Micha Streppel and Ke Yi

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Approximate Range Searching in External Memory

• Micha Streppel and Ke Yi
• Presented by Liu Nan

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Range Searching

• A set S of N points in Rd
• Build a data structure such that given a query
  range Q, S n Q can be returned efficiently

focus on range reporting, range aggregation in
paper
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Range Searching in External Memory

• 1D B-tree
• Size O(N/B), Query O(logB(N/B)k/B))
• 2D
• Half planes Agarwal et al. 2000
• Size O(N/B), Query O(logB(N/B)k/B))
• Orthogonal rectangles Arge et al. 1999
• Size O(N/B), Query T((N/B)ek/B)
• Query O(logB(N/B)k/B)) , Size T((N/B)
  log(N/B)/loglogBN)
• kdB-tree Robinson 1981
• Size O(N/B), Query O((N/B)½ k/B)

Exact range searching is difficult!
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Approximate Range Searching
radius e diam(Q)
Q

• Internal memory
• BBD-tree Arya and Mount, 1995
• BAR-tree Duncan et al. 2001
• Size O(N), Query O(log(N) 1/e ke) for any
  convex Q
• External memory this paper!

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Externalization
previously
Query bounds of linear structures in
internal/external memory
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Externalizing the kd-tree
B 3
Internal memory O(N½ k) External memory
O((N/B)½ k/B) for orthogonal rectangle ranges
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The BAR-Tree Duncan et al. 2001

• A space-partitioning scheme
• Similar to kd-tree
• But also use diagonal cuts
• All cells are convex and fat
• Some cuts have to be unbalanced
• But no two consecutive unbalanced cuts
• Height O(log N)
• Query range intersects O(log(N) 1/e ke)
  cells(any convex range)

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Blocking the BAR-Tree

• Top-down blocking
• Rules for u
• Check us two subtrees T1, T2
• Add u if both have B/2 nodes
• If T1 small, check if entire T1 fits
• then add T1
• else do not add u
• Not possible for both T1 and T2 to be small

B 8
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Blocking the BAR-Tree
Any subtree Tu is stored in O(Tu/B1) blocks
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Current Blocking Not Sufficient
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Regrouping Shallow Subtrees

• Identify shallow nodes top-down
• u is shallow if there is a path of length log(B)
  beneath u is stored in more than c blocks
• For such a u
• Do a BFS for log(B) levels
• Move these nodes from their original blocks to a
  new block

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I/O Analysis of a Query
organized in O(1/e)subtrees
Qe
Q
nodes completelyinside Qe
nodes intersectsboth Q and ?Qe
total O(ke)
total O(log N1/e) total I/O O(logBN1/e)
total I/O O(1/e ke/B)
Achieving the desired query I/O O(logB(N/B)
1/e ke/B)
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Construction and Update

• Construction O(N/B logM/B(N/B)) I/Os
• Same as sorting
• Insertions and deletions
• Use partial rebuilding
• O(logBN 1/B logM/B(N/B)log(N/B)) I/Os
  amortized

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Extension to Objects

• S a collection of objects
• The density of S is the smallest number ? such
  that any ball b is intersected by at most ?
  objects o in S with radius(o) radius(b) de
  Berg et al. 1997

low density
high density
high density
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Extension to Objects

• The object-BAR-tree (using guarding sets de Berg
  et al. 2003)
• Size O(?N/B)
• Query O(logB(N/B) ?/B1/e ?ke/B)
• Construction O(? N/B logM/B(N/B))

low density
high density
high density
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The END

• Thank you!