On Cooperative Local Repair in Distributed Storage

Transcription

1 On Coopeative Loca Repai in Distibuted Stoae Anit Sinh Rawat Depatment of ECE The Univesity of Texas at Austin Austin, TX 7871 Emai: Aya Mazumda Depatment of ECE Univesity of Minnesota Twin Cities Minneapois, MN Emai: Siam Vishwanath Depatment of ECE The Univesity of Texas at Austin Austin, TX 7871 Emai: Abstact Easue-coectin codes, that suppot oca epai of codewod symbos, have attacted substantia attention ecenty fo thei appication in distibuted stoae systems. In this pape we study a eneaization of the usua ocay ecoveabe codes. We conside such codes that any sma set of codewod symbos is ecoveabe fom a sma numbe of othe symbos. We ca this coopeative oca epai. We pesent bounds on the dimension of such codes as we as ive expicit constuctions of famiies of codes. Some othe esuts eadin coopeative oca epai ae aso pesented, incudin an anaysis fo the Hadamad codes. I. INTRODUCTION In this pape we expoe a new cass of codes that enabe efficient ecovey fom the faiue (easue) of a sma numbe of code symbos. In paticua, we study codes with (, )- ocaity which aow fo any faied codes symbos to be ecoveed by contactin othe intact code symbos. Ou study of such codes is motivated by thei appication in distibuted stoae systems a..a. coud stoae, whee infomation is stoed ove a netwo of stoae nodes (diss). In distibuted stoae systems, one has to intoduce edundancy in ode to potect the stoed infomation aainst inevitabe node (dis) faiues. The taditiona easue codes ae hihy sub-optima fo distibuted stoae settin as these codes incu hih cost of code epai [1]. Recenty, mutipe casses of easue codes have been poposed to optimize the pefomance of code epai pocess with espect to vaious metics. In paticua, the codes that minimize epai-bandwidth, i.e., the numbe of bits communicated duin a node epai, ae studied in [1] [4] and efeences theein. Anothe famiy of easue codes that focus on sma ocaity, i.e., enabin epai of sine code symbo by contactin a sma numbe of othe code symbos, ae pesented in [5] [9]. A code is said to have a-symbo ocaity of if evey code symbo is a function of at most othe code symbos. This ensues oca epai of each code symbo by contactin at most othe code symbos. In this pape we eneaize the notion of codes with a-symbo ocaity to codes with (, )- ocaity: any set of code symbos ae functions of at most othe code symbos. This aows fo coopeative oca epai of code symbos, whee a oups of code symbos is epaied by contactin at most othe code symbos. The appoach of coopeative code epai has been peviousy expoed in the context of epai-bandwidth efficient codes in [10], [11] and efeences theein. In this pape we addess two impotant issues eadin codes with (, )-ocaity: 1) obtainin tade-offs amon minimum distance, dimension, and ocaity paametes (, ) fo such code; and ) pesentin expicit constuctions fo codes with (, )-ocaity that ae cose to the obtained tade-offs. We pesent a foma definition of codes with (, )-ocaity in Section II. In Section III, we obtain an uppe bound on the minimum distance of a code with (, )-ocaity which encodes infomation symbos to n symbos on codewods. We then bound the best possibe ate fo a code with (, )-ocaity. In Section IV we pesent two constuctions fo the codes that have (, )-ocaity and comment on thei ates with espect to the bound obtained in Section III. In Section IV-D, we study the punctued Hadamad codes (a..a. Simpex codes) in the context of coopeative oca epai. In Section IV-E we eneaize the Patition code famewo fom Section IV-A. We biefy conside the settin of andom easues in Section IV-F. A shot note on notation: we use bod owe case ettes to denote vectos. Fo an intee n 1, [n] denotes the set {1,,..., n}. II. CODES WITH (, )-LOCALITY Definition 1. Let C be an (n, ) code. We ca C to be an (n, ) code with (, )-ocaity if fo each S [n] with S =, we have a set Γ S [n]\s such that 1) Γ S, ) Fo any codewod c = (c 1, c,..., c n ) C, the code symbos c S := {c i : i S} ae functions of the code symbos symbos c ΓS := {c i : i Γ S }. Note that Definition 1 ensues that any code symbos can be coopeativey epaied fom at most othe code symbos. This eneaizes the notion of codes with a-symbo ocaity [5], [6], [1], [13], whee ocaity is defined with espect to one code symbo, i.e., = 1. Rema 1. Fo a code C with a-symbo ocaity, we have the foowin bound on its minimum distance [5], [6]. d min (C) n +. (1) The code attainin the bound in (1) ae pesented in [6] [9] and efeences theein.

2 III. RATE VS. DISTANCE TRADE-OFF FOR CODES WITH (, )-LOCALITY In this section, fo iven and, we pesent a tade-off between the ate and the minimum distance of a code with (, )-ocaity (cf. Definition 1). We empoy the enea poof technique intoduced in [5], [14], [15] to obtain the foowin esut. Theoem 1. Let C be an (n, ) code (inea, o non-inea) with (, )-ocaity. Then, the minimum distance of C satisfies d min (C) n + 1 ( 1 ). () Aoithm: Constuction of sub-code C C. Input: (n, ) code C F n q with (, )-ocaity. 1: C 0 = C : = 0 3: whie C > q do 4: = : Choose i 1, i,..., i [n] such that, fo evey m [], thee exist at east two codewods in C 1 that diffe at i m-th coodinate. 6: Let R = Γ {i 1,...,i } be the index of at most code symbos that coopeativey epai code symbos indexed by {i 1,..., i }. 7: Let y F R q be the most fequent eement in the muti-set {x R : x C 1 F n q }. 8: Define C := {x : x C 1 F n q and x R = y}. 9: if 1 < C q then 10: C = C. 11: end whie 1: ese if C = 1 then 13: Pic a maxima subset R R such that C > 1, whee C := {x : x C 1 F n q, x R = ỹ } and ỹ F R q be the most fequent eement in the muti-set {x R : x C 1 F n q }. 14: C = C. 15: end whie. 16: end if 17: end whie Output: C. Fi. 1: Constuction of sub-code C C. Poof: The poof invoves constuction of a subcode C C such that a but a sma numbe of coodinates in evey codewod of C ae fixed. Note that we have d min (C) d min (C ). (3) Given C, one can obtain a code C with C = C by emovin fixed coodinates fom a the codewod in C. This impies that d min (C ) = d min (C ), which aon with (3) ive us the foowin. d min (C) d min (C ). (4) We descibe the constuction of the subcode C in Fi. 1. Befoe poceedin with the anaysis, we aue the coectness of the aoithm in Fi. 1. Note that it is aways possibe to find coodinates {i 1, i,..., i } at ine 5. When the aoithm eaches ine 5, the subcode C 1 has moe than q codewods. Theefoe, thee must be at east coodinates in the codewods in C 1 that ae not fixed in the pevious iteations. This aso impies that, fo m [], i m / I 1 := ) (R {i 1,..., i } [n]. (5) [ 1] Note that the code symbos indexed by I 1 ae fixed in C 1. Thus, we must have R = Γ {i 1,...,i } I 1. Othewise, {i 1,..., i } woud have been fixed in the pevious iteations. Assumin that the constuction in Fi. 1 ends in t-th iteation, we obtain in Appendix A that and t 1. (6) o q C I t + t. (7) Now, we define C = C It which denotes the code obtained by punctuin the codewods in C at the coodinates associated with the set I t. We have C = C and d min (C ) = d min (C ). Moeove, the enth of the codewods in C is n I. Next, appyin the Sineton bound on C ives us d min (C) d min (C ) n I t o q C + 1 n I t ( I t + t) + 1 = n t + 1, (8) It foows fom (8) and (6) that d min (C) n + 1 ( 1 ). (9) The anaysis of the minimum distance when the constuction in Fi 1 ends at ine 15 is pesented in Appendix A. This competes the poof. Note that an (n, ) code with (, )-ocaity has its minimum distance at east +1 as it can ecove fom the easue of any code symbos (cf. Definition 1). Combinin this obsevation with Theoem 1, we obtain the foowin esut. Cooay 1. The ate of an (n, ) code with (, )-ocaity is bounded as n +. (10)

3 Poof: It foows fom () and the fact d min (C) + 1 that + 1 d min (C) n + 1 ( 1 ). (11) Usin 1 1, we et o + n, n +. IV. CONSTRUCTION OF CODES WITH (, )-LOCALITY In this section we addess the issue of constuctin hih ate codes that have (, )-ocaity. In paticua, we descibe two simpe constuctions that ensue coopeative oca epai fo the faiue of any code symbos: 1) Patition code and ) Poduct code. In Patition code, we patition the infomation symbos in oups of symbo and encode each oup with an ( +, )-MDS code (cf. Section IV-A. On the othe hand, a poduct code is obtained by aanin = ( ) infomation symbos in an -dimensiona aay and then intoducin paity symbos aon diffeent dimensions of the aay (cf. Section IV-B). A. Patition Code Fo the ease of exposition, we assume that,, and ( ). Given infomation symbo ove Fq, a Patition code encodes these symbos into n = + symbos on codewods as foows (see Fi. ): 1) Patition infomation symbos into = oups of size each. ) Encode the symbos in each of the oups usin an ( +, )-MDS code ove F q. We efe to the + code symbos obtained by encodin infomation symbo in the i-th oup as i-th oca oup. As it is cea fom the constuction, Patition code has ate n = +. Moeove, a code symbo can be ecoveed fom any othe code symbo fom its oca oup. In the wost case, when faied code symbos beon to distinct oca oup, we can ecove a symbos fom = code symbos, downoadin symbos fom each of the oca oups containin one faied code symbo. B. Poduct Code Poduct codes ae a we nown famiy of codes in the codin theoy iteatue. Given = ( ) infomation symbos and, we fist aane = ( ) infomation symbos in an -dimensiona aay with index of each dimension of the aay anin in the set [ ]. These infomation symbos ae then encoded to obtain an n = ( + 1) symbos on code wod. In the foowin we descibe the encodin pocess fo = -dimensiona aay (see Fi. 3). The eneaization of the encodin pocess fo hihe dimensions is staihtfowad. 1) Aane = ( ) infomation symbos in an aay. ) Fo each ow of the aay, add a paity symbo by summin a symbos in the ow and append these symbos to thei espective ows. 3) Fo each of the + 1 coumns of the updated aay, add a paity by summin a symbos in the coumn. m 1 m m p 1 m +1 m + m p m γ+1 m γ+ m 4 p c 1 p c p c. p p c +1 Fi. 3: Desciption of Poduct code constuction with =. {p 1, p,..., p } denote the paity nodes associated with each ow of the two dimensiona aay of infomation symbos. {p c 1, p c,..., p +1} epesent the paity symbos fo the coumns of the updated aay. We use γ to denote ( 1). C. Compaison between Patition and Poduct code We now compae the ate of Patition code and Poduct code with the bound in (10). Fo any 1, we have ( ) + +. (1) Note that (1) foows fom the fact that ( + ) ( + 1 ) + ( ( ) i i) = ( + ). i Theefoe, Patition code appoach povides (, )-ocaity with a bette ate. Howeve, fo a system paametes, the ate of Patition code is smae than the nown bound (10), i.e., + +. Hee, we woud ie to note that the diffeence between the ate achieved by the Patition code and the bound in (10) ets smae as the paamete becomes ae as compaed to the paamete. It is an inteestin pobem to eithe tihten the bound in (10) o pesent a constuction fo codes with (, )- ocaity which have hihe ate than that of Patition code. i= D. Coopeative Loca Repai fo Hadamad Codes Hee, we comment on the oca epaiabiity of punctued Hadamad codes. Punctued Hadamad codes ae aso efeed to as Simpex codes, which ae the dua codes of Hammin codes. In paticua, we show that an [n = 1,, 1 ] punctued Hadamad code has ( = + 1, ) ocaity fo any n 1. An [n = 1,, 1 ] punctued Hadamad code encodes a bits on messae (m 1, m,..., m ) to an n = 1 codewod c = (c 1, c,..., c n= 1 such that c i = m b i (mod). =1

4 m = (m 1, m,..., m ) F q m 1 m m m +1 m + m m +1 m + m ( +, )-MDS code ( +, )-MDS code ( +, )-MDS code c 1 c c c + c ++1 c ++ c + c + c τ+1 c τ+ c n c n oca oup 1 oca oup Fi. : Iustation of the encodin pocess of Patition code. oca oup Hee b i = (b i 1, b i,..., b i ) F denotes the binay epesentation of the intee i [ 1]. In an [n = 1,, 1 ] punctued Hadamad code, we have c i + c = c i+, whee 1 1 and i [ 1]. Moeove, we note that an [n = 1,, 1 ] punctued Hadamad code has a paticua stuctua popety: fo any <, the pefix of enth 1 of each codewod is a codewod of the [ñ = 1,, 1 ] punctued Hadamad code which encodes the messae (m 1, m,..., m ). We now pesent the main esut of this subsection: Lemma 1. In an [n = 1,, 1 ] punctued Hadamad code, any 1 n 1 easues can be coected by contactin at most + 1 othe code symbos. Poof: We pove the Lemma by usin induction ove. Fo base case, we conside =, whee the [n = 3 = 1,, ] punctued Hadamad code encodes the messae (m 1, m ) to a codewod (c 1, c, c 3 ) = (m 1, m, m 1 +m ). In this case any = 1 easue can be ecoveed by contactin othe + 1 = code symbos. Fo exampe, one can ecove c = m fom (c 1, c 3 ) = (m 1, m 1 + m 3 ). Fo inductive step, we assume that the Lemma hods fo any punctued code of dimension up to 1. Conside the [n = 1,, 1 ] punctued Hadamad code of dimension, and two cases eadin the positions of eased code symbos. Case 1: Thee ae x 1 easues amon the fist n = 1 1 code symbos. Note that the fist n = 1 1 code symbos constitute a codewod of an [ n = 1 1, 1, ] punctued Hadamad code. Theefoe, fom the inductive hypothesis, one can coect the x easues amon the fist n code symbos by contactin x + 1 othe code symbos out of these n code symbos. Now, if the symbo c 1 in easue, we can ecove it by contactin one of the uneased symbo amon {c +1, c 1 +,..., c 1 n= 1} say c 1 + and the coespondin code symbo c fom the fist n code symbos. Now, we can epai the emainin eased symbos amon {c 1 +1, c 1 +,..., c n= 1} fom c 1 and the coespondin code symbo amon the fist n code symbos. Fo exampe, if we want to ecove the symbo c 1 +m, we can use c 1 and c m to econstuct c 1 +m. In the wost case, we contact +1 code symbos duin the epai of a easues. Case : Thee ae x easues amon the fist n = 1 1 code symbos. In this case, we fist ecove the code symbo c 1, if it is in easue. Without oss of eneaity we assume that c 1 is in easue. Note that thee ae n 1 = 1 distinct pais of code symbos {c i, c 1 +i} i [ 1 ] that can ecove c 1. Since we have at most n 1 1 = 1 1 easues apat fom c 1, one of the 1 pais {c i, c +i} 1 i [ 1 ] must be intact. This pai aows us to ecove c 1. Now that we now the symbo c 1 = m, we can emove the contibution of m fom any of the ast 1 code symbos {c 1 +1, c 1 +,..., c n= 1}. Simiay, we can add m to any of the fist n = 1 1 code symbos {c 1, c,..., c 1}. Theefoe, we can educe the Case to Case 1 of the poof, and epai any 1 easues by contactin at most code symbos. Combinin both cases competes the poof. E. Geneaization of Patition Code Appoach In the constuction of the Patition codes, as descibed in Section IV-A, we use an ( +, ) MDS code to encode disoint oups of messae symbos. Note that the ate of this MDS code ovens the ate of the ovea code. One can potentiay use anothe code C oca of minimum distance at east + 1 to encode disoint oups of messae symbos. Now, we use (x), x [] to denote the numbe of symbos that needs to be contacted to epai x easue in one oca oup. Fo the case when an ( +, ) MDS code is used, we have (x) = fo x []. Let (x) denote the uppe concave enveope of (x) on the inteva [1, ] R. Assume that we have disoint oca oups, then a patten of easues can be epesented by a vecto ( 1,,..., ). Hee, i denotes the numbe of easues within i-th oca oup. Note that we have i=1 i =. Fo a iven oca code C oca, one needs to access i=1 ( i) numbe of intact code symbos to epai the easue patten ( 1,,..., ). Now, we use concavity of ( ), the fact that (x) (x) fo x [], and Jensen s inequaity to obtain the

5 foowin. ( i ) i=1 ( i ) ( i=1 = ( ) i=1 i ) (13) Since the ate of the Patition code is anostic to the numbe of oca oups, we can use the vaue of which minimizes the R.H.S. of (13). This appoach optimizes the vaue of fo a iven choice of and C oca. F. Comment of Random Easues The constuctions in Section IV-A & IV-B ae desined to aow fo the coopeative oca epais in the case of advesaia easue pattens. One can conside the settin whee easues occu accodin to a andom mode. Hee, we biefy comment on the settin whee easues ae unifomy distibuted amon the code symbos. Moeove, we assume and to be ae enouh. We conside the constuction of Section IV-A, that is, the Patition codes. In this case, with easonaby hih pobabiity (dependin on and ), ( each ) oca oup (a tota of them) expeiences about t Θ numbe of easues. Theefoe, with hih pobabiity, one can pefom coopeative oca epai of andom easues even if an ( + t, )-MDS code in empoyed in the constuction of the Patition code (cf. Section IV-A). This tansates to a codin scheme with +t the ovea ate of. One can tae ae enouh to optimize this vaue. Indeed, usin the techniques of [16], it can be shown that, fo andom easues, the patition codes ae asymptoticay optima. REFERENCES [1] A. Dimais, P. Godfey, Y. Wu, M. Wainwiht, and K. Ramchandan. Netwo codin fo distibuted stoae systems. IEEE Tans. Inf. Theoy, 56(9): , 010. [] K. Rashmi, N. Shah, and P. Kuma. Optima exact-eeneatin codes fo distibuted stoae at the MSR and MBR points via a poduct-matix constuction. IEEE Tans. Inf. Theoy, 57:57 539, 011. [3] I. Tamo, Z. Wan, and J. Buc. Ziza codes: Mds aay codes with optima ebuidin. IEEE Tans. Inf. Theoy, 59(3): , 013. [4] D. Papaiiopouos, A. Dimais, and V. Cadambe. Repai optima easue codes thouh hadamad desins. IEEE Tans. Inf. Theoy, 59(5): , 013. [5] P. Gopaan, C. Huan, H. Simitci, and S. Yehanin. On the ocaity of codewod symbos. IEEE Tans. Inf. Theoy, 58(11): , 01. [6] D. Papaiiopouos and A. Dimais. Locay epaiabe codes. In Poc. of IEEE ISIT, 01. [7] A. S. Rawat, O. O. Koyuou, N. Sibestein, and S. Vishwanath. Optima ocay epaiabe and secue codes fo distibuted stoae systems. IEEE Tans. Inf. Theoy, 60(1):1 36, 014. [8] G. Kamath, N. Paash, V. Laitha, and P. Kuma. Codes with oca eeneation. CoRR, abs/ , 01. [9] I. Tamo and A. Ba. A famiy of optima ocay ecoveabe codes. CoRR, abs/ , 013. [10] N. Le Scouanec A.-M. Kemaec and G. Staub. Repaiin mutipe faiues with coodinated and adaptive eeneatin codes. In Poc. Intenationa Symposium on Netwo Codin, 011. [11] K. W. Shum and Y. Hu. Coopeative eeneatin codes. IEEE Tans. Inf. Theoy, 59(11):79 758, 013. [1] F. Oie and A. Datta. Sef-epaiin homomophic codes fo distibuted stoae systems. In Poc. of INFOCOM, 011. [13] N. Paash, G. Kamath, V. Laitha, and P. Kuma. Optima inea codes with a oca-eo-coection popety. In Poc. of IEEE ISIT, 01. [14] V. R. Cadambe and A. Mazumda. An uppe bound on the size of ocay ecoveabe codes. CoRR, abs/ , abs/ , 013. [15] M. Fobes and S. Yehanin. On the ocaity of codewod symbos in non-inea codes. CoRR, abs/ , 013. [16] A. Mazumda, V. Chanda, and G. W. Wone. Loca ecovey popeties of capacity achievin codes. In Infomation Theoy and Appications Woshop (ITA), 013, paes 1 3. IEEE, 013. APPENDIX A PART OF THE PROOF OF THEOREM 1 Fo the constuction of a subcode as descibed in Fi. 1, we define A = I \I 1 R {i 1,..., i } and a = A. Assumin that the whie oop in Fi. 1 ends with = t, fo [t], we have I = A, [] whee we tae union of the disoint sets A, []. Note that the code symbos indexed by the set {i 1,..., i } ae functions of the code symbos indexed by the set R = Γ {i 1,...,i }. Theefoe, at ine 7, thee ae at most qa possibiities fo y. This impies that C C 1 /q a. (14) The constuction of the subcode C can end at eithe ine 11 o ine 15. Hee we anyze ony the case when the constuction ends at ine 11. In this case, we have C t q, o 1 o q C t (a +1 ). (15) Now, usin that a A R {i 1,..., i } +, we et which ives us that 1 (a +1 ) t, (16) t Note that sub-code C = C t. Theefoe, o q C = o q C t 1. (17) o q C (a +1 ) = a +1 + t (a) = I t + t (18) whee (a) foows fom the fact that I t is union of the disoint sets A.