IOEfficient Structures for Orthogonal Range Max and Stabbing Max Queries

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I/O-Efficient Structures for OrthogonalRange Max
and Stabbing Max Queries

• Second Year Project Presentation
• Ke Yi
• Advisor Lars Arge
• Committee Pankaj K. Agarwal and Jun Yang

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Problem Definition Range Max Queries

• Range-aggregate queries range-count, range-sum,
  range-max
• N points in Rd
• Each point p is associated with a weight w(p)
• Query rectangle Q
• Compute maxw(p) p?Q
• Static and dynamic

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Problem Definition Stabbing Max Queries

• N hyper-rectangles in Rd
• Each rectangle ? is associated with a weight w(?)
• Query point q
• Compute maxw(?) q??

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Model

• I/O Model
• N Elements in structure
• B Elements per block
• M Elements in main memory
• n N/B
• Assumptions
• MgtB2
• Each word holds log2N bits
• Any coordinate or weight can be stored in one word

D
Block I/O
M
P
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Related Work Our Results Range Queries

• 1D range queries are easy B-tree
• O(n) space, O(logBn) query update
• 2D range queries
• Poly-logarithmic query CRB-tree AAG03
• O(nlogBn) space, O(log2Bn) query
• Linear space kdB-tree, cross-tree, O-tree
• query, O(logBn) update
• Our results

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Related Work Our Results Stabbing Queries

• 1D stabbing queries
• SB-tree YW01
• O(n) space, O(logBn) query insert
• Does not allow deletions!
• 2D stabbing queries
• No structures with worst-case guarantee
• Our results

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2D Range Max Queries

• The external version of Chazelles structure
  C88
• Linear space,
• Static O(log1eN) query
• Dynamic O(log3N log log N) query update
• Overall structure
• A normal B-tree F on y-coordinates of all the
  points
• A Fan-out base B-tree T on
  x-coordinates
• Pv all points stored in the subtree of v
• Each internal node v stores two secondary
  structures Cv, Mv storing information about Pv in
  a compressed manner
• Cv and Mv of size O(Pv / logBn) ? linear size
  in total
• Weights of points stored at leaves explicitly

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2D Range Max Queries

• Cv borrowed from CRB-tree
• Compute the ranks of the points one level down in
  O(1) I/Os
• Identify the weight of a point explicitly in
  O(logBn) I/Os
• Mv computes the maximum weight in a multislab
  inO(logBn) I/Os
• Answering a query
• Use F to compute the ranksin the root of T
• Use Mv to compute maximumat each level
• For a total of O(log2Bn) I/Os

v
v1
v2
v3
v4
v5
v6
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2D Range Max Queries Mv

• Divide Pv into chunks of BlogBN
• Divide each chunk into minichunks of size B
• Three-level structures
• Mv(?1, ?2, ?3)
• each of size O(Pv / logBn)

v
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2D Range Max Queries Mv

• Basic idea encode the range max information in a
  compressed manner, identify the maximum point
  using Cv once its rank is found
• ?3l for each minichunk, stores a (slab index,
  weight rank) pair for each point inside the
  minichunk
• Find the rank of the maximum-weight point in O(1)
  I/Os
• Identify it in O(logBN) I/Os.
• ?2k for each chunk, encode a Cartesian tree on
  the O(logBN) minichunks for each of the O(B)
  multislabs
• Find the minichunk containing the maximum-weight
  point in O(1) I/Os
• Use ?3 to find the exact point in O(logBN) I/Os
• ?1 A fanout B-tree on the O(Pv / (BlogBn))
  chunks
• Find the maximum-weight point in O(logBN) I/Os.

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2D Range Max Queries

• Static structures
• O(n) size, O(log2BN) query, O(nlogBN)
  construction
• O(n) size, O(logB1eN) query, O(NlogBN)
  construction
• Dynamization
• Throw away ?2 and expand?3
• O(nlogBlogBN) size
• O(log3BN) query, worst case
• O(log2BN logM/BlogBN) insert, amortized
• O(log2BN) delete, amortized
• Extending to d-dimension
• Standard technique
• Pay an extra O(logd-2BN) factor to all these
  bounds

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1D Stabbing Max Queries

• Modify the external interval tree AV96 to
  support max
• Fan-out base B-tree on
  x-coordinates
• Interval stored in highest node v where it
  contains slab boundary
• In one left (right) slab structure and the
  multislab structure
• Answering a query
• Search down tree and visit O(logBN)nodes
• Compute the maximum weight in left (right)slab
  structure and the multislab structure

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1D Stabbing Max Queries

• Slab structures are implemented using B-trees
• Query and update O(logBN) I/Os
• Multislab structure Fan-out B-tree
• At each internal node, we store the maximum
  weight for each of the slabs and for each of
  the children
• Query O(1) I/Os (only look at the root)
• Update O(logBN) I/Os
• Rebalancing the base tree O(logBN) I/Os
• Weight-balanced B-trees
• Overall cost size O(n), query O(log2BN), update
  O(logBN).

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1D Stabbing Max Queries

• Space-time tradeoff
• O(nlogBeN) size
• O(nlogB2-eN) query
• Can handle the general semigroup queries
• A semigroup (S, )
• Each weight w(?) ?S
• Want to compute ? q?? w(?)
• Ideas can also be used to improve the internal
  memory algorithm
• Linear size, O(log2N / log log N) query and update

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2D Stabbing Max Queries

• Extend our 1D stabbing query structure
• Use our 2D range query structure as a building
  block
• Extending to d-dimension
• Standard technique
• Pay an extra O(logd-2BN) factor to all these
  bounds

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Conclusions and Open Problems

• In this project, we developed I/O-efficient
• linear space structures with poly-logarithmic
  query cost for the static 2D range max queries
• near linear space structures with
  poly-logarithmic query update cost for the
  dynamic 2D range max queries
• linear space structures with poly-logarithmic
  query cost for the dynamic 1D stabbing max
  queries
• near linear space structures with
  poly-logarithmic query update cost for the
  dynamic 2D stabbing max queries
• Open problems
• Linear size dynamic structures for the 2D range
  stabbing max queries?
• General semigroup queries?

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THE END

• Thank you!