Exact-Regenerating Codes between MBR and MSR Points

Transcription

1 Exact-Regeneratng Codes between BR SR Ponts arxv: v1 [cs.dc] 19 Apr 2013 Abstract In ths paper we study dstrbuted storage systems wth exact repar. We gve a constructon for regeneratng codes between the mnmum storage regeneratng SR the mnmum bwdth regeneratng BR ponts show that n the case that the parameters n, k, d are close to each other our constructons are close to optmal when comparng to the known capacty when only functonal repar s requred. We do ths by showng that when the dstances of the parameters n, k, d are fxed but the actual values approach to nfnty, the fracton of the performance of our codes wth exact repar the known capacty of codes wth functonal repar approaches to one. A. Regeneratng Codes I. INTRODUCTION In a dstrbuted storage system a fle s dspersed across n nodes n a network such that gven any k < n of these nodes one can reconstruct the orgnal fle. We also want to have such a redundancy n our network that f we lose a node then any d < n of the remanng nodes can repar the lost node. We assume that each node stores the amount α of nformaton, e.g., α symbols over a fnte feld, n the repar process each reparng node transmts the amount β to the new replacng node called a newcomer hence the total repar bwdth s γ dβ. We also assume that k d. The repar process can be ether functonal or exact. By functonal repar we mean that the nodes may change over tme,.e., f a node v old s lost n the repar process we get a new node v new nstead, then we may have v old v new. If only functonal repar s assumed then the capacty of the system, denoted by C k,d α, γ, s known. Namely, t was proved n the poneerng work by Dmaks et al. [1] that k 1 C k,d α, γ mn α, d j } d γ. j0 If the sze of the stored fle s fxed to be B then the above expresson for the capacty defnes a tradeoff between the node sze α the total repar bwdth γ. The two extreme ponts are called the mnmum storage regeneraton SR pont the mnmum bwdth regeneraton BR pont. The SR pont s acheved by frst mnmzng α then mnmzng γ to obtan α B k db γ kd k+1. 1 Ton Ernvall Turku Center for Computer Scence & Department of athematcs Statstcs FI Unversty of Turku Fnl Emal: tmernv@utu.f. By frst mnmzng γ then mnmzng α leads to the BR pont α 2dB k2d k+1 2dB γ k2d k+1. 2 In ths paper we are nterested n codes that have exact repar. The concepts of exact regeneraton exact repar were ntroduced ndependently n [2], [3], [4]. Exact repar means that the network of nodes does not vary over tme,.e., f a node v old s lost n the repar process we get a new node v new, then v old v new. We denote by n,k,dα, γ the capacty of codes wth exact repar wth n nodes each of sze α, wth total repar bwdth γ, for whch each set of k nodes can recover the stored fle each set of d nodes can repar a lost node. We have by defnton that n,k,dα, γ C k,d α, γ. It was proved n [5], [7], [8] that the codes wth exact repar acheve the SR pont n [5] that the codes wth exact repar acheve the BR pont. The mpossblty of constructng codes wth exact repar at essentally all nteror ponts on the storage-bwdth tradeoff curve was shown n [6]. B. Contrbutons Organzaton In Secton II we gve a constructon for codes between SR BR ponts wth exact repar. In Secton III we derve some nequaltes from our constructon. Secton IV provdes an example showng that, n the specal case of n k + 1 d + 1, our constructon s close to optmal when comparng to the known capacty when only functonal repar s requred. In Secton V we show that when the dstances of the parameters n, k, d are fxed but the actual values approach to nfnty, the fracton of performance of our codes wth exact repar the known capacty of functonal-repar codes approaches to one.

2 II. CONSTRUCTION Assume we have a storage system DSS 1 wth exact repar for parameters n, k, d wth a node sze α the total repar bwdth γ dβ. In ths secton we propose a constructon that gves a new storage system for parameters n n + 1, k k + 1, d d + 1. Let DSS 1 consst of nodes v 1,..., v n, let the stored fle F be of maxmal sze Cn,k,d exact α, γ. Let then DSS 1+ denote a new system consstng of the orgnal storage system DSS 1 one extra node v n+1 storng nothng. It s clear that DSS 1+ s a storage system for parameters n + 1, k + 1, d + 1 can store the orgnal fle F. Let σ j be the permutatons of the set 1,..., n+1} for j 1,..., n + 1!. Assume that DSSj new s a storage system for j 1,..., n + 1! correspondng to the permutaton σ j such that DSSj new s exactly the same as DSS 1+ except that the order of the nodes s changed correspondng to the permutaton σ j,.e., the th node n DSS 1+ s the σ j th node n DSSj new. Usng these n + 1! new systems as buldng blocks we construct a new system DSS 2 such that ts jth node for j 1,..., n+1 stores the jth node from each system DSS new for 1,..., n + 1!. It s clear that ths new system DSS 2 works for parameters n + 1, k + 1, d + 1, has exact repar property, stores a fle of sze n + 1!Cn,k,d exact α, γ has a node sze total repar bwdth α 2 n + 1! n!α n n!α γ 2 n + 1! n!γ n n!γ. oreover, because of the symmetry of the constructon we have β 2 n n!β. that s, Ths constructon mples the nequalty n+1,k+1,d+1n n!α, n n!γ n + 1! n,k,dα, γ, Cn+1,k+1,d+1α, exact γ n + 1 n Cexact n,k,dα, γ. 3 Example 2.1: If we relax on the typcal requrement of a DSS to be homogeneous, meanng that each node s transmttng the same amount β of nformaton n the repar process, nstead only requre that the total repar bwdth γ s constant.e., β may take dfferent values dependng on the node, then we can buld our constructon a lttle easer. Let n, k, d 3, 2, 2 DSS 1 be a dstrbuted storage system wth exact repar. Let DSSj new be a storage system wth 4 nodes for j 1,..., 4 where the jth node stores nothng, the th node for < j stores as the th node n the orgnal system DSS 1, the th node for > j stores as the 1th node n the orgnal system DSS 1. That s, n the jth subsystem DSSj new the jth node stores nothng whle the other nodes are as those n the orgnal system DSS 1. Usng these four new systems as buldng blocks we construct a new system DSS 2 such that ts jth node for j 1,..., 4 stores the jth node from each system DSS new for 1,..., 4. Hence each node n DSS 2 stores 4 1α 3α the total repar bwdth s 4 1γ 3γ. For example, f the orgnal system DSS 1 conssts of nodes v 1 storng x, v 2 storng y, v 3 storng x + y then DSS1 new conssts of nodes u 11 storng nothng, u 12 storng x 1, u 13 storng y 1, u 14 storng x 1 +y 1. Smlarly DSS2 new conssts of nodes u 21 storng x 2, u 22 storng nothng, u 23 storng y 2, u 24 storng x 2 +y 2 so on. Then n the resultng system the frst node w 1 conssts of nodes u 11 storng nothng, u 21 storng x 2, u 31 storng x 3, u 41 storng x 4. The stored fle s x 1, x 2, x 3, x 4, y 1, y 2, y 3, y 4. w 1 : w 2 : w 3 : w 4 : x 1 y 1 x 1 +y 1 x 2 y 2 x 2 +y 2 x 3 y 3 x 3 +y 3 x 4 y 4 x 4 +y 4 Fg. 1. The fgure llustrates the DSS bult n Example 2.1. It conssts of nodes w 1, w 2, w 3, w 4. III. INEQUALITIES FRO THE CONSTRUCTION Next we wll derve some nequaltes for the capacty n the case of exact repar. Usng Equaton 3 nductvely we get Theorem 3.1: For an nteger j [0, k 1] we have Cn,k,d exact α, γ n n j Cexact n j,k j,d jα, γ. It s proved n [5], [7], [8] that the SR pont can be acheved f exact repar s assumed. As a consequence of ths Theorem 3 we get the followng bound. Theorem 3.2: For ntegers 1 k we have Cn,k,d exact d k + α nα α, d k + 1 n k +. Proof: Wrte n n j, k k j, d d j, α B k, γ. It s proved n [5], [7], [8] that.e., d B k d k +1 Cn exact,k,d α, γ B, Cn j,k j,d j exact d jα α, k jα. d k + 1

3 Hence by Theorem 3.1 we have Cn,k,d exact d jα nk jα α, d k + 1 n j Now a change of varables by settng k j gves us the result. IV. EXAPLE: CASE n k + 1 d + 1 In ths secton we study the specal case n k +1 d+1 compare t to the known capacty wth the assumpton of functonal repar, n 2 C n 1,n 1 α, γ Now our bound gves so we can wrte for ntegers 1,..., k. j0 mn α, n 1 j n 1 γ n,n 1,n 1α, α nα 1 + f n nα 1 +. }. Notce that now n the extreme ponts our lower bound acheves the known capacty,.e., for the BR pont for the SR pont. n,n 1,n 1α, α f n 1 nα 2 n,n 1,n 1α, kα f n k n 1α As an example we study the fracton f n C n 1,n 1 α, α nα } n 2 j0 mn α, n 1 j n 1 α for ntegers [1, k]. Wrtng t out we see that f n C n 1,n 1 α, α n T j0 1 + n 2 T n 1 j jt +1 n 1 n 2n 1 where T n 11 1 n T 1n T 2, For large values of n ths s approxmately for all 1,..., k Fg. 2. The fgure shows the performance of our constructon dotted curve between the capacty of functonally reparng codes uppermost curve the trval lower bound gven by nterpolaton of the known SR BR ponts when n, k, d 51, 50, 50, α 1, γ [1, 50]. V. THE CASE WHEN n, k AND d ARE CLOSE TO EACH OTHER Next we wll study the specal case where n, k d are close to each other. We wll do ths by settng n n +, k k + d d+ lettng, then examne how the capacty curve asymptotcally behaves. The example n the prevous secton showed us that n that specal case our bound s qute close to the capacty of functonally regeneratng codes. However, n the prevous secton we fxed to be an nteger then assumed that n s large. In ths secton we te up the values together to arrve at a stuaton where the total repar bwdth stays on a fxed pont between ts mnmal possble value gven by the BR pont ts maxmal possble value gven by the SR pont. For each the bound from Theorem 3.2 gves Cn exact,k,d α, d k + α n α d k + 1 n k + for 1,..., k, hence n ths secton we wrte g n α n k + for ntegers 1,..., k extend ths defnton for x [1, k] such that g x s the pecewse lnear curve defned by g. Let s 0, 1] be a fxed number 1 + sk 1. We wll study how the fracton g α, d k +α behaves as we let. Informally ths tells how close our lower bound curve the known capacty curve are to each other when s large,.e., values n, k, d are close to each other. Remark 5.1: In the SR pont we have γ SR d α d k + 1

4 n the BR pont Hence γ BR α. α d k + d k + 1 sγ SR + 1 sγ BR. Theorem 5.1: Let s 0, 1] be a fxed number 1 + sk 1. Then lm g α, d k +α 1. Proof: Let 1 + sk 1. We study the behavor of the fracton for large, so we have 1. Thus, to smplfy the notaton, we may assume that acts as an nteger. We also use the notaton d sk 1 t d k sk 1. We have g 1 + sk 1 n 1 + sk 1α n k + α, d k + α d k + 1 t k 1 α d j 1 + d k + d j0 jt+1 d k + 1 α t k t 12d + k td k +, 2d d k whence where g α, d k +α h 1 h 2 h 3 + h 4, h 1 2n 1 + sk 1d d k + 1, h 2 n k sk 1, h 3 2t + 1d d k + 1, h 4 k t 12d k+ td k+1+sk 1. Now t s easy to check that h 1 3 2sd k + 1, h 2 s, h 3 2 2d k as. Note that when s large hence h 4 2 t d k + 1 ds s k t 12d k + t 0 s 0 as. Fnally, g α, d k +α h 2 h 1 3 h3+h4 2 2sd k + 1 s2d k as, provng the clam. d k sk 1 As a straghtforward corollary to Theorem 5.2 we have Theorem 5.2: Let s [0, 1] be a fxed number let γ SR d α d k +1 γ BR α. Then Cn exact lm,k,d α, sγ SR + 1 sγ BR 1. α, sγ SR + 1 sγ BR VI. CONCLUSIONS We have shown n ths paper that when n, k, d are close to each other, the capacty of a dstrbuted storage system when exact repar s assumed s essentally the same as when only functonal repar s requred. Ths was proved by usng a specfc code constructon explotng some already known codes achevng the SR pont on the tradeoff curve by studyng the asymptotc behavor of the capacty curve. However, when n, k, d are not close to each other then the bound our constructon gves s not good. So as a future work t s stll left to fnd the precse expresson of the capacty of a dstrbuted storage system when exact repar s assumed, especally to study the behavor of the capacty when n, k, d are not close to each other. VII. ACKNOWLEDGENTS Ths research was partly supported by the Academy of Fnl grant # by the Eml Aaltonen Foundaton, Fnl, through grants to Camlla Hollant. Dr. Salm El Rouayheb at the Prnceton Unversty s gratefully acknowledged for useful dscussons. Dr. Camlla Hollant at the Aalto Unversty s gratefully acknowledged for useful comments on the frst draft of ths paper. 7 8

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