Clustering Objects in Large Spatial Network

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Clustering Objects in Large Spatial Network

• DB seminar
• Speaker Ken Yiu
• Date 31/10/2003

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Outline

• Motivation
• Problem definition and related work
• Disk based representation of the model
• K-medoid and Density based algorithms
• Applications in wider domain
• Experimental results
• Conclusions

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Motivation

• Euclidean distance of two points does not
  represent the actual distance
• Shortest path of two points constrained by
  transportation network
• Real examples road network, river network, plane
  network, rail network,
• Conventional clustering algorithms cannot be
  applied without distance matrix
• Require computing expensive all pairs distance

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Problem definition

• Network weighted graph G(V,E,W)
• V set of nodes
• E set of edges
• W E?? associate each edge an edge weight
• Point a position on a edge (ni, nj, pos) where
  pos ? 0,W(e) and e(ni, nj)
• the point is pos units away from the node ni
  along the edge (ni, nj)
• How to ensure unique representation
• Require in the triplet that niltnj

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Example model
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Problem definition

• Direct distance dL(x,y)
• 8 if x and y are on different edges
• x.pos-y.pos (dist. along edge), otherwise
• Network distance shortest distance
• Between nodes
• Can be computed by Dijkstras algorithm
• Between points x on (na, nb) and y on (nc, nd)
• Different edges
• Same edge minimum of previous expression and
  dL(x,y)
• Example using previous slide

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Dijkstras algorithm

• Shortest path algorithm
• One source, all destination(s)
• Many variations (by using different ADT)
• We choose to use priority queue/heap

Set dist for all nodes to ? Qlt(ns,0)gt //
node,dist pair for src. node while (Q ?
?) Entry B Dequeue(Q) distB.nodeB.dist
for each non visited adjacent node nv of
B.node Create new entry B B.node
nv B.distB.distW(e(B.node,nv)) Enqueue(Q,
B)
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Dijkstras algorithm
Network expansion from node 1 to all nodes
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Properties

• Network distance is a metric
• Satisfies triangular inequality
• d(x,z) ? d(x,y) d(y,z) (min. dist)
• With dist. matrix for all pairs of nodes, dist.
  between points found in O(1) time
• Only need to calculate all pairs dist. of nodes
  which are incident to some edges with some points
  (Let V be these set of nodes)
• Time complexity discussed later
• Space for dist. matrix O(V2) instead of
  O(V2)
• Possible that V ltlt V

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Properties

• Planar graphs
• Graph drawn in plane with no edge crossings
• Most real-life road networks are planar
• E ? 3V - 6 ( by Eulers formula )
• Time complexity
• Dijkstras algorithm
• O(E logE ) ( i.e. O(V logV ) for planar
  graphs)
• Prev. node dist. matrix (for planar graph)
• Apply Dijkstras algorithm on each node in V
• O(VV logV ) ( at most O (V2 logV ) )

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Related work

• K-medoid algorithms PAM, CLARA,CLARANS
• Density based algorithm DBSCAN
• C2P
• Finding NN is efficient in spatial DB but not in
  our model
• e.g. expensive to find the NN for an outlier
• Classical graph partitioning/clustering
  algorithms
• Single link, Complete link
• Too expensive
• Minimum spanning tree, then remove long edges
• Sensitive to outliers

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Related work

• Networking field (NTC,CCAM)
• Partition nodes to facilitate efficient query
  processing in terms of I/O cost
• Do not consider point/object lying on edges
• Spatial Network Databases (SNDB)
• Only discuss simple query processing such as
  point query, range query,
• Do not discuss about clustering

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Transformation for graph alg.

• Our interest is in the points but not the nodes
• Need transformation for graph clus. alg.
• Transformed graph may not be planar
• More complex model ? more expensive clustering

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Conventional clustering alg.

• Many clustering alg. can be applied
• Disadvantage They require N2 distance matrix
  between every pairs of objects
• Expensive to compute
• Too large to be fit in memory and even in disk
  for moderate N
• Better methods
• Our approach cluster the points and compute the
  distances at the same time
• By integrating Dijkstras algorithm with the
  clustering algorithm(s)
• Should have similar complexity as Dijkstras
  algorithm

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Disk-based representation

• Point group
• Avoid storing the edge redundantly
• Group of points on the same edge
• Fixed fields edge, of pts
• Variable fields ref. pos. of each pt.
• Adjacency list
• Fixed fields size of adj. List
• Variable fields adjacent node ID, edge weight,
  pointer to its point group
• Btree index
• Efficient search of adjacency lists and points

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Disk-based representation
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K-medoid

• Frequent operation finding the nearest medoid
  for each point p
• p on edge (ni, nj), Mu,Mv nearest to ni,nj
  respectively
• With node distance matrix, above comparison can
  be done in O(2k) time (Avg O(1) time ?)
• Node dist. matrix still expensive (to compute and
  store) for large networks
• Need to use more efficient method
• Concurrent expansion from the medoids
• Incremental replacement of node distances
• Medoid pool

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K-med (concurrent expansion)

• Network expansion (Dijkstras algorithm) for each
  medoid too expensive
• Apply it concurrently for all k medoids
• By sharing the same heap/priority queue

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K-med (concurrent expansion)
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K-med (incremental replacement)

• Only few medoids (e.g. 1 medoid) are changed in
  the next (k-medoid) iteration
• Reuse result of previous iteration
• Drill hole(s)
• Expand from border of the hole(s) and all medoids

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K-med (incremental replacement)
ChgC set of medoid index to be changed (cluster
ID in 1,k) Generalize the case for changing
arbitrary number of medoids Common case ChgC1
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K-med (Medoid Pool)

• Choose Ak points as the medoid pool
• A trade off between quality and speed
• Medoid can only be chosen from the pool
• Apply network expansion for each medoid
• Store in the AkV dist. array (in memory/disk)
• Find the nearest medoid for each node
• More benefit for more iterations, and vice versa

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Density based algorithms

• Disadvantages of k-medoid
• Convergence of k-medoid depends on the choice of
  medoid
• May take many iterations
• Cannot identify outliers easily
• Density based clustering methods
• Easily identifies outliers
• Fast
• obtain the optimal solution (by its density
  definition)
• do not need to improve result by iterations
• ?-Link (new) , DBSCAN

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?-Link

• One parameter ?
• Points within distance ? in the same cluster
• Method (similar to network expansion)
• Only one heap is used
• Store NNdist of each node to a clustered point
  (in current cluster)
• Update NNdist of the nodes
• Whenever NNdistnz ? and NNdistnz? ?, expand
  from that node i.e. Enqueue entry (nz,
  NNdistnz)
• Result is a special case of DBSCAN
• No need to perform expensive range search
• Cheap clustering cost

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DBSCAN

• The DBSCAN algorithm is adapted to our model
• Two parameters MinPts and ?
• Only need to discuss about the range search with
  ? distance
• Straight forward range search may introduce
  duplicates which affect the count of ?-neighbors
• A simple check can avoid (not eliminate) these
  duplicates but we do not discuss it here

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Comparison of the techniques

• K-medoid (common point assignment, heap space)
• Dijkstras algorithm for each medoid
• Time O(k VlogV) per iteration
• Space O(kV) for storing node dist. (label) to
  medoids
• Concurrent Expansion
• Time O(VlogV) per iteration
• Space O(V) for storing node dist. to medoids
• Incremental Medoid Replacement
• Better average case, worst case same as
  concurrent expansion
• Space O(V) for storing node dist. to medoids
• Medoid Pool
• Time O(AkVlogV) for overhead, O(V) per
  iteration
• Space O(AkV) for pool dist, O(V) for node
  dist.

Note complexity for planar graph (or sparse
graph) only !
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Comparison of the techniques

• Density based methods
• ?-Link
• Time O(VlogV) for transversing network
• Better average case, only transverse the part of
  network containing points
• Space O(V) for storing node dist. to medoids
• DBSCAN
• Time total O(N range-search(?))
• Space only space for ? range search, much
  smaller than O(V)

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Applications in wider domain

• Edge weight interpretation
• Different users can interpret edge weights in
  different ways
• Set edge weight as the edges traveling time
• Obtain clusters based on traveling time (t)
• Set edge weight as the edges traveling cost
• Obtain clusters based on traveling cost ()
• Set edge weight as
• Obtain clusters based on

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Applications in wider domain

• Combination of networks
• Useful for discovering clusters across different
  networks
• Modeling methods
• Transition node
• Node with (at least) an edge to another network
• Transition edge weights
• Assigned as the cost of transition
• The combined network may not be planar but it is
  still a sparse graph with low average degree
• Still efficient for performing clustering

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Preliminary results

• Real network
• Main roads in North America
• 175,813 nodes and 179,179 edges
• Generate synthetic clusters in the network using
  expansion
• N number of points
• k number of clusters
• V number of nodes
• Default N100K, k10, V100K

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Preliminary results
Elements pop from heap as a function of k,
V100K
Heap Pop size
k
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Preliminary results
Cost of k-med(CE) as functions of N (fix V100)
Cost of k-med(CE) as functions of V (fix N1K)
Cost of different algorithms as functions of V
(fix N100K)
Cost of density based methods as functions of ?
(fix N100K)
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Future Work

• Avoid scanning points many times
• Cluster the weighted nodes instead of points
• Scan points once and store summary statistics
  (pt. count, seg. dist.) in each node
• Space O(V) acceptable
• For each edge e(ni, nj)
• Count the of points of pos 0, ½W(e))
• Increment point count of the node ni
• Count the of points of pos ½W(e),W(e)
• Increment point count of the node nj
• Increment the segment distance of both nodes ni
  and nj by W(e)/2

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Future Work

• k-medoid
• CE can still be applied
• Only put nodes with non-empty point counts in
  clusters
• Density based methods
• The density for the region near node ni is
  defined as
• Point count of ni / Segment distance of ni
• Adapt the definitions for facilitating the
  discovery of these clusters

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Conclusion

• A new clustering problem formulated
• Propose a disk based representation for large
  datasets
• Reduce or amortize the effort of finding shortest
  distance
• Propose three clustering algorithms
• Efficient and scalable
• Applicable for some other related problems

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References

• 1) E. W. Dijkstra. A note on two problems in
  connection with graphs. Numeriche Mathematik,
  1269271, 1959.
• 2) A. K. Jain and R. C. Dubes. Algorithms for
  Clustering Data. Prentice Hall, 1988.
• 3) E. Martin, H. P. Kriegel, J. Sander, and X.
  Xu. A density-based algorithm for discovering
  clusters in large spatial databases with noise.
  In ACM SIGKDD, 1996.
• 4) A. Nanopoulos, Y. Theodoridis, and Y.
  Manolopoulos. C2P Clustering based on closest
  pairs. In VLDB, 2001.
• 5) D. Papadias, J. Zhang, N. Mamoulis, and Y.
  Tao. Query processing in spatial network
  databases. In VLDB, 2003.

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References

• 6) S. Shekhar and D. Liu. CCAM A connectivity
  clustered access method for networks and network
  computations. Journal of Computational Biology,
  19(1)102119, 1997.
• 7) S. H. Woo and S. B. Yang. An improved network
  clustering method for I/O-ecient query
  processing. In ACM GIS, 2000.