Nearest Neighbor Search in Spatial and Spatiotemporal Databases

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Nearest Neighbor Search in Spatial and
Spatiotemporal Databases

• Dimitris Papadias
• Hong Kong University of Science and Technology

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Spatial and spatiotemporal databases

• Spatial databases manage large collection of
  multi-dimensional objects.
• Important query types
• Window query Retrieve all rivers in CA
• Nearest neighbor Find my nearest gas station
• Spatial join Report pairs of (city C, river R)
  such that R crosses C
• Spatiotemporal databases deal with the same
  queries assuming, however, moving objects
• Mobile computing
• Traffic supervision
• Flight control
• Weather forecasting

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R-trees Guttman SIGMOD 84. Sellis et al VLDB 87,
Beckman et al SIGMOD 00
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TPR-trees Saltenis et al., SIGMOD 00, our group
VLDB 03

• Extends the R-tree by introducing the velocity
  bounding rectangle (VBR) in non-leaf entries.
• Objects are grouped together based on both their
  location and velocities.

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Conventional NN search with R-(TPR-) trees

• Depth-first Roussopoulos et al., SIGMOD 95
• Best-first traversal Hjaltason and Samet TODS
  99, incremental and optimal

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NN search - other approaches

• Several algorithms and theoretical performance
  bounds have been devised for exact and
  approximate processing in main memory. Here we
  care about I/O efficiency (minimization of node
  and page accesses) as well as cost models about
  the practical performance (suitable for query
  optimization).
• Several approaches for NN in high-dimensional
  spaces (but the problem is different due to the
  dimensionality curse). Here we consider low
  dimensional spaces (spatial and spatiotemporal
  databases).
• Ferhatosmanoglu et al SSTD 01 discover the NN
  in a constrained area of the data space (e.g.,
  find the NN to the south of the query point).
• Korn and Muthukrishnan SIGMOD 00 discuss
  reverse nearest neighbor queries, where the goal
  is to retrieve the data points whose nearest
  neighbor is a specified query point.
• Korn et al. VLDB 02 study the same problem in
  the context of data streams, where the data are
  not known in advance.

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NN search for mobile queries

• Zheng and Lee, SSTD 01 return the current NN
  and the validity time of the result.
• Restrictions (i) assumes a maximum speed (ii)
  applicable only to single NN (iii) requires
  voronoi diagrams.

• Song and Roussopoulos, SSTD 01 minimize the
  number of queries for moving clients by returning
  mgtk NNs.
• Problem how to determine m.

IF 2?dist(q,q') ? dist(q,b)-dist(q,a), THEN the 2
NN at q' be among the 4 NN of the first query.
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Time parameterized NN (our group, SIGMOD 02)

• Assuming a constant and known velocity, a TPNN
  returns
• The current query result R
• The validity period T of R
• The change C of the result at the end of T

Result
Ri, T2, Cj
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TP NN queries Influence Time

• Some objects have infinite influence time.
• The object that will become the next nearest
  neighbor is the one with the minimum influence
  time.

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Processing TP NN with R- (TPR-) trees

• Influence time of a MBR the earliest possible
  time that any object in the MBR will become the
  new NN.

• Algorithm traverse the R-tree using depth-first
  or best-first traversal using the influence time
  instead of the mindist .
• Cost of TPNN queries about the same as that of
  conventional queries because we have to visit the
  influencing nodes anyway (to find the NN).

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Continuous Nearest Neighbors (CNN) (our group,
VLDB 02)

• Given a line segment qs,e, find the NN of
  every point on q.
• Result representation s(.NNa), s1(.NNc),
  s2(.NNf), s3(.NNh), e.
• The points (s, s1, s2, s3, e) are the split
  points.

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Main idea

• Maintain the set of split points incrementally.

After processing a
After processing c
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Processing TP NN with an R- (TPR-) tree

• Avoid examination of all points.
• Given an MBR E and query segment q, E must be
  searched if and only if there exists a split
  point si?SL such that dist(si,si.NN) gt
  mindist(si, E).

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Location Based NN queries (LBNN) (our group,
SIGMOD 03)

• A location-based kNN query q returns
• The current k NNs
• A validity region such that the result remains
  the same as long as q remains in the region.
• The validity region of q is the Voronoi Cell (VC)
  of the NN o.

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Computing the Voronoi Cell on-the-fly

• Step 1 Find the current NN
• Step 2 Use time TP NN queries to tighten the
  validity region

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NN queries in road networks (our group, VLDB 03)

• Find my nearest gas station in terms of driving
  distance.
• Answer Hotel b (the Euclidean NN is d)

• Assumptions
• We can incrementally compute Euclidean NN using
  conventional NN algorithms.
• We can compute the network distance between the
  query and any point (i.e., the length of the
  shortest path connecting them) using Dijkstra's
  algorithm.

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Euclidean Restriction Algorithm
1st Euclidean NN
2nd Euclidean NN
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Network Expansion Algorithm
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NN in the presence of obstacles (not published)

• The NN of q in terms of obstructed distance is b,
  although the Euclidean NN is a.

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Visibility graphs

• Have been used widely in Computational Geometry
  for shortest path problems (e.g., find the
  shortest path from pstart to pend that does not
  cross any obstacle).

• Problem We cannot maintain the entire visibility
  graph in memory for real spatial datasets.
• Solution We only need the obstacles and objects
  that affect the result of the query.

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Obstacle nearest neighbor algorithm

• Idea Similar to the Euclidean Restriction
  algorithm for road networks.

• BUT how do we perform the obstructed distance
  computations?

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Obstructed distance computation

• Goal compute the obstructed distance between p
  and q.
• First retrieve obstacles o1, o2 in the Euclidean
  range.
• Compute a provisional distance d1(p,q) using only
  o1, o2.
• d1(p,q) is not enough because the shortest path
  is obstructed by o3.
• Perform a second Euclidean range query on the
  obstacle R-tree using d1(p,q) and retrieve o3,
  o4.
• Compute a new obstructed distance d2(p,q) taking
  o3, o4 into account.
• Repeat the process until the obstructed distance
  remains the same for two consecutive iterations.

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Other related work

• By our group Similar concepts to the ones
  presented here, apply to several other spatial
  queries, i.e., TP spatial joins, Continuous
  window queries.
• Cost Models for TP and continuous queries TODS
  03.
• Analysis of predictive NN (and other) queries
  TODS to appear.
• An Efficient Cost Model for Optimization of
  Nearest Neighbor Search in Low and Medium
  Dimensional Spaces TKDE to appear.
• By other groups increasing interest for novel
  types of NN search in the context of mobile
  computing and data streams applications
• Iwerks et al VLDB03 discuss continuous NN in
  the presence of object updates.
• Shekhar et al ACM GIS 03 discuss the in-route
  nearest neighbor query, which, given a
  trajectory, retrieves the single NN (e.g., gas
  station) that results in the minimum diversion
  from the trajectory.
• Jensen et al ACM GIS 03 discuss NN for objects
  moving on road networks.

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Group NN queries (our group, ICDE 04)

• Input a set Pp1,,pN of static data points in
  multidimensional space and a group of query
  points Qq1,,qn.
• Output the k (?1) data point(s) with the
  smallest sum of distances to all points in Q. The
  distance between a data point p and Q is defined
  as dist(p,Q)?i1npqi, where pqi is the
  Euclidean distance between p and query point qi.
• Example three users at locations q1, q2 and q3
  want to find a meeting point (e.g., a
  restaurant) the corresponding query returns the
  data point p that minimizes the sum of Euclidean
  distances pqi for 1?i?3
• Assumption the data points are indexed by an
  R-trees. Q may or may not fit in main memory.

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Multiple Query Method (MQM)

• Idea Perform incremental NN queries for each
  point in Q and combine their results.

• ltp10, 7gt, ltp11, 6gt, T5 (23)
• ltp11, 7gt
• T6 (33)
• MQM terminates

• Problem MQM may visit the same node and discover
  the same data point many times (for different
  query points).

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Minimum Bounding Method (MBM)

• Applies the MBR of Q to prune the search space.
• Heuristic 1 Let M be the MBR of Q, and best_dist
  be the distance of the best GNN found so far. A
  node N cannot contain qualifying points, if

• Heuristic 2 A node N cannot contain qualifying
  points, if

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File Multiple Query Method (F-MQM)

• What happens if Q does not fit in memory.
• F-MQM sorts query points according to their
  Hilbert value and splits Q into blocks Q1, ..,
  Qm that fit in memory.
• For each block, it computes the GNN using one of
  the main memory algorithms
• It finally combines their results using MQM.
• Complication once a NN of a group has been
  retrieved, we cannot compute its global distance
  (i.e., with respect to all data points)
  immediately.

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F-MQM (cont)

• Solution lazy evaluation
• First we find the GNN p1 of the first group Q1
• Then, we load in memory the second group Q2 and
  retrieve its NN p2. At the same time, we also
  compute the distance between p1 and Q2.
• Similarly, when we load Q3, we update the current
  distances of p1 and p2 taking into account the
  objects of the third group.
• After the end of the first round, we only have
  one data point (p1), whose global distance with
  respect to all query points has been computed.

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File Minimum Bounding Method (F-MBM)

• First, the points of Q are sorted by their
  Hilbert value and are assigned to groups (that
  fit in memory) according to this order.
• For each group Qi, F-MBM keeps in memory its MBR
  Mi and cardinality ni (but not its contents).
• F-MBM descends the R-tree of P (in depth-first or
  best-first traversal), only following nodes that
  may contain qualifying points.

Heuristic Let best_dist be the distance of the
best GNN found so far. A node N can be safely
pruned if