1. Voronoi-based Geospatial Query Processing with MapReduce
Afsin Akdogan, Ugur Demiryurek,
Farnoush Banaei-Kashani, Cyrus Shahabi
Integrated Media System Center,
University of Southern Califonia Los Angeles, CA 90089
Cloud Computing Technology and Science (CloudCom),
2010 IEEE Second International Conference on
(MON)
Regular seminar
TaeHoon Kim

2. Contents Introduction Background
Constructing Voronoi Diagram with MapReduce
Query processing
Performance Evaluation
Conclusions

3. 1. Introduction
With the recent advances in location-based services, the amount of geospatial data is rapidly growing
Geospatial queries are computationally complex problems which are time consuming to solve, especially with large datasets
On the other hand, we observe that a large variety of geospatial queries are intrinsically parallelizable.
Cloud computing enables a considerable reduction in operational expenses
by providing flexible resources that can instantaneously scale up and down
The big IT vendors including Google, IBM, Microsoft and Amazon are ramping up parallel programming infrastructures
ramp something up ~을 늘리다[증가시키다]

4. 1. Introduction Google’s MapReduce programing model
Parallelization for processing large datasets
Can do that at a very large scale
Google processes 20 petabytes of data per day with MapReduce
A variety of geospatial queries can be efficiently modeled using MapReduce
Reverse Nearest Neighbor query(RNN)
RNN
Given a query point q and a set of data points P, RNN query retrieves all the data points that have q as their nearest neighbors p1 p2 p4 p3 p5
Example of RNN Query
p5: {p2, p3, p4}

5. 1. Introduction
Parallel spatial query processing has been studied in the contexts of parallel databases, cluster systems, and peer to peer systems as well as cloud platforms
However, the points in each partition are not locally indexed
Meanwhile, several approaches that use distributed hierarchical index structures
R-tree based P2PR-Tree, SD-Rtree, kd-tree based k-RP, quad-tree based hQT
Problems of distributed hierarchical index structures
Do not scale due to the traditional top-down search that overloads the nodes near the tree root, and fail to provide full decentralization

6. 1. Introduction R-Tree index with MapReduce
Without supporting types of query
Motivate the problem of geospatial query processing without specifically explaining how to process the queries
Hierarchical indices like R-tree are structurally not suitable for MapReduce
In this paper, we propose a MapReduce-based approach
Both constructs a flat spatial index, Voronoi diagram (VD), and enables efficient parallel processing of a wide range of spatial queries
Spatial queries
reverse nearest neighbor (RNN),
maximum reverse nearest neighbor (MaxRNN)
k nearest neighbor (kNN) queries

7. 2. Background Voronoi diagrams
A voronoi diagram decomposes a space into disjoint polygons based on the set of generators(i.e., data points)
Given a set of generators S in the Euclidean space, Voronoi diagram associates all locations in the plane to their closest generator
Each generator s has a Voronoi polygon consisting of all points closer to s than to any other site S1 S2 S3 S6 S5 S4
: Data point
: A set of generators
{S1, S2, S3, S4, S5 } >

8. 2. Background
MapReduce
Hadoop first each mapper is assigned to one or more splits depending on the number of machines in the cluster
Secondly, each mapper reads inputs provided as a <key, value> pair at a time, applies the map function to it and generates intermediate <key, value>pairs as the output
Finally, reducers fetch the output of the mappers and merge all of the intermediate values that share the same intermediate key, process them together and generate the final output

9. 3. Constructing Voronoi Diagram with MapReduce
Voronoi Diagram construction is inherently suitable for MapReduce modeling
Because a VD can be obtained by merging multiple partial Voronoi diagrams (PVD)
VD is built with MapReduce using divide and conquer method
The main idea behind the divide and conquer method
1. Given a set of data points P as an input in Euclidean space, first the points are sorted in increasing order by X coordinate
2. P is separated into several subsets of equal size
3. A PVD is generated for the points of each subset, and then all of the PVDs are merged to obtain the final VD for P

10. 3. Constructing Voronoi Diagram with MapReduce
Map phase
Given a set of data points P = { p1,p2 … p13,p14 } as input, first the points are sorted, then they are divided into two splits equal in size
The first mapper reads the data points in Split 1, generates a PVD and emits <1, PVD1>
Likewise, the second mapper emits, <1, PVD2>
Reduce phase
The reducer aggregates these PVDs and merges them into a single VD
The final output of the reducer is in the following <key, value> format : <pi, VNs(Pi) > where VNs(Pi) represents the set of Voronoi neighbors of pi

11. 3. Voronoi Generation: Map phase
Generate Partial Voronoi Diagrams (PVD)
Split 1
Split 2
right
left
mapper reads the data points in Split 1, generates a PVD and emits
Output :
<1, PVD1> p7 p1 p3 p5 p8 p4 p2 p6
right
left
emit
emit
<key, value>: <1, PVD(Split 1)>
<1>, <p2,p3,p4>
<key, value>: <1, PVD(Split 2)
<1>, <p6,p7,p8>

12. 3. Voronoi Generation: Reduce phase
Remove superfluous edges and generate new edges
Split 1
Split 2
right
left
The reducer aggregates these PVDs and merges them into a single VD
Final Output
<pi, VNs(Pi) > p7 p1 p3 p5 p8 p4 p2 p6
right
left
emit
<key, value>: <point, Voronoi Neighbors>
<p1, {p2, p3}>
<p2, {p1, p3, p4}>
<p3, {p1, p2, p4, p5}>
…..

13. 4. Query Processing : RNN Suppose the input the map function is
<point, Voronoi Neighbors(VNs)>, ∀𝑝∈𝑃
Map phase
Each point p finds its Nearest Neighbor
Emit : <NN(pn), pn>
Reduce phase
The reducer aggregates all pairs with the same key pK
Emit : <point, Reverse Nearest Neighbors>

14. 4. RNN query processing Mapper emit <p5, p2> <p5, p3>
Reduce
Reducer
<p5, {p2, p3, p4}>
emit

15. 4. Query Processing : MaxRNN
Nearest Neighbor Circle(NNC)
Given a point p and its nearest neighbor p’, NNC of p is the circle centered at p’ with radius |p, p’|
MapReduce
Finds the NN of every point and computes the radiuses of the circles
Finds the overlapping circles first, Then, it finds the intersection points that represent the optimal region

16. 4. Max RNN query processing
Mapper 1st step
p1 finds p3 as the nearest neighbor among its VNs {p2,p3} and sets |p1,p3| as the radius of its NNC
<(p1, |p1,p3|)>, <(p2,r2),(p3 r3)>
Mapper 2nd step
p1 checks its VNs<p2,p3>, finds that is overlaps <p2,p3> and
computes the intersection points {i2,i3,i5,i6}
Each intersection point i, then computes its weight by counting the NNCs covering I <1, i,w(i)> p1 p3 p2 r2 r3 p1 p3 p2 i5 i1 i2 i3 i4 i6
<1, (i2,3)>, <1, (i3,3)> <1, (i5,2)> <1,(i6,2)>
emit
Map
Reducer
Finds the intersection points with the heightest weights who are emitted as the final ouput
i1,i2,i3
emit

17. 4. Query Processing : kNN K Nearest Neighbor Query(kNN)
Given a query point q and a set of data points P, k Nearest Neighbor(kNN) query finds the k closest data point pi∈𝑃 to q where d(q, pi) ≤ d(q, p)
Suppose we answer a 3NN query with two mappers and one reducer for query point q1
Map phase
Each mapper reads its split
Simultaneously process the query point q2 and other queries as well
Reduce phase
Receives the query point as key and its neighbors as value, and emits them as they are

18. 4. kNN query processing Map
NN is among the VNs of p1, { p2, p3, p4, p5, p6, p7, p8 }
Map Output <q1>, <{p1, p2, p5}>
Reduce
Receives the query point as key and its neighbors as value, and emits them as they are
<q1,> <p1, p2 … pi>
<q2,> <p1, p2 … pi>
we answer a 3NN query with two mappers and one reducer for query point q1

19. Performance Evaluation
Competitors
Voronoi Diagram
R-tree
Factor
Index generation time
Query response time
Parameter
Node number
kNN : K
Datasets
BSN: all businesses in the entire U.S., containing approximately 1,300,000 data points.
RES: all restaurants in the entire U.S., containing approximately 450,000 data points.
Environment
Hadoop on Amazon EC
64bit Fedora 8 Linux Operating System with 15GB memory
4 virtual cores, and disks with 1,690 GB storage

20. Performance Evaluation
VD & R-Tree construction time
Construction of the R-tree index takes less time than that of VD
Because, the mapper generating PVDs execute expensive computations and there is only one reducer merging the PVDs in the cluster

21. Performance Evaluation
NN query on VD & R-Tree
Our experiments show that VD outperforms R-tree in query response time
This is because with VD, points can immediately find there VNs in the map phase

22. Performance Evaluation
RNN
The average response time slightly decreases with the increasing number of data points; hence the overall throughput increases
Single Node without parallelism
Multi Node with parallelism

23. Performance Evaluation
MaxRNN
As shown, parallelization reduces the response time at linear rate for both datasets
We observe a performance increase from %30 up to 50%
Effect of node number on MaxRNN

24. Performance Evaluation
kNN
This is because, for given number of data points P and given number of query point Q, (MapReduce based kNN search)MRK always outputs P X Q <key, value> pairs in the map phase independent of k
Due to the massive amount of data generated by the mapper, the response time is high

25. Conclusions
In this paper, we investigated the problem of handling and efficient querying of massive spatial data in a cloud architecture
We have presented a distributed Voronoi index and techniques to answer three types of geospatial queries
RNN, MaxRNN, kNN
Future Work : Supporting other types of queries
Bichromatic reverse nearest neighbor
Skyline
reverse k nearest neighbor