Reuse of Original Redundant Data in .NET

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15.4.2 Reuse of Original Redundant Data
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To understand how we can reduce the redundant data update overhead, we rst need to study the computation of the redundant blocks using the RSE code. Let B be the total number of xed-size media data blocks in the system and denote the jth block of a media object by v j . For simplicity we consider only one media object in the rest of the section. The results can be readily extended to multiple media objects.
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Algorithms for System Expansion
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Let (N h) and h be the number of data nodes and redundant nodes in the system respectively. Assuming the number of redundant nodes in the system is xed, then we can apply the (N , h)-RSE code to compute the h redundant data blocks in each group of (N h) data blocks using di,0 f 1,1 f 1,2 f 1,3 . . . f 1,N h f 2,1 f 2,2 f 2,3 . . . f 2,N h di,1 F D= . . . . . . . . . . . . . . . f h,1 f h,2 f h,3 . . . f h,N h di,N h 1 1 1 1 ... 1 di,0 1 2 3 ... N h di,1 = . (15.2) . . . . . . . . . . . . . . 1 2h 1 3h 1 ci,0 ci,1 = . =C . . ci,h 1 . . . (N h)h 1 di,N h 1
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where the F, D, and C are the Vandermonde matrix [4], the media data vector, and the redundant data vector respectively; and di, j , ci,k represent data block j ( j = 0, 1, . . . , N h 1) and redundant block k (k = 0, 1, . . . , h 1) of parity group i. Elements in F are constants computed from f i, j = j i 1 . Note that the matrix multiplication in equation (15.2) is computed over Galois Fields of 2w where N < 2w . For example, by setting w = 16, then the code can support up to 65,535 nodes. Reconsider the generation of a redundant data block before and after addition of one data node in Figure 15.11 and Figure 15.12 respectively. We can observe that in many cases, the reorganized parity group still comprises some data blocks from the original parity group before reorganization. For example, in expanding a system from N nodes to N + 1 nodes, the rst parity group will be reorganized from the composition of {v0 , v1 , . . . , v N h 1 } to {v0 , v1 , . . . , v N h 1 , v N h }, which differs by only one data block v N h . This means that the resultant matrix computations in equation (15.2) will differ by only one variable in the data vector. This raises the question of whether we can reuse the original redundant data in computing the new redundant data, discussed next. Consider the original con guration in Figure 15.11. The rst two redundant data in redundant node r1 , denoted by c0,1 and c1,1 , are computed from
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c0,1 =
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f 2, j+1 v j
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c1,1 =
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respectively according to equation (15.2).
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After a new node is added, the system will be reorganized to that in Figure 15.12. Now the two new redundant data block, denoted by c0,1 and c1,1 , are computed from
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c0,1 =
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f 2, j+1 v j
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Comparing equation (15.5) with equation (15.3) we can observe that they share four common terms in v j v0 , v1 , v2 , v3 . Hence we can rewrite equation (15.5) as follows:
c0,1 =
f 2, j+1 v j + f 2,5 v4 (15.7)