈ {0,1, , Nτ } with δ , ω ∉ {(u ′)τ +1 , (v′)τ +1 , λ , (x ′)τ +1 , ( y )τ +1} .

)

∪ G3 . By Lemma 6, there exist a disjoint path Pφ′

covering all nodes in G3 and a disjoint path Pτ′+1 covering all nodes in G4 . Moreover, let Pτ +1 = 〈 x, Pτ′+1 〉 and 〈 P( Pϕ , sφ , u ), Pϕ′ , P( Pφ , v, tφ )〉 . Then, P1 , P2 , , Pr +1 are r+1 disjoint paths in Dτ +1,n , Pi joining si and ti , 1 ≤ i ≤ τ + 1 , Pφ = that cover all the nodes of Dτ +1,n . Case 3.2.2. (u ′)τ +1 = ( y )τ +1 or (v′)τ +1 = ( y )τ +1 or (x ′)τ +1 = ( y )τ +1 . The case is similar to the Case 3.2.1, so we skip it. (u ′)τ +1 = ( y )τ +1 . Case 3.3. m′ = 2 . The case is similar to the Case 3.2, so we skip it. In summary, given a set of r+1 distinct sources S = {s1 , s2 , , sr +1} and a set of r+1 distinct sinks T = {t1 , t2 , , tr +1} in ∅ ,there always exist r+1 disjoint paths P1 , P2 , , Pr +1 in Dτ +1,n , Dτ +1,n such that S ∩ T =

joining si and ti , 1 ≤ i ≤ τ + 1 , such that V ( ir=+11 Pi ) = V ( Dτ +1, n ) . Theorem 10.

For k ≥ 0 and

n ≥ 2 , Dk , n is a paired many-to-many r-disjoint path coverable with

1 ≤ r ≤ (n + k − 1) / 2  except for D1,2 .

Proof. By Lemma 4, the theorem holds with k = 0 . By Lemma 8, the theorem holds with D1,n . For k > 1 and n ≥ 2 , the theorem holds for n + k is even by Lemma 9 and the theorem holds for n + k is odd is similar to that of Lemma 9 and thus omitted. Clearly, Theorem 10 implies the following theorem. Theorem 11.

For k ≥ 0 and

n ≥ 2 , Dk , n is (unpaired) many-to-many r-disjoint path coverable with

1 ≤ r ≤ (n + k − 1) / 2  except for D1,2 .

Conclusions DCell has been proposed for one of the most important data center networks and can support millions of servers with outstanding network capacity and provide good fault tolerance by only using commodity switches. In this paper, we give a construction scheme for many-to-many r-disjoint path covers in a k-dimensional DCell with nport switches, Dk , n , with 1 ≤ r ≤ (n + k − 1) / 2  . ACKNOWLEDGMENT

This work is supported by National Natural Science Foundation of China (No. 61170021, 61303205), Graduate Training Excellence Program Project of Soochow University (No. 58320235), Graduate innovation project of Jiangsu Province (No. CXZZ13\_0814), Soochow University Outstanding Doctoral Thesis Funding (No. 233200413), and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 12KJB520016). REFERENCES

[1]

Al-Fares, Mohammad, and Loukissas, Alexander, eds. “A scalable, commodity data center network architecture“. ACM SIGCOMM Computer Communication Review, 38 (2008): 63—74.

Information Engineering (IE) Volume 4, 2015

[2]

www.seipub.org/ie

Guo, Chuan X., and Wu, Hai T., eds. “DCell: A scalable and fault-tolerant network structure for data centers“. Proceedings of SIGCOMM'08, (2008): 75—86.

[3]

Guo, Chuan X., and Lu, Guo H., eds. “BCube: A high performance, server-centric network architecture for modular data centers“. Proceedings of SIGCOMM'09, (2009): 63—74.

[4]

Park, Jung H., and Kim, Hee C., eds. “Many-to-many disjoint path covers in the presence of faulty elements“. IEEE Transactions on Computers, 58 (2009): 528—540.

[5]

Chen, Xie B. “Paired many-to-many disjoint path covers of hypercubes with faulty edges“. Information Processing Letters, 112 (2012): 61—66.

[6]

Reinhard Diestel. “Graph Theory, 4th Edition“. Springer-Verlag, Heidelberg, 2012.

[7]

Wang, Xi, and Fan, Jian X., eds. “One-to-one Disjoint Path Covers in DCell“, Proceedings of NPC’13, (2013): 61—70.