Spatial Join Algorithms

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Spatial Join Algorithms

• Mohamed F. Mokbel

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Why we need a Spatial Join ?

• Spatial data are commonly found in applications
  like cartography, CAD, and GIS
• In spatial join, two spatial relations are
  combined together based on some spatial criteria.
• Examples of spatial join
• Only spatial join is required
• Find all forests which are in a city
• Find all cities that are crossed by river
• Find all cities that are affected by the fire
  region.
• Find all buildings that overlap with a park.
• Spatial join with a selection criteria
• Find all government-owned buildings that overlap
  with a park.
• Find all forests in USA that receive more than 20
  inches of average rainfall by year.
• Find all day cares in lafayette that are within 3
  miles from houses with rent less than 600.

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Why not using Relational Join Algorithms ?

• Nested Loop Join
• Every object of one relation has to be checked
  against all objects of the other relation. Since
  we consider a very large relations of spatial
  objects, the performance of the nested loop is
  not acceptable
• Hash-Based Join
• Hash-based joins are suitable for equi-joins but
  not for spatial joins.
• Sort-Merge Join
• There is no total ordering of spatial objects.
• However, adaptation of these algorithms to deal
  with the spatial properties can be applied.

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Is the Spatial Join Problem is Already Solved in
Another Domain?

• Geometric domain
• The abstraction of the spatial join is finding
  the intersection between two sets of geometric
  shapes.
• Many solutions are provided in the context of
  geometry.
• Geometric solutions considers only CPU cost.
• Geometric solutions are accepted if the data set
  can fit in memory.
• VLSI domain
• Spatial-Access methods (e.g., R-tree) are also
  defined in the VLSI context.
• Divide-and-conquer algorithm (Gutting et al, IS
  93) is designed for rectangle intersection
  problem with large sizes.
• However, still the I/O time is not well
  addressed. I/O is essential for spatial join
  algorithms due to the massive amount of spatial
  data.

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What is Special about Spatial ?

• There exists no total ordering among spatial
  objects that preserves spatial proximity
• Space-filling curves can be used, but not with an
  accurate ordering.
• Many spatial operators are not closed
• The intersection of two polygons may return any
  number of single points, dangling edges, or
  disjoint polygons.
• Spatial operates are more expensive than standard
  relational operators
• Examples of Spatial operators are overlap,
  contained, include
• Spatial databases tend to be large
• The cities, rivers, restaurants, gas stations,
  forests, highways in USA.
• Spatial data have a complex structure
• Imagine representing the boundaries of Lafayette
  in a database.

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Filter Step and Refinement Step

• Filter Step
• An approximation of each spatial object (e.g.,
  the minimum bounding rectangle) is used to
  eliminate tuples that cannot be part of the
  result. This step produces candidates that are a
  superset of the actual result.
• Refinement Step
• Each candidate is examined to check if it is a
  part of the result. I/O cost due to fetching the
  exact object from the disk and a CPU-intensive
  computational geometry algorithm.
• An intermediate step (geometric filter) is used
  in some algorithms.

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The Filter Step

• Transformation approaches
• There is no index in any of the relations
• The two relations are indexed.
• There exists only one index for only one relation
• Unified approaches regardless of the index
  existence

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Transformation Approaches for Spatial Join

• Map to the one-dimensional space (Orenstein et
  al, TSE 88)
• Rectangles are sorted according to the Z-order.
• Two one-dimensional spatial join algorithms are
  proposed (spatial-merge, and spatial-filter).
• Both algorithms are later enhanced in (Aref et
  al, SDH 94) with the linear-scan and
  estimate-based spatial join algorithms.
• Map to higher-dimensions. (Becker et al, ICDE 93)
• D-dimensional rectangles are transformed into
  points in the 2D-dimesional space (corner
  transformation).
• A multi-dimensional join algorithm is used with
  the support of grid files.
• Transformation-Based Spatial Join (Song et al
  CIKM99, TKDE 99)
• Corner transformation are used
• A special algorithm for spatial join that does
  not rely on indexing is proposed.

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Spatial Join algorithms without Indexing Support
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PBSM (Partition-Based Spatial-Merge Join) (Patel
et al, SIGMOD 96)

• The spatial universe is divided into disjoint P
  partitions.
• The MBRs from the two relations are mapped to
  their partitions. One MBR can be clipped to
  several partitions.
• An in-memory spatial join algorithm is used for
  each partition using the plane-sweep algorithm.
• The number of partitions is chosen to allow each
  partition to fit in memory.
• Additional techniques are provided to handle data
  skew.
• If data inside one partition is still cannot fit
  in memory, a recursive partitioning may be used.
• The output data may contain duplicates.
• A sorting step need to be done in the Refinement
  step to remove the duplicates.

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Spatial Hash-Join (Lo et al, SIGMOD 96)

• A general framework for extending a relational
  hash join algorithms.
• Can produce duplicate results

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PBSM as an instance of the Spatial Hash-Join
framework
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Spatial Hash Join Designers Choice
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Other Spatial join algorithms for non-indexed
relations

• Seeded tree approach (Lo et al, SSD 95)
• Two seeded trees are built for both relations
  using spatial sampling techniques.
• A depth-first traversal algorithm is used for
  joining the two seeded trees.
• Size Separation Spatial Join, S3J (Koudas et al,
  SIGMOD 97)
• For each entry in both data sets A, and B
• Compute the Hilbert value H of the centroid.
• Determine the level at which this entry belongs
  to (Similar to the Filter Tree) and place the
  entry in this level file
• For each level file, sort entries by the Hilbert
  value
• Perform a synchronized scan over the pages of
  each level.
• Scalable Sweeping-Based Spatial Join, SSSJ (Arge
  et al, VLDB 98)
• Similar to PBSM, it is a partition-based and
  plane-sweep based approach.
• The main contribution is that it utilize the
  foundations of computational geometry algorithms
  to improve the in-memory plane-sweep algorithm.
• This requires changing the partitioning function
  to partition over only on-dimension.

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Spatial Join algorithms For Both Indexed Relations
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Spatial Join using Depth-Traversal R-Tree
(Brinkhoff et al, SIGMOD 93)

• The sketch of the proposed algorithm for the case
  of equal heights and intersection operator is
• Procedure spatialJoin (R, S R-Tree Node)
• For all entries ER in R and all entries ES in S
  where ER.rect intersects ES.rect
• If (R, S are leaf pages)
• Output (ER, ES)
• Else
• SpatialJoin (ER.ref, ES.ref) // ER.ref,
  ES.ref is the node referenced by ER, ES
• End
• The main idea is to synchronously traverse the
  two R-trees in a depth-first traversal.
• Enhancements are proposed to tune
• The CPU time
• The I/O time

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Spatial Join using Depth-Traversal R-Tree (Cont.)

• Tuning CPU-Time
• Restricting the search space. Among all the nodes
  in R,S, we only check the entries that intersect
  with R?S.
• Spatial sorting and plane-sweep. Use the
  plane-sweep algorithm for the set of candidate
  from nodes R, S.
• I/O time tuning
• Local plane-sweep order. Use the plane-sweep
  order for fetching pages from disk.
• Local plane-sweep order with pinning. In addition
  to the previous approach, we pin the rectangles
  that have maximal degree. The degree of a
  rectangle is the number of its intersected
  rectangles.
• Local Z-order with pinning. Instead of doing
  plane-sweep order for reading the disk pages, we
  use the Z-order of the centroids of the
  rectangles.

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Other Spatial Join algorithms for both indexed
relations

• BFRJ Breadth-First R-tree Join (Huang et al,
  VLDB 97)
• Synchronous traversal of two R-trees in a
  breadth-first traversal.
• Unlike the depth-first traversal, where a local
  optimization is achieved for a node by node join,
  in BFRJ, a global optimization is achieved for
  each level.
• The main idea is that, based on the global
  optimization, we can take decisions as which
  nodes need to be joined to each other.
• Notice that these cannot be done using
  depth-first traversal, because the limitation of
  the current scope.
• Both relations are indexed using PMR quadtree
  (Hoel et al, VLDB 95)
• Performs a synchronized tree traversal at the
  leaf level.

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Spatial Join algorithms for one indexed relation

• Spatial join using seeded trees (Lo et al, SIGMOD
  94, TKDE 98)
• A seeded tree is built for the non-indexed
  relation.
• The steps to build the seeded tree is guided by
  the existing R-Tree
• The R-tree and the seeded tree are joined using
  the dept-traversal approach.
• Sort and Match (Papadopoulos et al, SSD 99)
• The STR bulk loading algorithm is applied for the
  non-indexed relation.
• Instead of building the packed tree, it directly
  matches in-memory created leaf nodes with the
  existing R-tree index.
• Slot Index Spatial join, SISJ (Mamoulis et al,
  TKDE 03)
• SISJ combines the ideas of the seeded tree join
  with the spatial hash join.
• The key idea is to define the spatial partitions
  of the spatial hash join using the existing
  R-tree.

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A Unified Approach for Indexed and Non-Indexed
Spatial Joins (Arge et al, EDBT 00)

• An extension for SSSJ to deal with indexed
  relations.
• Similar to SSSJ, non-indexed data are sorted
  according to their MBRs, and fed into the
  plane-sweep algorithm.
• For indexed data, an additional pre-processing
  step is required to exploit the index structure
  and directly extract the data in a sorting order
  according to the plane-sweep algorithm.
• A main conclusion of this paper is that using an
  index-based approach for spatial join whenever
  indexes are available does not always lead to the
  best execution time.
• A cost model is proposed to decide whether to
  follow an index-based approach or the unified
  approach.

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The Refinement Step
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Multi-Step Processing (Brinkhoff et al, SIGMOD 94)

• The refinement step is divided into two steps.
• Identifying more false and true hits.
• In this step, more accurate approximations other
  than the MBR is investigated to identify false
  and true hits.
• Exact geometry intersection.
• Eventually all the remaining pairs of candidates
  are examined at this stage. This the most time
  consuming step, where it requires CPU time to
  compute the exact intersection test, and I/O time
  to read the spatial object from disk.
• It is important to notice that improvements in
  the exact geometry intersection step has the
  lowest impact, since its effect can be canceled
  by the improvements in the previous two steps.

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Multi-Step Processing (Cont.)

• Removing more false hits

• Identifying true hits

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Exact Geometry Processing in Multi-Step Processing
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Other Work for the Refinement Step

• Approximations other than MBR (Veenhof et al,
  BNCOD 95)
• Approximations of spatial objects are constructed
  by rotating two parallel lines around the object.
• Symbolic Intersection Detection (SSD 97,
  Geoinformatice 98)
• Concerned with the exact geometry computation.
• Enumerates all the possible situations that two
  clipped polygon segments can have inside an MBR.
• Raster approximation (Zimbrao et al, VLDB 98).

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Other Problems Related to the Spatial Join

• Non-blocking spatial join (Luo et al, ICDE 02)
• Multiway Spatial join
• (Mamoulis et al, GIS 98), (Mamoulis et al, SIGMOD
  99), (Papadias et al, PODS 99), (Papadias et al,
  EDBT 02).
• Selectivity Estimation
• (Faloutsos et al, SIGMOD 00), (An et al, ICDE
  01), (Mamoulis et al, SSTD 01), (Sun et al, EDBT
  02)
• Cost models
• For R-tree based indexed relation (Gunther, ICDE
  93)
• Parallel spatial Join
• PMR-Quadtree (Hoel et al, VLDB 94). R-tree
  (Brinkhoff et al, ICDE 96). Non-indexed relation
  (Patel et al, GIS 00)
• Cascaded Spatial Join (Aref et al, GIS 96)
• Caching strategies (Abel et al, GeoInformatica
  99)
• Duplicate Detection (Dittrich et al, ICDE 00)
• High-dimension spatial Join (Koudas et al, ICDE
  98)

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Summary

• Spatial Join algorithms are performed in two
  steps Filter Step and Refinement Step.
• In Filter Step, five approaches are used based
  on
• Transformation approaches
• No index is available
• Only one index is available
• Two indices are available
• A unified Approach.
• In the Refinement step This step can be further
  divided into Geometric filter step and exact
  geometry processing step.