Collaboration coding to multi-node repair program under the twin-mds codes framework in cloud storage systems

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1 36 Z1 Vol.36 No.Z Journal on Communications November 2015 doi: /j.issn x ,2 1 ( ) MDS 3 P302 A Collaboration coding to multi-node repair program under the twin-mds codes framewor in cloud storage systems XIE Xian-zhong 1, HUANG Qian 1, 2, WANG Liu-su 1 (1. Institute of Broadband Access echnologies, Chongqing University of Posts and elecommunications, Chongqing , China; 2. Department of Computer Science, College Mobile elecommunications, Chongqing University of Posts and elecommunications, Chongqing , China) Abstract: A multi-node exact repair code scheme, which can repair multiple system nodes or redundant nodes simultaneously, was shown and proved to against the disadvantages of the existing multi-node repair model in cloud storage. he multi-node exact repair code was combined with a twin-mds codes framewor with health cooperative nodes. In this way, repair bandwidth, the number of repair lins and the amount of data to be treated in an intermediate node were reduced, while multi-node were repaired. Finally, numerical simulation results show that this scheme has greater improvements. In particular, it reduces the load in an intermediate node. And the advantages was more obvious with the more storage nodes in cloud storage. Key words: cloud storage; multi-node exact repair code; collaboration repair; twin-mds codes model 1 / 2007 Dimais [1] / regenerating codescloud storage networ codes [2,3] Suh [4] - MDS, maximum distance separable [4] MDS E-MDS, exact MDS CSC2011jjA40006 CSC2010BB2415KJ KJ Foundation Items: he National Natural Science Foundation of China ( , ); he Natural Science Foundation of Chongqing (CSC2011jjA40006, CSC2010BB2415); he Scientific and echnological Research Program of Chongqing Municipal Education Commission (KJ120501, KJ110530)

2 2 36 [5~12] [5] [6,7] [8] =2 [9,10] MBRC, minimum-bandwidth regenerating codes MSRC, minimum-storage regenerating codes MBRC MSRC Shum [11] MCR, multi-node cooperative regenerating MCR [12] [13] MDS (twin-mds codes) MDS [12] [4] E-MDS MER, multi-node exact repair 2 MDS 2.1 r d d m m A 1 A 2 B 1 B MDS m 2 A 1 A 2 m m m d 1 β r α γ=(d 1)β+rα

3 Z1 3 m MDS 2.2 MDS MDS n 1 n 1 2 n 2 2 i n i A i G i i l(1 l n i ) l A i g (i,l) (1 l n i ) i l G i g (i,l) g (i,l) 3 3 MDS 2 2 MDS 1 C 1 2 C 2 2 C 1 C 2 MDS MDS F q M i=0 1 C i [n i, ] F q MDS G i G i l g (i,l) (1 l n i ) 2 2 A 1 A 2 =A t 1 t C i A i MDS g (1,1) g (2,4) MDS 2.3 MDS MER MDS MDS

4 4 36 [12,13] MDS 5 2 C 1 C m 1 m 2 2 r 2 1 d 1 m 1 d r 1 m 2 d 2 1 α ββ=α MDS MER 3 MDS MER Suh [4] (n,, d)=(2,, 2 1) E-MDS E-MDS M=β (d +1) β=1 α = M = E-MDS 5 MDS r C 1 C 2 2 r 1 r 2 d 1 d m 1 m 2 m 1 m 2 d i 1 β i r i α γ=γ 1 +γ 2 γ i =(d i 1)β i +r i α m d=d 1 1+d 2 1=d 1 +d 2 2 d=d 1 +d 2 1 E-MDS [4] X=[x ij ], Y=[y ij ], Ψ=[ψ ij ], Φ =Ψ 1 =[φ ij ] X = YΨ, Y = XΦ (1) Ψ Φ X x 1,, x Y y 1,, y K={1,,} i K(1) x = ψ y + ψ y + + ψ y i 1i 1 2i 2 i y = ϕ x + ϕ x + + ϕ x i 1i 1 2i 2 i (2) X =(X ) 1 Ỹ =(Y ) 1 X Y X Ỹ x 1,, x ỹ 1,, ỹ Θ=[θ 1 θ ] D=[δ 1 δ ]θ i i δ i +i [4] E-MDS i +i i K

5 Z1 5 (3) D = ωyθ X + σθψ ω σ GF(q) ΘΨ j λ j = = 1 ψ ε ε j ε θ 3 δ = j ω y x θ + σ i j i λj i= 1, j K (4) Θ = ω XD Y + σ DΦ (5) ω σ GF(q) DΦ j λ j = = 1 ϕ ε ε j ε δ 5 θ =, j ω x y δ + σ j K i j i λ (6) j i= 1 35 ω σ ω σ ωω + σσ = 1, σω + ωσ = 0 (7) E-MDS F(Θ)= ωyθ X +σθψ G(D)= ω XD Y + σ DΦ G(F(Θ))=Θ F(G(D))= D E-MDS MDS E-MDS MER, multi-node exact repair E-MDS 1 ( ) [14] m n 1 Ψ= a i b j GF(q) a b i j a i b j 0 a i a j b i b j 1 i m 1 j n C Cauchy 1 E-MDS Y Ψ ω σ ω σ 1) Y GF(q) 2) Ψ GF(q) 3) ω σ ω σ 7 4) i j K ψ ij φ ij 1 r MER E-MDS i K Θ i D +i MER r MER d=n r i y i i +i x i (+i) r 2 r r Θ D 2 r +1 +r (+i) x i θ 1,,x i θ x i θ j =ωx j λ i +σx i λ j i {1,,r } j {r+1, r+2,, }4 (+i) δ = i ω y x θ + σ ε i ε λ (8) i ε = 1 8 (x i θ 1,, x i θ ) r+1 j (+i) x i δ j =ωx j λ i +σx i λ j x λ = i i ψ x θ εi i ε ε = 1 (9) 1 σ x λ = x δ ψ j i i j ε j i ε x θ ω ω ε = 1 1 j r (+i) r 1 x λ i j j i x 1,,x λ i r=2 2 u +v u v K u +v u y θ u i y δ = ω x θ + σ y λ u j j u u j (10)

6 6 36 (+v) x θ v i x δ = ω x λ + σ x λ v j j v v j i K \{u} j K \{v} (11) ε ψ iε ϕ εj δ ij i=j ε ψ iε ϕ εj =1 i j ε ψ iε ϕ εj =0 (+v) 11 ϕ j = 1, j v ju v j x δ +(ω+σ) i= 1, y u λ v x v θ u ψ ϕ x θ i u iv vu v i ( ( ) ) ω y λ + σ σ + ω ψ ϕ x θ (12) u v uv vu v u (+v) 12 u u ωψ iv y θ u i 10 ( ωψ ( σ ( σ ω) ψ ϕ ) ) y + + x θ ( ω x + σψ y j uj u ) θu uv u uv vu v u ( σ ( σ + ω) ψ ϕ uv vu ) x + ωψ y v uv u ω x + σψ y 1 u1 u Α = ω x + σψ y ω x ω x + σψ y u u v 1 u, v 1 u + σψ y v+ 1 u, v+ 1 u (13) (14) θ u u MER A σ (σ+ω)ψ uv ϕ vu 2 σ (σ+ω)ψ uv ϕ vu =0 14 ωψ y uv u uv u ωψ ϕ x ε ε ε = 1 Β = = ω K \{ v} X ω X K \{ v} (15) X K\{v} =[x 1 x v 1 x v+1 x ] 15 B σ (σ+ω)ψ uv ϕ vu 0 2 y u = = 1 C = C C ϕ ε ε u ε 1 2 x 14 σ σψ ϕ ωψ Φ uv vu uv \{ } x, v K v u = (16) σϕ Ψ ωι + σψ Φ vu u, K \{ v} u, K \{ v} K \{ v}, u X K \{ v} Ψ u, K\{v} =[ψ u1 ψ u,v 1 ψ u,v+1 ψ u ] Φ K\{v}, u= [ϕ 1u ϕ v 1,u ϕ v+1,u ϕ u ] I ( 1) ( 1) A 16 C C 1 C 1 C = Λ + gf 1 ( ) σ σ + ω ψ ϕ 0 uv vu = + 0 ωι ωψ uv ϕvu K \{ v}, u σ u, K \{ v Φ Ψ } (17) σ (σ+ω)ψ uv ϕ vu ω Λ Sherman-Morrison = 1+ f Λ 1 g 0C 1 ψ ϕ ε 1 iε ε j δ ij 1 ψ ϕ 1 ε = uε εu ( )( uv vu ) σ ( σ + ω) ψ ϕ σ σ + ω 1 ψ ϕ 1 1+ f Λ g = (18) 7 2 (ω 2 σ 2 )(ω 2 σ 2 )=1 1 ω 2 σ 2 σ 0 ψ uv ϕ vu 1 1+ f Λ 1 g 0 14 A MER u θ u MER E-MDS (+v) u ω x λ u +(σ (σ +ω )ϕ vu ψ uv ) y δ v +v v u 1 MER E-MDS E-MDS r 2 4 uv vu

7 Z1 7 MDS [12,13] [1] [15] [11] [12] =d,r =n n<2 (n,, d)(α, B) 4.1 MDS γ 1 1 (n,, d)(α, B) (4,2,2) (2,4) γ =4 γ =3 γ =3 γ =3 γ =3 (7,4,4)(21,84) γ =84 γ =51.3 γ =42 γ =42 γ =35 (14,8,8)(80,640) γ =640 γ =261.3 γ =173.3 γ =173.3 γ = f 2 2 (n,, d)(α, B) (4,2,2)(2,4) f=6 f=3 f=3 (7,4,4)(21,84) f=18 f=6 f=5 (14,8,8)(80,640) f=78 f=13 f= MDS 2 m 1 m 2 m 1 m 2 [12] 3 3 (n,, d)(α, B) (4,2,2)(2,4) 4 2 (7,4,4)(21,84) (14,8,8)(80,640) MDS MER E-MDS 1 MER MDS 5 MDS MDS E-MDS MER

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