Optimal Distributed Declustering using Replication

1. Optimal Distributed Declustering using Replication Keith Frikken Purdue University Jan 5, 2005 ICDT 2005

2. Declustering Data • Declustering data over multiple disks to improve performance for range queries has been well studied • Applications include: • Spatio-temporal databases • Image and video data • Scientific simulation datasets ICDT 2005

3. Goal • Divide data uniformly along dimensions to create tiles • Put records contained in each tile on different disks so that I/O can be parallelized • Assumptions • Data can be tiled in such a way • Disks have constant retrieval times • Assigning tiles to disks is similar to a coloring problem (disks are colors) • A range query can be answered optimally if the # of I/O retrievals for any specific disk is: # of tiles/# of disks • Two approaches: • Coloring schemes • Replication ICDT 2005

4. Notations • k is number of disks • m is number of tiles in queries • r is level of replication (i.e., is 2) • Q is the set of all range queries • ret(q) is the actual retrieval time of q • Optimal retrieval time for a query q is oq=m/k • Additive error ε, maxqQ{ret(q)-oq} ICDT 2005

5. Coloring schemes • Disk Modulo (DM) [Du and Sobolewski, 1982] • Fieldwise XOR (FX) [Kim and Pramanik, 1988] • Cyclic Schemes (RPHM, GFIB, EXH) – [Prabhakar et al, 1998] • Golden Ratio Sequences (GRS) – [Bhatia et al, 2000] ICDT 2005

6. Other schemes • [Atallah and Prabhakar, 2000] developed a scheme in two dimensional grids for k=2n disks the has additive error of O(log k) • [Sinha et al, 2001] proved lower bounds on the additive error of Ω(log k) and Ω(log(d-1)/2 k) for 2 dimensions and d (>2) dimensions respectively • [Chen and Cheng, 2002] showed that an additive error of O(log(d-1) k) is achievable for any # of dimensions (>2) ICDT 2005

7. Replication • Placing records on multiple disks can further improve performance of declustering schemes • Two Problems: • How to schedule a query (i.e., what tiles are retrieved from each disk) • How to use replication to balance load • Approaches: • Chained Declustering [Hsiao and DeWitt, 1990] • Random Duplication Allocation [Sanders et al 2000], [Sanders, 2001], and [Czumaj and Scheidler, 2003] ICDT 2005

8. Replication Results • Chained Declustering • Fast Scheduling Algorithm O(m+k) time to test if a specific retrieval time is possible [Aerts et al, 2000] • RDA • If m≥ck(log k) then optimal with high prob [Czumaj and Scheideler, 2003] • “Fast” scheduling algorithm” O(ΔkO(1)) time [Czumaj and Scheideler, 2003] • Hybrid techniques [Chen and Cheng, 2002] • Use GRS with second random disk ICDT 2005

9. Our Results • We define a new class of schemes called the shift schemes • Deterministic • Any query with at least k(k-1)ε tiles can be answered in an optimal fashion • Queries can be scheduled in O(m+k(log ε)) time • If a single disk fails, then any query with at least k(k-1)ε tiles can be answered optimally • Experimental performance similar to RDA (better for many cases) ICDT 2005

10. Shift Scheme Definition • Use any strong coloring scheme • Use a modified chain declustering • Defined by shift value s (where gcd(s,k)=1) • Base scheme is defined by function f(x,y) • Second color is (f(x,y)+s mod k) ICDT 2005

11. Shift Scheme Definition • Use any strong coloring scheme • Use a modified chain declustering • Defined by shift value s (where gcd(s,k)=1) • Base scheme is defined by function f(x,y) • Second color is (f(x,y)+s mod k) ICDT 2005

12. Scheduling • Can use modification of chain declustering scheduling algorithm to schedule queries in O(m+k(log ε)) time • Essentially, use previous algorithm to test if a specific load is possible and do a binary search on the possible loads ICDT 2005

13. Bound(1) • There are k disks (D0,…,Dk-1) • Disk Di has ti tiles initially (as the primary disk) • The number of tiles is m=t0+…+tk-1 • Di shifts di tiles to Di+1 • di≤ ti • The goal is to minimize the most tiles at a disk, i.e., max0≤i≤k-1{di-1+ti-di} ICDT 2005

14. Bound(2) • Recall, • o=m/k • max0≤i≤k-1{ti} ≤ o+ε • Suppose m≥k(k-1)ε • Then, • o ≥ (k-1)ε • Surplus ( ) is bounded by (k-1)ε • max0≤i≤k-1{di} ≤ (k-1)ε ≤ o • Two cases: • If disk has a surplus • If disk has a shortage ICDT 2005

15. 32 disks ICDT 2005

16. 64 disks ICDT 2005

17. 128 disks ICDT 2005

18. 32 disks, 3 dimensions ICDT 2005

19. Generalizations • Permutations • Higher levels of replication • Survivability • If the level of replication is r, can handle any r-1 failures • When r=2, and a single disk fails then: • Fast scheduling still possible • Large queries still optimal ICDT 2005

20. Summary • Shift schemes are a new class of schemes • Optimal for “large enough” queries • Efficient scheduling algorithm • Resilient to disk failures • Future Work • Better analysis of scheme • Choosing shift values ICDT 2005