Computer science department university of
Southern California Los Angeles.
Songhua Xing
a PhD candidate at Computer Science Department
now at IBM T.J. Watson Research Center
Country: China
Cyrus Shahabi
Director of the Information Laboratory (InfoLAB)
Professor of Computer Science and Electrical Engineering at USC
Director of the Integrated Media Systems Center
Chief Technology Officer and Founder of Geosemble Technologies
Ph.D. in Computer Science
Bei Pan
a PhD candidate at Computer Science Department
Microsoft Research Asia, Autodesk, University of Southern California
Country:China
New type kNN queries
Surface k Nearest Neighbor(skNN) queries
Extend to constrained third dimension
Two exact methods for highly dynamic
environment as arbitrary movement
SE(Surface Expansion)-Tree
Fat and short - not efficiency
an analogous counterpart of the Dijkstra Expansion
Tree on land surface
ASI(Angular Surface Index)-Tree
Thin and tall – low cpu and I/O overhead
Related research
Static and dynamic query
Problem & Preliminaries
kNN methods only on Euclidean and road networks.
skNN is for surface distance, represented as triangular meshes.
CskNN can be used on continuous queries and the complexities of land
surfaces.
CskNN algorithm
monitor and coordinate among the moving objects.
One of CkNN for road map
Dijkstra Expansion tree
SE-Tree – for Static query
More effective – ASI-Tree
Details of our surface index(ASI) and its corresponding CskNN
Dynamic query
Experiments
Summary and future work
ASI- Tree outperforms SE-Tree
Static(snapshot) query
Different constrained environment
Road networks
Land surface
Dynamic query
The paper three types of updates
Object movements
Query movements
Fluctuations of edge weights
Assumption and problem definition
Moving object
Point Of Interest
Static query point
Three distance metrics
Euclidean distance
Lower bound of surface distance
Network distance
Upper bound of surface distance
Surface distance
Problem definition
query consists of two steps
snapshot skNN query
continuously monitoring and updating the result
sets as the objects move
Shortest surface path computation
Chen-Han algorithm
costs
Unfolding process - expensive
Surface expansion tree
Dijkstra Expansion tree, based on the ChenHan algorithm
Definition
Surface Expansion Tree is the final result of
Chen-Han algorithm and there is only one path
from the source to the vertice
Surface expansion tree
Observation 1 makes partitioning these
surface shortest paths of an SE-Tree
possible.
Observation 2 Drawback
SE-Tree in general is fat and short
Surface Expansion Tree
Initial query processing – two areas
Three categories
Within the result boundary
Ignore this case
Result set remains the same
Incoming movement
Outgoing movement
Two scenarios
More Outgoing movement
More Incoming movement
Expansion phase, the complexity is
In the shrinking phase, there is no surface
distance computation
Complexity is mlog(m)
Similarity
All these methods built an expansion tree rooted at the
query point
The result boundary and expansion boundary are the
same on road networks.
Different
This naïve approach could be fast during the phase
when the SE-Tree shrinks
Expansion Two problems:
Surface path computation is extremely high
Expansion areas of SE-Tree could be large.
Overcome by Surface Shortest Path Container
store partial results of pre-computation
build a novel index schema(Angular Surface Index(ASI))
Angular Surface Index (ASI)
Thin and tall
Two data structure
Surface Shortest Path Container
Surface Equidistant Line
Surface Shortest Path Container
To pre-compute a complete SE-Tree offline and store its
shortest path.
Two Steps
locate the data object using a spatial index
retrieve the shortest path directly from disk
Drawbacks
a data object lays on the face rather than a vertex, this
approach cannot find the exact shortest path and the accurate
distance
storing all these shortest paths is
per site
The search time is almost linear.
How to speed up
take advantage of partial results based on geometric property to
speed up the online process.
The advantage is to minimize the search
area of Dijkstra algorithm.
A new concept of Cover Set and redefine
the concept of Shortest Path Container for
surface, and then discuss their spatial
properties.
According to Observation 1, we can always
find a polyline sp from the source s to a
point p on the margin of T, which is
immediately left to the leftmost shortest
path to CS(e) and do not cross any
shortest paths, hence sp constitutes the
left part of the boundary b.
Container’s boundary consists of
the left boundary line,
the right boundary line
the end boundary line (which only exists if the left
and right boundary lines do not converge)
Propose an algorithm to create a surface
shortest path container
In Line 6, the end boundary can be NULL if left
and right boundaries do not intersect the margin.
The time complexity of Algorithm 3 is O(NlogN)
due to the sort operation in Line 3. However, since
the pre-computation of shortest paths takes
,
the overall time complexity is
.
Designed to partition along the horizontal (latitude)
direction
These lines are sorted by their increasing distance
value to the source point and this order is termed
as levels
Based on surface shortest path containers
and surface equidistant lines
Each partition of is called a surface chunk.
With this ASI-Tree, each node represents a
container.
Compared with SE-Tree, ASI-Tree has the
following advantages
Experiment setup
Model
BH: Bearhead (BH) area in WA, USA which covers an area around 10.7km×14km and 2)
EP: Eagle Peak (EP) area in WY, USA with similar size as BH.
Create five synthetic surface models with the same size (10km×10km)
Device
PC with Intel 6420 Dual CPU
2.13G Hz and 3.50 GB RAM
The operating system is Windows XP SP2
The parameters
100 CskNN queries, each query is 50 timestamps.
The first 6 parameters are tested on both BH and EP (Surface Roughness RA is only for
synthetic data sets.)
The Impact of k
ASI based algorithm outperforms the naïve algorithm both in query
efficiency and I/O operations (least a factor of two for k > 4.)
Performance in I/O by an average factor of two because the search is
localized to avoid unnecessary access to surface vertices.
Uniform or Gaussian distributions
the ASI based algorithm has a slightly better
performance for objects with Gaussian
distribution than objects with uniform
distribution.
The Impact of Object Distribution and DO
Both query processing time and I/O cost
decrease for both algorithms as DO increases.
The Impact of a and v
(a)(b)both query processing time increases slightly as well
because the possibility to enlarge the search area is increased.
(c)(d)both algorithms are practically unaffected by object speed
because the core of both algorithms only concern whether there
are object updates rather than how far the objects move.
The Impact of DC
the performance is enhanced as more containers are created for both BH and EP.
The Impact of RA
ASI-based algorithm keeps outperforming the naïve Algorithm
rougher terrains could probably generate a larger search area than smooth terrains.
Propose two algorithm
naïve algorithm
surface index (ASI) based algorithm
ASI-based algorithm outperforms the
naïve algorithm under all circumstances
Simplified problem setting (pre-defined static
query points)
Further studying these complex settings,
where queries move arbitrarily.