1. Spatio-Temporal Databases
Moving Objects

2. Introduction
Spatiotemporal Databases: manage spatial data whose geometry changes over time
Geometry: position and/or extent
Global change data: climate or land cover changes
Transportation: cars, airplanes
Animated movies/video DBs

3. Spatio-temporal Queries
Historical Queries: Store the past the history of a spatio-temporal evolution.
R-tree, MV3R-tree (PPR-tree)
“Future” Queries: Find the future positions of moving objects.
Indexing?

4. Indexing moving objects
Database stores the current location of each object and the velocity vector.
Example: cars moving in a highway system. GPS can provide position/velocity

5. Moving Objects: Queries
Range Queries
NN queries
Aggregation queries Q

6. Moving Objects:Representation
Consider the 1-d case (objects moving on a line)
Storing the locations of moving objects is a challenge:
Update the database with the new locations
Use a function of time f(t) to store a location
Update overhead is reduced; update the database only when velocity changes

7. Space-time
Trajectories are plotted as lines in the time-location space (y, t); p(t) = vt+a
trajectories o 1 o 2 o 3 o 4
(t) time

8. Indexing Use R-tree to index the lines Large MBRs, extensive overlap
Use a Quadtree approach (or a grid)
Partition the space into cells, store for each cell the lines that intersect it
Disk space is increased

9. Dual space-time Idea: map a line to a point trajectories o o o o o o o
(y) location
intercept
trajectories o o 1 1 o 2 o 3 o 4 o 2 o 3
(t) time
slope
intercept

10. Dual space-time indexing
Query must be transformed. [(y1q, y2q), (t1q, t2q)]
a + t2qv >= y1q and a+ t1q v <= y2q , for v>0
a + t1qv >= y1q and a+ t2q v <= y2q , for v<0

11. Dual space-time indexing
Another transformation (Hough-Y) is:
The difference is that we compute the intercept over a horizontal line
Queries in the dual space are similar with the previous transformation

12. Hough-Y space

13. Querying the dual space
Use a PAM to index the dual points, change the search function to find the points inside the query
Problem: Partitioning is not aligned with the queries  many I/Os
An idea is to try to store multiple structures, one for each set of queries with similar slope

14. Improving the query In the Hough-Y, the slope of the queries is:
y1q – yr (or y2q – yr)
location y3 y2
query y1
time

15. Improving the query Compute the dual using multiple y-lines
Store an R-tree for each line
Given a query, find the line that is closer to the query and then use the corresponding index
Thus, the query will appear as vertical as possible  better performance

16. Indexing in 2-dimensions
The dual transformation can be extended to 2 dimensional points
Map the trajectories in a point in 4-d using the transformations on x-t and y-t planes
Use the 1-d structures to answer a query

17. Dual for 2-D Moving Objects
Using Hough-X:
Map a moving point p with location (px, py) and velocity (vx, vy) to the 4D point (vx, ax, vy, ay)
Query is also transformed to a linear constraint query:
Q=[(x1q, x2q), (y1q, y2q), (t1q, t2q)]
ax + t2qvx >= x1q and ax+ t1q vx <= x2q
ay + t2qvy >= y1q and ay+ t1q vy <= y2q
x1qvy - y2qvx <= axvy – ayvx<= x2qvy - y1q vx

18. TP R-tree Time-Parametrized R-tree Store the MBRs as functions of time
The MBRs grow with time, at any time instant in the future we can compute the “MBR”

19. Motion function For each object, the database stores
Its minimum bounding rectangle (MBR) at the reference time 0
Its current velocity bounding rectangle (VBR)
Examples: MBR(a)={2,4,3,4}, VBR(a)={1,1,1,1}; MBR(c)={8,9,8,9}
An update is necessary only when an object’s VBR changes.

20. TPR – MBRs (a) The boundaries at current time 0
(b) The boundaries at future time 1

21. Insertion and Deletion
Insertion and Deletion similar to R*-tree
The only difference is that you have to compute the values: margin, overlap, volume over time
Trick: try to optimize the structure for the next H time instants.
Another optimization: when you update an object, re-compute the MBR at the current time

22. An example of update and re-computation of MBR (1D)y o1 o2 o3 o4 t
update
time
An example of update and re-computation of MBR (1D)
Reference:
Simonas Saltenis, Christian S. Jensen, Scott T. Leutenegger, Mario A. Lopez: Indexing the
Positions of Continuously Moving Objects. SIGMOD Conference 2000: