SDRtree: A Scalable Distributed Rtree

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SD-Rtree A Scalable Distributed Rtree

• Witold Litwin
• Cédric du Mouza Philippe Rigaux

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Plan

• Introduction
• SDDS
• R-tree
• SD-Rtree Evolution
• Balancing
• Spatial Rotations
• Overlapping
• Redundant Coverage
• Queries
• Performance
• Conclusion

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SDDS Principles (1993)

• Data are at server nodes
• Communicating through point-to-point messaging
• Overloaded servers split over new servers
• Queries go to client nodes use local images of
  the SDDS
• No central addressing component
• A node can be client and server (peer)

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SDDS Principles (1993)

• An outdated image may send a query an incorrect
  server
• Servers forward such a query to the correct
  server
• Image gets adjusted
• Image Adjustment Message (IAM) comes back
• Client does not repeat the same error twice
• Data are basically in the RAM of the servers

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SD-Rtree a Spatial SDDS
Distributed Spatial Data
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SD-Rtree a Spatial SDDS

• Distributed Index
• No central component

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SD-Rtree a Spatial SDDS

• Point Window Queries
• kNN queries (future)

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SD-Rtree Generalizes R-tree

• R-tree
• Nodes are minimal bounding boxes
• Leaf nodes point to data
• Internal nodes bound subtrees
• May overlap
• Split when overflow
• Generate balanced m-ary tree

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SD-Rtree Generalizes R-tree

• R-tree
• An insert may go through multiple paths
• Ends up in the smallest bounding box
• If there is any
• One of the boxes gets enlarged
• Box may split

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SD-Rtree Generalizes R-tree

• R-tree
• Search may go through multiple paths
• All paths may bring relevant objects

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Distribution issues

• First issue adapt the structure to the context
• Cost model based on messages
• No paging ! The degree M of the tree is not a
  constraint
• gt split and balancing algorithms must be
  reviewed
• Second issue distribute the tree over the
  servers
• Balance evenly the load
• Do not overload the root node
• gt search algorithms must be reviewed as well

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SD-Rtree a Balanced Binary Tree

• Each split generates a new edge
• Half of data moves to the new server
• Each server hosts exactly one leaf and one
  internal node of SD-Rtree

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SD-Rtree a Balanced Binary Tree

• The SD-Rtree is a balanced binary tree,
  distributed on a set of servers, such that
• Each internal node (or routing node) has exactly
  two sons
• Each leaf node stores a subset of the indexed
  dataset
• At each node, the height of the subtrees differ
  by at most one
• Each server stores one data node and one routing
  node

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Sd-tree Binary Tree Structure

• di data node (leaf)
• ri routing node (internal node)

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Sd-tree Tree Distribution
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Sd-tree Evolution
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SD-Rtree Balancing

• The binary tree should be height-balanced
• The heights of the two subtrees rooted at any
  node should not differ by more than 1 (cf. AVL
  trees)
• The tree height is then logarithmic in the number
  of leaves

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SD-Rtree Balancing

• SD-Rtree balancing occurs during splits
• Messages are sent bottom-up to adjust the height
  of the ancestor nodes
• Rotation occurs if an ancestor is imbalanced
• SD-Rtree rotation are spatial
• change rectangles of internal nodes
• Best rotation minimizes rectangle overlapping
• Tie breaking minimizes the dead space

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SD-Rtree Spatial Rotations
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Rotation Pattern

• Properties
• The sons of a node are not ordered
• gt more freedom for reorganizing the tree
• Any imbalanced node matches a rotation pattern

• A rotation pattern is a subtree a(b(e(f,g),d),c)
  such that
• h(c) h(d) h(f ) n - 1 (n gt 0)
• h(g) max(0, n - 2)

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SD-Rtree Spatial Rotation
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Rotation Cost

• Constant number of messages (3 or 6, depending on
  the choice)
• Few rotations in practice
• In particular when the dataset is uniformly
  distributed
• See our experiments

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SD-Rtree Images

• Each image defines the addressing structure
• Resides as cache on a client or on a peer
• Starts with the address of the contact server
• IAMs make it a subtree
• Splits make images outdated
• IAMs adjust it incrementally

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Image Adjustment

• Client contacts a server with a query
• Each incorrect server initiates a traversal of
  the tree
• During the traversal, the description of the
  nodes is collected
• The correct server sends the up-to-date tree
  structure
• The client updates its image

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Image Construction

• Using the image
• The client first searches its local image and
  chooses the servers that best corresponds to the
  query
• The correct server is found in O(log n) in the
  worst case

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Out-of-range situation
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Insertion of objects
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Overlapping management

• The directory rectangles in an Rtree may overlap
• Local subtree does not suffice for locating all
  the nodes that contains the point (point query)
  or the window (window query) searched for.
• SD-Rtree servers maintain data on node
  overlapping
• Redundant Coverage
• It avoids to systematically access the root node.

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Redundant Coverage

• Example
• The region common to A and B is stored on both
  nodes
• If a point query sent to A falls in the region
  shared with B A sends a point query message to B
• For D we must keep the intersection with C or B
  here empty.

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Queries

• Point queries and window queries. The technique
  is similar to the insertion algorithm
• Search in the client image a server whose mbb
  contains the point or intersects the window
• Send the query to this server
• If the server actually covers the point or the
  window it answers to the client else it sends
  the query to its parent node
• A server uses the overlapping information to
  transmit the query

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Experiments

• Synthetic data (points and rectangles) generated
  with GSTD
• 50.000 to 500.000 objects
• 0 to 3.000 queries
• Server capacity 3 000 objects
• Comparison of three SD-Rtree variants
• BASIC no image every query is processed
  top-down from the root
• IMSERVER no IAMs among the servers
• IMCLIENT client images

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Cumulative Insert cost
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Per Insert Cost
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Cost of balancing
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Image convergence
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Distribution of messages
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Cost per Query
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Conclusion

• SD-Rtree is an efficient scalable distributed
  Rtree
• For very large spatial data collections
• Can be processed in distributed RAM
• Access time much faster than to disk data
• Load balancing
• Spatial rotations
• Overlapping management
• Redundant coverage
• O(log n) worst insert cost
• Future work
• kNN-queries
• Objects distribution balancing on servers

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SD-Rtree

• Thank You
• for
• Your Attention
• Questions First.Last_at_dauphine.fr