Codes between MBR and MSR Points with Exact Repair Property

Transcription

1 Codes between MBR and MSR Points with Exact Repair Property 1 Toni Ernva arxiv: v1 [cs.it] 18 Dec 2013 Abstract In this paper distributed storage systems with exact repair are studied. A construction for regenerating codes between the minimum storage regenerating MSR and the minimum bandwidth regenerating MBR points is given. To the best of author s knowedge, no previous construction of exact-regenerating codes between MBR and MSR points is done except in the work by Tian et a. On contrast to their work, the methods used here are eementary. In this paper it is shown that in the case that the parameters n, k, and d are cose to each other, the given construction is cose to optima when comparing to the known functiona repair capacity. This is done by showing that when the distances of the parameters n, k, and d are fixed but the actua vaues approach to infinity, the fraction of the performance of constructed codes with exact repair and the known capacity of codes with functiona repair, approaches to one. Aso a simpe variation of the constructed codes with amost the same performance is given. A. Regenerating Codes I. INTRODUCTION In a distributed storage system a fie is dispersed across n nodes in a network such that given any k < n of these nodes one can reconstruct the origina fie. We aso want to have such a redundancy in our network that if we ose a node then any d < n of the remaining nodes can repair the ost node. We assume that each node stores the amount α of information, e.g., α symbos over a finite fied, and in the repair process each repairing node transmits the amount β to the new repacing node caed a newcomer and hence the tota repair bandwidth is γ = dβ. We aso assume that k d. The repair process can be either functiona or exact. By functiona repair we mean that the nodes may change over time, i.e., if a node v od i v od i is ost and in the repair process we get a new node v new i instead, then we may have vi new. If ony functiona repair is assumed then the capacity of the system, denoted by C k,d α, γ, is known. Namey, it was proved in the pioneering work by Dimakis et a. [2] that k 1 { C k,d α, γ = min α, d j } d γ. j=0 Part of this paper was presented at 2013 IEEE Information Theory Workshop, Sevie, Spain [1]. T. Ernva is with the Turku Centre for Computer Science, Turku, Finand and with the Department of Mathematics and Statistics, FI-20014, University of Turku, Finand e-mai:tmernv@utu.fi.

2 2 If the size of the stored fie is fixed to be B then the above expression for the capacity defines a trade-off between the node size α and the tota repair bandwidth γ. The two extreme points are caed the minimum storage regenerating MSR point and the minimum bandwidth regenerating MBR point. The MSR point is achieved by first minimizing α and then minimizing γ to obtain α = B k 1 db γ = kd k+1. By first minimizing γ and then minimizing α eads to the MBR point α = γ = 2dB k2d k+1 2 2dB k2d k+1. In this paper we wi study codes that have exact repair property. The concepts of exact regeneration and exact repair were introduced independenty in [3], [4], and [5]. Exact repair means that the network of nodes does not vary over time, i.e., if a node v od i denote by is ost and in the repair process we get a new node vi new, then vi od C exact n,k,dα, γ = vi new. We the capacity of codes with exact repair with n nodes each of size α, with tota repair bandwidth γ, and for which each set of k nodes can recover the stored fie and each set of d nodes can repair a ost node. We have by definition that C exact n,k,dα, γ C k,d α, γ. B. Reated Work It was proved in [6], [8], [9], and [10] that the codes with exact repair achieve the MSR point and in [6] that the codes with exact repair achieve the MBR point. The impossibiity of constructing codes with exact repair at essentiay a interior points on the storage-bandwidth tradeoff curve was shown in [7]. Other papers studying exact-regenerating codes in MSR point incude e.g. [11], [14], [12], and [13]. Locay repairabe codes that achieve repair bandwidth that fas beow the time-sharing trade-off of the MSR and MBR points are studied in [15]. To the best of author s knowedge, no previous construction of exact-regenerating codes between MBR and MSR points is done except in [17]. Our construction is very different to that. We do not use compex combinatoria structures but instead expoit some optima codes in MSR point. However, we require in our construction that storage symbos can be spit into a sufficienty arge number of subsymbos. Tian has shown in [16] that there exists a non-vanishing gap between the optima bandwidth-storage tradeoff of the functiona-repair regenerating codes and that of the exact-repair regenerating codes by characterizing the rate region of the exact-repair regenerating codes in the case n, k, d = 4, 3, 3.

3 3 C. Organization and Contributions In Section II we give a construction for codes between MSR and MBR points with exact repair. In Section III we derive some inequaities from our construction. Section IV provides an exampe showing that, in the specia case of n = k + 1 = d + 1, our construction is cose to optima when comparing to the known capacity when ony functiona repair is required. In Section V we show that when the distances of the parameters n, k, and d are fixed but the actua vaues approach to infinity, the fraction of performance of our codes with exact repair and the known capacity of functiona-repair codes approaches to one. In Section VI we give another construction with quite simiar performance. The main differences of this construction when compared to the construction of Section II is its easiness as advantage and reaxation of assumption of symmetric repair as its disadvantage. In Section VII we give yet two other constructions that have some simiarities with the construction of Section II. However, the performance of these constructions is reativey bad and the main interest of this section is the comparison of these constructions with the construction of Section II. To make it easier to compare our constructions we use notions Pn,k,d 1 α, γ, P n,k,d 2 α, γ, P n,k,d 3 α, γ, and Pn,k,d 4 α, γ to denote the performances of constructions of Section II, Section VI, Subsection VII-A, and Subsection VII-B, respectivey. It is cear that for j = 1, 2, 3, 4. P j n,k,d α, γ Cexact n,k,dα, γ II. MAIN CONSTRUCTION Assume we have a storage system DSS 1 with exact repair for parameters n, k, d with a node size α and the tota repair bandwidth γ = dβ. In this section we propose a construction that gives a new storage system for parameters n = n + 1, k = k + 1, d = d + 1. Let DSS 1 consist of nodes v 1,..., v n, and et the stored fie F be of maxima size Cn,k,d exact α, γ. Let then DSS 1+ denote a new system consisting of the origina storage system DSS 1 and one extra node v n+1 storing nothing. It is cear that DSS 1+ is a storage system for parameters and can store the origina fie F. n + 1, k + 1, d + 1 Let {σ j j = 1,..., n + 1!} be the set of permutations of the set {1,..., n + 1}. Assume that DSS new j storage system for j = 1,..., n + 1! corresponding to the permutation σ j such that DSS new j is a is exacty the same

4 4 as DSS 1+ except that the order of the nodes is changed corresponding to the permutation σ j, i.e., the ith node in DSS 1+ is the σ j ith node in DSS new j. Using these n + 1! new systems as buiding bocks we construct a new system DSS 2 such that its jth node for j = 1,..., n + 1 stores the jth node from each system DSS new i for i = 1,..., n + 1!. It is cear that this new system DSS 2 works for parameters n + 1, k + 1, d + 1, has exact repair property, and stores a fie of size n + 1!Cn,k,d exact α, γ. By noticing that there are n! such permutated copies DSSnew j, where the ith node is empty, we get that the node size of the new system DSS 2 is α 2 = n + 1! n!α = n n!α. Simiary, since an empty node does not need any repair we aso find that the tota repair bandwidth of the new system is γ 2 = n + 1! n!γ = n n!γ. Definition 2.1 Symmetric repair: By symmetric repair we mean that in the repair process of a ost node, each heper node transmits the same amount β of information. Let us fix some repairing scheme for subsystems. Namey, define β ijs {0, β} to be the amount of information when the ith node repairs the jth node and the other heper nodes have indices from the set S. Now γ if j n + 1 β ijs = 0 if j = n + 1 and hence β 2 = = = i S n+1 j=1 n j=1 n j=1 S [n+1]\{j} S =d+1 S [n+1]\{j} S =d+1 n d + 1 β ijs n d 1!d! i S γ n d 1!d! γ n d 1!d! d = n n! d + 1 β. This proves that our construction has symmetric repair property. The distributed storage system DSS 1 that we used as a starting point in our construction is not yet expicity fixed. We have just fixed that the used storage system is some optima system. To make it easier to foow our 1, in progress construction we use the notation Pn+1,k+1,d+1 α, γ to denote the performance of our incompete construction. The above reasoning impies the equaity 1, in progress Pn+1,k+1,d+1 n n!α, n n!γ = n + 1!Cexact n,k,dα, γ. 3 Dividing both sides by n n! gives 1, in progress Pn+1,k+1,d+1 n + 1 α, γ = n Cexact n,k,dα, γ. 4

5 5 Exampe 2.1: If we reax on the requirement of a DSS to have symmetric repair then the construction becomes a bit simper. Now, require instead ony that the tota repair bandwidth γ is constant i.e., β may take different vaues depending on the node. Let n, k, d = 3, 2, 2 and DSS 1 be a distributed storage system with exact repair. Let DSS new j be a storage system with 4 nodes for j = 1,..., 4 where the jth node stores nothing, the ith node for i < j stores as the ith node in the origina system DSS 1, and the ith node for i > j stores as the i 1th node in the origina system DSS 1. That is, in the jth subsystem DSS new j nodes are as those in the origina system DSS 1. the jth node stores nothing whie the other Using these four new systems as buiding bocks we construct a new system DSS 2 for parameters 4, 3, 3 such that its jth node for j = 1,..., 4 stores the jth node from each system DSS new i in DSS 2 stores 4 1α = 3α and the tota repair bandwidth is 4 1γ = 3γ. for i = 1,..., 4. Hence each node For exampe, if the origina system DSS 1 consists of nodes v 1 storing x, v 2 storing y, and v 3 storing x + y then DSS new 1 consists of nodes u 11 storing nothing, u 12 storing x 1, u 13 storing y 1, and u 14 storing x 1 + y 1. Simiary DSS new 2 consists of nodes u 21 storing x 2, u 22 storing nothing, u 23 storing y 2, and u 24 storing x 2 + y 2 and so on. Then in the resuting system the first node w 1 consists of nodes u 11 storing nothing, u 21 storing x 2, u 31 storing x 3, and u 41 storing x 4. The stored fie is x 1, x 2, x 3, x 4, y 1, y 2, y 3, y 4. w 1 : x 2 x 3 x 4 w 2 : x 1 y 3 y 4 w 3 : y 1 y 2 x 4 +y 4 w 4 : x 1 +y 1 x 2 +y 2 x 3 +y 3 Fig. 1. The figure iustrates the DSS buit in Exampe 2.1. It consists of nodes w 1, w 2, w 3, and w 4. III. BOUNDS FROM THE CONSTRUCTION Next we wi derive some inequaities for the capacity in the case of exact repair. Using Equation 4 inductivey we get Theorem 3.1: For an integer j [0, k 1] we have Cn,k,d exact α, γ n n j Cexact n j,k j,d jα, γ. It is proved in [6], [8], [9], and [10] that the MSR point can be achieved if exact repair is assumed. As a consequence of this and Theorem 4 we get the foowing bound.

6 6 Theorem 3.2: For integers 1 i k we have α, C exact n,k,d d k + iα Proof: Write n = n j, k = k j, d = d j, α = B k, and γ = in [6], [8], [9], and [10] that i.e., Hence by Theorem 3.1 we have Cn exact,k,d α, γ = B, niα n k + i. Cn j,k j,d j exact d jα α, = k jα. Cn,k,d exact d jα nk jα α,. n j Now a change of variabes by setting i = k j gives us the resut. d B k d k +1. It is proved for the MSR point Our construction is now ready since we have decided to use MSR optima codes as a starting point for our construction. So et us use the notion P 1 n,k,d α, d k + iα = for integers i = 1,..., k, to note the performance of our construction. niα n k + i Exampe 3.1: Tian characterized the rate region of the exact-repair regenerating codes in the case n, k, d = 4, 3, 3 in [16]. In this exampe we wi compare our construction to this. In [16] the stored fie is assumed to be of size 1 and then the rate-region of exact-regenerating codes is characterized by foowing pairs of α, β: 1 3, 1 3, 3 8, 1 4, and 1 2, 1 6. These correspond to foowing pairs of α, γ: 1 3, 1, 3 8, 3 4, and 1 2, 1 2, i.e., 1 Cexact 4,3,3 3, 1 = 1, C4,3,3 exact 3 8, 3 4 = 1, and C exact 1 4,3,3 2, 2 1 = 1. Theorem 3.2 gives in this same specia case for integers i = 1, 2, 3. P 1 4,3,3α, iα = 4iα 1 + i Hence P 1 4,3,3 α, α = 2α, P 1 4,3,3 α, 2α = 8α 3, and P 1 4,3,3 α, 3α = 3α. By substituting into these α = 1 2, 3 8, 1 3, respectivey, we get exacty the same performances as in [16]. 5 IV. EXAMPLE: CASE n = k + 1 = d + 1 In this section we study the specia case n = k + 1 = d + 1 and compare the resuting capacity with exact repair to the known capacity with the assumption of functiona repair, n 2 { C n 1,n 1 α, γ = min Our construction gives codes with performance j=0 α, n 1 j n 1 γ P 1 n,n 1,n 1α, iα = niα 1 + i }.

7 7 for integers i = 1,..., k. Notice that now in the extreme points our performance P 1 n,n 1,n 1 achieves the known capacity, i.e., C exact n,n 1,n 1α, α = P 1 n,n 1,n 1α, α = nα 2 for the MBR point and C exact n,n 1,n 1α, kα = P 1 n,n 1,n 1α, kα = n 1α for the MSR point. As an exampe we study the fraction P 1 n,n 1,n 1α, iα C n 1,n 1 α, iα = for integers i [1, k]. Writing it out we see that where T = n 11 1 i. = P 1 n,n 1,n 1α, iα C n 1,n 1 α, iα ni 1+i T j=0 1 + n 2 j=t +1 = T For arge vaues of n this is approximatey for a i = 1,..., k. with i 2n 1 niα 1+i { } n 2 j=0 min α, n 1 j n 1 iα n 1 j n 1 ni 1+i i n T 1n T 2, 2i 2 2i 2 + i Notice that if we had chosen n = k + 2 = d + 2 instead of n = k + 1 = d + 1, then we woud have ended up 2i 2 2i 2 + 3i 2. Simiary, if we had chosen n = k + 3 = d + 3 then we woud have ended up with 2i 2 2i 2 + 5i 3. These both are aso cose to 1 when i is not too sma. For this reason we wi study the asymptotic behavior of the capacity curve more carefuy in the next section. 6 V. THE CASE WHEN n, k AND d ARE CLOSE TO EACH OTHER Next we wi study the specia case where n, k and d are cose to each other. We wi do this by setting n M = n + M, k M = k + M and d M = d + M and etting M, and then examine how the capacity curve asymptoticay behaves. The exampe in the previous section showed us that in that specia case the performance Pn,k,d 1 α, γ is quite cose to the capacity of functionay regenerating codes. However, in the previous section we

8 8 TABLE I THE PERFORMANCE OF CONSTRUCTION OF SECTION II n, k, d = 100, 99, 99 n, k, d = 100, 96, 98 n, k, d = 100, 95, n, k, d = 100, 90, 90 n, k, d = 100, 85, 90 n, k, d = 100,, 85 The figures show the performance Pn,k,d 1 of codes from construction of Section II dotted curve between the capacity of functionay repairing codes uppermost curve and the trivia ower bound given by interpoation of the known MSR and MBR points with different n, k, d. Here α = 1, and γ [1, d d k+1 ]. fixed i to be an integer and then assumed that n is arge. In this section we tie up the vaues i and M together to arrive at a situation where the tota repair bandwidth stays on a fixed point between its minima possibe vaue given by the MBR point and its maxima possibe vaue given by the MSR point. For each M our construction gives a code with performance Pn 1 M,k M,d M α, d M k M + iα d M k M + 1 for i = 1,..., k M, hence Pn 1 M,k M,d M α, d M k M +xα these points. d M k M +1 = n M iα n k + i with x [1, k] is the piecewise inear curve connecting Let s 0, 1] be a fixed number and i = 1 + sk M 1. We wi study how the fraction Pn 1 M,k M,d M α, d M k M +iα d M k M +1 C km,d M α, d M k M +iα d M k M +1 behaves as we et M. Informay this tes how cose our performance curve and the known capacity curve are to each other when M is arge, i.e., vaues n M, k M, d M are cose to each other. Remark 5.1: In the MSR point we have γ MSR = d M α d M k M + 1

9 9 and in the MBR point γ MBR = α. Hence α dm k M + i d M k M + 1 = sγ MSR + 1 sγ MBR. Theorem 5.1: Let s 0, 1] be a fixed number and i = 1 + sk M 1. Then Pn 1 M,k M,d M α, d M k M +iα d M k M +1 im M C km,d M α, d M k M +iα d M k M +1 = 1. The proof is rather technica and is hence postponed to Appendix. As a straightforward coroary to Theorem 5.1 we have Theorem 5.2: Let s [0, 1] be a fixed number and et γ MSR = d M α d M k M +1 and γ MBR = α. Then Cn exact im M,k M,d M α, sγ MSR + 1 sγ MBR = 1. M C km,d M α, sγ MSR + 1 sγ MBR VI. A SIMPLER CONSTRUCTION In this section we wi give a construction of a distributed storage system that again uses optima codes at the MSR point as buiding bocks. There are two important differences to the main construction in Section II of this paper. The first difference is the easiness of the construction of this section. The second is that this construction has no symmetric repair. We ony require that the tota repair bandwidth is fixed to be γ but it can consist of varying βs. A. Construction We are interested in to design a storage system for given parameters n, k, d and α, γ. Write ɛ = n k and Choose Z + integers n 1,..., n such that δ = n d. n j ɛ + 1 for a j = 1,..., and n = n n. For this choice, write k j = n j ɛ and d j = n j δ for a j = 1,...,.

10 10 Assume we have storage systems DSS 1,..., DSS corresponding parameters n 1, k 1, d 1,..., n, k, d, respectivey. Each of these systems has node size α and tota repair bandwidth γ. Suppose we put these systems together to get a new bigger system DSS big with n n = n nodes and storing the same fies that origina systems DSS 1,..., DSS store. This is indeed a distributed storage system for parameters n, k, d and α, γ: It is cear that we have n nodes, each of size α. Each set of k nodes can recover the fie: Indeed, there are ɛ = n k nodes that are not part of the reconstruction process. Hence of each subsystem DSS j we have at east n j ɛ = k j nodes that are part of the reconstruction process and hereby we can recover the corresponding fie and hence the whoe fie. By the same argumentation as above we notice that contacting any d of the nodes we can repair a ost node. Hence we ony have to downoad the same amount of information in the repair process of this new bigger system as in the repair process of the corresponding subsystem the tota repair bandwidth is indeed γ. Remark 6.1: The main disadvantage of constructed storage systems of this section is that they do not have symmetric repair. By shuffing the nodes corresponding to each permutation on set {1,..., n} as in the construction of Section II woud give a DSS with symmetric repair and same performance. However, this woud destroy the main advantage of this construction, namey its easiness. v 11 DSS 1 DSS 2 DSS v 12 v 1n1 v 21 v 22 v 2n2 v 1 v 2 v 11 v 12 v 1n1 v 21 v 22 v 2n2 v 1 v 2 v n DSS big v n Fig. 2. The figure iustrates the construction of Section VI. First we have storage systems DSS 1,..., DSS and then we just put them together to get a new storage system DSS big.

11 11 B. The Performance of the Construction In the construction we did not stick to any fixed type of subsystem. Hence we have the foowing genera inequaity. Proposition 6.1: Given positive integers n, k, d with k d < n and the decomposition of n to positive integers n 1,..., n with n = n n and n j n k + 1 for a j = 1,...,. Define aso integers k j = n j n + k and d j = n j n + d for j = 1,...,. Then we have C exact n,k,dα, γ j=1 Proof: The setup is just as in the construction of subsection VI-A. To make it easier to foow, et us use the notation P construction. By above, we have 2, in progress Pn,k,d α, γ = C exact n j,k j,d j α, γ. 7 2, in progress n,k,d j=1 α, γ for the performance of this incompete C exact n j,k j,d j α, γ. 8 Next we wi fix the subsystems DSS 1,..., DSS and then derive another ower bound for the performance of our construction of exact-regenerating codes. Let n n j = for j = 1,..., 1 and Then n n = n 1. k 1 = n 1 n + k, d 1 = n 1 n + d, n k = n n + k = k 1 and n d = n n + d = d 1. By substituting these into the equaity 8 we get P 2, in progress n,k,d α, γ = 1Cn exact 1,k 1,d 1 α, γ + Cn exact,k,d α, γ.

12 12 To finish our construction we again use MSR optima codes as buiding bocks and substitute in the above giving C exact n 1,k 1,d 1 α, By noticing that d 1α d k+1 C exact n,k,d and then defining α new = d k+1γ d γ = = k 1 α and hence α, d 1 α α = = d1α d, i.e., d 1 α d 1 k = d 1 α 1k 1 α + C exact n,k,d α, γ γ d 1 d d 1 α. we find that C exact n,k,d α, γ = d α new = d α new d k + 1 d 1 α Cn exact,k,d α new, d 1 α = Cn exact d α new,k,d α new, d k = k α new = k d 1 α d. Here the second to the ast equaity was again because of the fact that we know that the MSR point can be achieved by exact-regenerating codes. In the cacuation above giving the inequaity Cn exact d,k,d α, 1α d k+1 k d 1α d we just adapted the biggest possibe MSR code when the upper bounds for node size and tota repair bandwidth was given. The reason for this is that we are eager to give a very simpe construction by using aready known MSR codes as buiding bocks. So now we are ready to give a new ower bound for the capacity of exact-regenerating codes. n Theorem 6.2: For integers 1 n+1 k we have Cn,k,d exact d 1 α α, 1k 1 + k d 1 α 10 d with n k 1 = n n + k and d 1 = n + d and n k = k 1 n and d = d 1. Proof: By the above reasoning we have the inequaity 10 for given if we can spit our n, k, d storage system into pieces by the above way. This is possibe if we have 1 k 1 d 1 n 1 1 and 1 k d n 1. The first chain of inequaities is proved by noticing that d 1 = n 1 n + d n 1 1,

13 13 and n k 1 = n 1 n + k = n + k n n n+1 k n + k = 1 d 1 k 1 = n 1 n + d n 1 n + k = d k 0. The second chain of inequaities is proved by noticing that and k k 1n d = n n d n 1, = k n + n k n + n n n+1 k d k = n n + d n n + k = d k 0. = 1 Hence the performance of our construction is α, for 1 n n+1 k. P 2 n,k,d d 1 α = 1k 1 + k d 1 α 11 d Exampe 6.1: Let n, k, d = 3, 2, 2. Suppose a system with the first node storing x, second node storing y and third node storing x + y. This MSR-optima code storing a fie x, y has node size α = 1 and tota repair bandwidth γ = 2β = 2. Take three copies of this system to form a bigger system with nine nodes: x 1, y 1, x 1 + y 1, x 2, y 2, x 2 + y 2, x 3, y 3, x 3 + y 3. Simiary as in our construction this is a storage system with n, k, d = 9, 8, 8, node size α = α = 1, and tota repair bandwidth γ = 2. It stores a fie x 1, y 1, x 2, y 2, x 3, y 3 of size 6. C. Connection to the Construction of Section II i.e., Consider equaity 11 in the case divides n, i.e., n = n 1. In that case we have k 1 = k and d 1 = d and hence Pn,k,d 2 d 1 α α, = k 1 α, P 2 n,k,d Let j = k n + n. Since 1 α, n n + dα n we have j k n + n+1 k = n n + kα. 12 n n n+1 k = 1 and j k n + n = k. Hence

14 14 TABLE II THE PERFORMANCE OF CONSTRUCTION OF SECTION VI n, k, d = 100, 99, 99 n, k, d = 100, 96, 98 n, k, d = 100, 95, n, k, d = 100, 90, 90 n, k, d = 100, 85, 90 n, k, d = 100,, 85 The figures show the performance Pn,k,d 2 of codes from construction of Section VI dotted curve between the capacity of functionay repairing codes uppermost curve and the trivia ower bound given by interpoation of the known MSR and MBR points with different n, k, d. Here α = 1, and γ [1, d d k+1 ]. we can use Equation 5 with this vaue. We get Pn,k,d 1 α, n n + dα = Pn,k,d 1 so the performances Pn,k,d 1 and P n,k,d 2 are same in this case. = α, njα n k + j = nk n + n α n d k + jα = n n + kα = Pn,k,d 2 α, n n + dα This tes us that the performance of the construction of Section II and the performance of the construction of Section VI are exacty the same whenever divides n, i.e., whenever the atter construction is buit using optima MSR codes of equa size n. The expanation for the simiarity of the performances of these two constructions is that the main idea of the both constructions is to increase vaues k and d but to restrain the vaues α and γ. 13

15 15 VII. COMPARISON TO SIMILAR CONSTRUCTIONS The main idea in our construction of Section II was to add a new empty node in the storage system. The benefit of this was the reduction of the average node size and the average tota repair bandwidth. The drawback was that we had to increase parameters k and d. In this section we study what happens if we add something ese than an empty node in the system. We try out what happens when adding an exact copy of some existing node and when adding the stored fie itsef. We wi see that these variations are not very usefu. The performance of the construction of Subsection VII-A is moderate but the performance of the construction of Subsection VII-B is not good. The key differences wi be summarized in Subsection VII-C. A. Construction by Copying Nodes Assume we have a storage system DSS 1 with exact repair for parameters n, k, d with the node size α and the tota repair bandwidth γ = dβ. In this section we propose a construction that gives a new storage system for parameters n = n +, k = k +, d = d + for integers = 1,..., k 1. Let DSS 1 consist of the nodes v 1,..., v n, and et the stored fie F be of maxima size Cn,k,d exact α, γ. Let then DSS 1+ denote a new system consisting of the origina storage system DSS 1 and extra nodes v n+1,..., v n+ such that v n+j is the exact copy of the node v j for j = 1,...,. It is cear that DSS 1+ is a storage system for parameters and can store the origina fie F. n +, k +, d + Again we use permutations just simiary as in the construction of Section II: et {σ j j = 1,..., n +!} be the set of permutations of the set {1,..., n + }. Assume that DSS new j corresponding to the permutation σ j such that DSS new j is a storage system for j = 1,..., n +! is exacty the same as DSS 1+ except that the order of the nodes is changed corresponding to the permutation σ j, i.e., the ith node in DSS 1+ is the σ j ith node in DSS new j. Using these n +! new systems as buiding bocks we construct a new system DSS 2 such that its jth node for j = 1,..., n + stores the jth node from each system DSS new i for i = 1,..., n +!. It is cear that this new system DSS 2 works for parameters n +, k +, d +, has exact repair property, and stores a fie of size n +!C exact n,k,d α, γ. The node size of the new system DSS 2 is α 2 = n +!α. When repairing a node there are 2d + n + 2! subsystems in which the exact copy of the ost node is one of the heper nodes. Hence there are n +! 2d + n + 2! subsystems in which this not the case. So

16 16 the tota repair bandwidth is γ 2 = 2d + n + 2!α + n +! 2d + n + 2!γ Hence the performance of this incompete construction is 3, in progress Pn+,k+,d+ α 2, γ 2 = n +!Cn,k,dα, exact γ that is 3, in progress Pn+,k+,d+ α, γ 3 = Cn,k,dα, exact γ 14 for = 1,..., k 1 with γ 3 = γ 3 = γ n+!, that is, 2d + n + n + 1 α + 1 2d + n + n + 1 γ. By the change of variabes n = n +, k = k +, d = d + we obtain for = 1,..., k 1 2 with γ 4 = P 3, in progress n,k,d α, γ 4 = Cn,k,d α, exact γ 15 2d nn 1 α + 1 2d nn 1 γ. Finish again the construction by using MSR-optima codes as a starting point. The performance we obtain is P 3 n,k,dα, γ 4 = k α 16 with γ 4 = 2d nn d d α. nn 1 B. Construction by Adding the Fie Assume we have a storage system DSS 1 with exact repair for parameters n, k, d with the node size α and the tota repair bandwidth γ = dβ. In this section we propose a construction that gives a new storage system for parameters n = n + 1, k = k, d = d. Let DSS 1 consist of the nodes v 1,..., v n, and et the stored fie F be of maxima size Cn,k,d exact α, γ. Let then DSS 1+ denote a new system consisting of the origina storage system DSS 1 and one extra node v n+1 storing the whoe fie F. It is cear that DSS 1+ is a storage system for parameters and can store the origina fie F. n + 1, k, d Again we use permutations just simiary as in the construction of Section II: et {σ j j = 1,..., n + 1!} be the set of permutations of the set {1,..., n + 1}. Assume that DSS new j is a storage system for j = 1,..., n + 1!

17 17 TABLE III THE PERFORMANCE OF THE CONSTRUCTION OF SECTION VII-A n, k, d = 100, 99, 99 n, k, d = 100,, 85 The figure shows the performance Pn,k,d 3 of codes from the construction of Subsection VII-A dotted curve between the capacity of functionay repairing codes uppermost curve and the trivia ower bound given by interpoation of the known MSR and MBR points with different n, k, d. corresponding to the permutation σ j such that DSS new j is exacty the same as DSS 1+ except that the order of the nodes is changed corresponding to the permutation σ j, i.e., the ith node in DSS 1+ is the σ j ith node in DSS new j. Using these n + 1! new systems as buiding bocks we construct a new system DSS 2 such that its jth node for j = 1,..., n + 1 stores the jth node from each system DSS new i for i = 1,..., n + 1!. It is cear that this new system DSS 2 works for parameters n + 1, k, d, has exact repair property, and stores a fie of size n + 1!Cn,k,d exact α, γ. By noticing that there are n! such permutated copies DSSnew j where the ith node is storing the whoe fie we get that the node size of the new system DSS 2 is α 2 = n + 1! n!α + n!c exact n,k,dα, γ = n!nα + C exact n,k,dα, γ Since to repair a node storing the whoe fie can be done by bandwidth of size kα and repairing a node when the whoe fie is one of the heper nodes requires bandwidth α, we find that the tota repair bandwidth of the new

18 18 system is γ 2 =nn 1 n d n d!γ + ndn 1 n d + 1 n d!α + n!kα 17 =n!n dγ + d + kα Hence the performance of this incompete construction is that is P P 4, in progress n+1,k,d 4, in progress n+1,k,d =n + 1C exact n,k,dα, γ. α 2, γ 2 = n + 1!C exact n,k,dα, γ nα + Cn,k,dα, exact γ, n dγ + d + kα Substituting MSR point into above gives a code with performance dn d Pn+1,k,d 4 n + kα, + d + k α = n + 1kα i.e. P 4 n+1,k,d α, nd + d k2 + kα = n + k n + 1kα. n + k However, this construction is useess because it is easy to verify that this performance is stricty worse than the trivia ower bound by timesharing when d > k and it ies on the timesharing ine when k = d. 18 C. Summary of Differences of Different Approaches Despite the cear simiarities of the construction techniques, there is a huge difference on the performances Pn,k,d 1 α, γ, P n,k,d 3 α, γ, and P n,k,d 4 α, γ of codes constructed using these different approaches. In the cases where the performance Pn,k,d 1 α, γ of the construction of Section II is very poor, the construction of Section VII-B performs better. However, the performance Pn,k,d 4 α, γ of the construction of Section VII-B is sti worse than the one achieved by timesharing of MSR and MBR points. Comparing to the trivia ower bound given by timesharing MBR and MSR points one can summarize that the construction of Subsection VII-B is useess, the construction of Subsection VII-A is in certain cases quite good, and the construction of Section II is in certain cases very good. VIII. CONCLUSIONS We have constructed exact-regenerating codes between MBR and MSR points. To the best of author s knowedge, no previous construction of exact-regenerating codes between MBR and MSR points is done except in [17]. Compared to that construction, our construction is very different. We have shown in this paper that when n, k, and d are cose to each other, the capacity of a distributed storage system when exact repair is assumed is essentiay the same as when ony functiona repair is required. This was

19 19 proved by using a specific code construction expoiting some aready known codes achieving the MSR point on the trade-off curve and by studying the asymptotic behavior of the capacity curve. A very easy aternative construction for the main construction of this paper was presented. Its performance is amost as good as the performance of the main construction and it is simpe to buid up. The drawback of this construction was that it has no symmetric repair. Aso we have constructed two constructions in a simiar manner as the main construction. These were to be compared to the main construction. Despite the cear simiarities of these three constructions their performances vary hugey. However, when n, k, and d are not cose to each other then the performance of our main construction is not good when compared to the capacity of functionay repairing codes. However, there is no evidence that the capacity of a distributed storage system when exact repair is assumed is generay cose to the capacity of functionay repairing codes. So as a future work it is sti eft to find the precise expression of the capacity of a distributed storage system when exact repair is assumed, and especiay to study the behavior of the capacity when n, k, and d are not cose to each other. IX. ACKNOWLEDGMENTS This research was party supported by the Academy of Finand grant # and by the Emi Aatonen Foundation, Finand, through grants to Camia Hoanti. Prof. Saim E Rouayheb at the Iinois Institute of Technoogy is gratefuy acknowedged for usefu discussions. Prof. Camia Hoanti at the Aato University is gratefuy acknowedged for usefu comments on the first draft of this paper. have i i and APPENDIX The proof of Theorem 5.1: Let i = 1 + sk M 1. We study the behavior of the fraction for arge M, so we We have 1. Thus, to simpify the notation, we may assume that i acts as an integer. We aso use the notation t = d M sk M 1 + sk M 1. Pn 1 M,k M,d M α, d M k M sk M 1α d M k M + 1 = n M 1 + sk M 1α n k + i C km,d M α, d M k M + iα d M k M + 1 t =α 1 + =α k M 1 d M j d k + i d j=0 j=t+1 M t k M t 12d + M k td k + i 2d M, 19 20

20 20 whence where and P 1 n M,k M,d M C km,d M α, d M k M +iα d M k M +1 α, d M k M +iα d M k M +1 h 1 M = h 2 Mh 3 M + h 4 M, h 1 M = 2n M 1 + sk M 1d M, h 2 M = n k sk M 1, h 3 M = 2t + 1d M, h 4 M = k M t 12d k + M t + sk M and Now it is easy to check that as M. Note that when M is arge and hence as M. Finay, as M, proving the caim. h 4 M M 2 h 1 M M 3 2s, h 2 M M s, h 3 M M 2 2 M t ds s = k M t 12d k + M t M 0 s = 0 = P 1 n M,k M,d M C km,d M h 2M M + sk M 1 M α, d M k M +iα d M k M +1 α, d M k M +iα d M k M +1 h 1M M 3 h3m+h4m M 2 2s s2 + 0 =

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