Optimal Distributed Declustering using Replication

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Optimal Distributed Declustering using Replication

• Keith Frikken
• Purdue University
• Jan 5, 2005

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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

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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

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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 e, maxq?Qret(q)-oq

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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

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Other schemes

• Atallah and Prabhakar, 2000 developed a scheme
  in two dimensional grids for k2n disks the has
  additive error of O(log k)
• Sinha et al, 2001 proved lower bounds on the
  additive error of ?(log k) and ?(log(d-1)/2 k)
  for 2 dimensions and d (gt2) dimensions
  respectively
• Chen and Cheng, 2002 showed that an additive
  error of O(log(d-1) k) is achievable for any of
  dimensions (gt2)

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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

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Replication Results

• Chained Declustering
• Fast Scheduling Algorithm O(mk) time to test if
  a specific retrieval time is possible Aerts et
  al, 2000
• RDA
• If mck(log k) then optimal with high prob
  Czumaj and Scheideler, 2003
• Fast scheduling algorithm O(?kO(1)) time
  Czumaj and Scheideler, 2003
• Hybrid techniques Chen and Cheng, 2002
• Use GRS with second random disk

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Our Results

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

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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)

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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)

0,3 1,4 2,0 3,1 4,2
2,0 3,1 4,2 0,3 1,4
4,2 0,3 1,4 2,0 3,1
1,4 2,0 3,1 4,2 0,3
3,1 4,2 0,3 1,4 2,0
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Scheduling

• Can use modification of chain declustering
  scheduling algorithm to schedule queries in
  O(mk(log e)) time
• Essentially, use previous algorithm to test if a
  specific load is possible and do a binary search
  on the possible loads

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Bound(1)

• There are k disks (D0,,Dk-1)
• Disk Di has ti tiles initially (as the primary
  disk)
• The number of tiles is mt0tk-1
• Di shifts di tiles to Di1
• di ti
• The goal is to minimize the most tiles at a disk,
  i.e., max0ik-1di-1ti-di

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Bound(2)

• Recall,
• o?m/k?
• max0ik-1ti oe
• Suppose mk(k-1)e
• Then,
• o (k-1)e
• Surplus ( ) is bounded by
  (k-1)e
• max0ik-1di (k-1)e o
• Two cases
• If disk has a surplus
• If disk has a shortage

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32 disks
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64 disks
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128 disks
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32 disks, 3 dimensions
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Generalizations

• Permutations
• Higher levels of replication
• Survivability
• If the level of replication is r, can handle any
  r-1 failures
• When r2, and a single disk fails then
• Fast scheduling still possible
• Large queries still optimal

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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