1. Chapter 4: Probabilistic Query Answering (2)
Probabilistic Data Management
Chapter 4: Probabilistic Query Answering (2)

2. Objectives In this chapter, you will:
Learn the definition and query processing techniques of a probabilistic query type
Probabilistic Group Nearest Neighbor Query
X. Lian and L. Chen, "Probabilistic Group Nearest Neighbor Queries in Uncertain Databases," In IEEE Trans. on Knowledge and Data Eng. (TKDE), vol. 20, no. 6, pp , June, 2008.

3. Recall: Probabilistic Query Types
Probabilistic Spatial Query
Uncertain/probabilistic database
Probabilistic range query
Probabilistic k-nearest neighbor query
Probabilistic group nearest neighbor (PGNN) query
Probabilistic reverse k-nearest neighbor query
Probabilistic spatial join /similarity join
Probabilistic top-k query (or ranked query)
Probabilistic skyline query
Probabilistic reverse skyline query
Probabilistic Preference Query 3 3

4. Probabilistic Group Nearest Neighbor Queries in Uncertain Databases
In IEEE Trans. on Knowledge and Data Engineering (TKDE), 2008

5. Group Nearest Neighbor Query
Group Nearest Neighbor (GNN) Search [ICDE04]
Given a database D and a set Q of query objects q1, q2, …, and qn, a GNN query retrieves an object o D that has the smallest summed distance to Q, i.e. i=1~n dist(qi, o)
find a data object o that minimizes
i=1~3 dist(qi, o)
D. Papadias, Q. Shen, Y. Tao, and K. Mouratidis, “Group Nearest Neighbor Queries,” Proc. 20th Int’l Conf. Data Eng. (ICDE), 2004.

6. An Example of GNN Query In a city, there are many restaurants
Three people want to have lunch together at a restaurant
Problem:
To find a restaurant in the city that minimizes its total distances to these 3 people

7. Other GNN Applications
Image Retrieval
Find an image in the database that is similar to a group of user-specified query images
Geographic information system
Mobile computing applications

8. Data Uncertainty
In many applications, real-world data are usually uncertain and imprecise
Image database
Noises in image features
Location-based services (LBS)
Measurement errors (GPS, RFID, etc.)
Trajectory data of mobile users are blurred for the sake of privacy preserving
Sensor networks
Sensory data inherently contain noises due to the environmental factor, network latency, or fluctuation of battery power …

9. Data Uncertainty (cont.)
object o
uncertainty region UR(o)
traditional database
uncertain database

10. GNN in Uncertain Databases
Probabilistic Group Nearest Neighbor (PGNN) Query in Uncertain Database [TKDE08]

11. Motivation Example of PGNN
In the mixed-reality games (like counter strike (CS)), 3 players may want to find a moving enemy to attack who can minimize their total travelling distance
Position of each enemy is uncertain (due to the movement or the network delay)

12. Motivation Example of PGNN (cont'd)
When a forest has several places on fire, at least 3 firefighters can collaborate to put out a place on fire
The positions that are on fire are uncertain
A PGNN can be issued to find those places on fire such that this group of firefighters can arrive as early as possible

13. Contributions
The proposal of probabilistic group nearest neighbor (PGNN) query in the uncertain database
Two effective pruning methods
Spatial pruning
Probabilistic pruning
Efficient PGNN query processing approach
Variants of PGNN query

14. Introduction Uncertain database Query processing over uncertain data
Uncertain objects are represented by uncertainty regions rather than precise points
Distances between uncertain objects are variables instead of fixed values
Query processing over uncertain data
Re-define traditional query types over precise data
Consider unique characteristics (i.e. uncertainty) of uncertain data
Guarantee the accuracy of query answers

15. Definition of PGNN Problem
Probabilistic Group Nearest Neighbor (PGNN)
Given an uncertain database D, a set of n query points, Q = {q1, q2, …, qn}, and a user-specified probability threshold a  (0, 1], a PGNN query retrieves a set of uncertain objects o  D such that they are expected to be GNN of query set Q with probability greater than a, that is,
where adist(o, Q) is defined as a monotonically increasing function adist(o, Q) = f (dist(o, q1), dist(o, q2), …, dist(o, qn))
aggregate function like
SUM, MIN, or MAX
Euclidean distance

16. Computation of PGNN Answers
Straightforward Approach, Linear Scan
For each uncertain object o  D
Sequentially scan the entire database
Compute the complex probability integration (via numerical method)
Output object o, if its probability is greater than a
Time Complexity -- O(|D|2)

17. Two Pruning Techniques
Spatial Pruning
Probabilistic Pruning

18. select the smallest distance upper bound as threshold
Spatial Pruning
Basic idea
Compute the lower/upper bounds of the aggregate distance, adist(o, Q), from each uncertain object o to query set Q, at a low cost
Use lower/upper bounds to filter out false alarms
select the smallest distance
upper bound as threshold
candidates

19. Spatial Pruning (cont'd)
Recall from the PGNN definition:
In fact, the spatial pruning method discards those objects with expected probability of being GNN equal to 0

20. Derivation of Distance Bounds
Probabilistic minimum bounding method (PMBM)
Bound all query points with an MBR
Compute the minimum/maximum distances between query MBR and uncertain objects
max
min

21. Derivation of Distance Bounds (cont'd)
Probabilistic single point method (PSPM)
Select a geometric centroid of query points as the representative
Compute the minimum/maximum distances via triangle inequality
|dist(q1, q3) - dist (q3, Co)| - ro ≤ dist(q1, o) ≤ dist(q1, q3)+ dist (q3, Co)+ro

22. Probabilistic Pruning
Intuition of probabilistic pruning
Prune those data objects that have the expected PGNN probability (LHS of inequality) smaller than or equal to 

23. Probabilistic Pruning (cont'd)
(1-b)-Hypersphere,
For any uncertain object p, we can pre-compute a hypersphere, namely , within its uncertainty region UR(p), such that object p reside in with probability (1-b), where b  [0, a]

24. Probabilistic Pruning (cont'd)
Use (1-b)-hypersphere to obtain lower/upper bounds of distance adist(p1-b, Q), say [LB_adist(p1-b, Q), UB_adist(p1-b, Q)]
Any object o can be safely pruned if it holds that:
UB_adist(p1-b, Q) <LB_adist(o, Q)

25. PGNN Query Processing
Construct an R-tree over uncertainty regions of data objects
Nodes are recursively bounded by minimum bounding rectangle (MBR) until finally one node (root) is obtained

26. PGNN Query Processing (cont'd)
Pruning Intermediate Nodes
Define lower/upper bounds of aggregate distances between nodes and query points
PMBM (bounding query points with an MBR)
PSPM (using centroid of query points and triangle inequality)
An intermediate node e can be pruned, if it holds that LB_adist(e, Q)  UB_adist(o, Q) for a candidate o
PMBM

27. PGNN Query Procedure
Traverse the nodes of R-tree in a best-first manner by maintaining a minimum heap H (with key the lower bound of distance from node/object to query set Q)
Every time we encounter a node/object, we compute its lower/upper bounds of aggregate distance, and apply the spatial pruning to filter out false alarms
For uncertain objects that cannot be pruned by spatial pruning, apply probabilistic pruning
After tree traversal, we refine the obtained candidate set by calculating the actual PGNN probability

28. Variants of PGNN Query PGNN query with uncertain query objects
Re-define the lower/upper bounds of aggregate distances

29. Variants of PGNN Query (cont'd)
PGNN with different aggregate functions
SUM, MIN, MAX
AVG, weighted SUM, etc.
k-PGNN query
To retrieve k uncertain objects that are expected to be closest to a query set Q with probability greater than a

30. Variants of PGNN Query (cont'd)
Any other variants of PGNN?
PGNN on road networks? …

31. Experimental Evaluation
Experimental Settings
Synthetic data sets
Generate center location Co of uncertain object o in a data space [0, 1,000]d
Produce radius ro  [rmin, rmax] for uncertainty region UR(o)
Four types of data sets: lUrU, lUrG, lSrU, lSrG
Measures
wall clock time (the filtering time of index traversal)
speed-up ratio (total time cost compared with that of the linear scan method)

32. Query Performance vs. b the time cost of spatial pruning (upper part)
the time cost of probabilistic pruning (lower part)
data size |D| = 30K, dimensionality d = 3, the number of query points n = 4, probability threshold a = 1, SUM aggregate function

33. Scalability Test on Data Size |D|
dimensionality d = 3, the number of query points n = 4, probability threshold a = 1, SUM aggregate function

34. Summary
We formulated probabilistic group nearest neighbor (PGNN) query in the context of uncertain databases
We proposed two effective pruning methods, spatial and probabilistic pruning, to reduce the search space of PGNN query, which can be seamlessly integrated into an efficient query procedure
We further discussed some variants of the PGNN query
We demonstrated through extensive experiments the efficiency and effectiveness of our proposed pruning methods as well as query processing approaches, in terms of wall clock time and speed-up ratio compared with linear scan