1. Aggregate-Max Nearest Neighbor Searching in the Plane
Haitao Wang
Utah State University
CCCG 2013

2. Nearest neighbor searching
Input: a set P of n red points
Query: a query blue point q
report the nearest red point to q

3. Aggregate/Group nearest neighbor searching
Input: a set of n red points
Query: a set Q of m query blue points
report the “nearest” red point to Q

4. What is the “nearest neighbor” to Q?
For any red point p in P,
the aggregate distance from p to Q: the farthest distance from p to the points of Q
that’s why it is called Aggregate-max
The nearest neighbor of Q: the red point whose aggregate distance to Q is the minimum p

5. An application A group of people want to meet together
given a set of candidate meeting locations
find a location that minimizes the longest distance to all people

6. Previous work Heuristic algorithms given in the database area
Minimum bounding method (Papadias et al, TODS 2005)
R-trees (Li et al, TKDE 2011)
Ω(n+m) time in the worst case
(1+ε)-approximation result (Li et al, TKDE 2011)
Aggregate-sum version
3-approximation (Li et al, SIGMOD 2011)
(1+ε)-approximation (Agarwal et al, PODS 2012)

7. Our results Preprocessing: O(n log n) time and O(n log log n) space
Query: O(mn1/2 logO(1) n) time
Or,
Preprocessing: O(n2+ε) time and space for any ϵ > 0
Query: O(m log n) time
n: the number of input points
m: the number of query points
m<<n

8. Our results – the L1 distances
Preprocessing: O(n log n) time and O(n) space
Query: O(m+logn) time
Top-k queries: Report the k nearest neighbors to Q
Query: O(m+klogn) time
vertical distance e
horizontal distance
The L1 distance of e = the horizontal distance + the vertical distance

9. A simple observation For any red point p,
how to determine the farthest point q of Q to p?
An observation: q is the point whose farthest Voronoi region contains p q p

10. A query algorithm
For each blue point q that defines a farthest Voronoi region FVR(q)
find the nearest neighbor of q in FVR(q)
Return the shortest distance q
Difficulty: How to find the nearest neighbor of q in FVR(q)?
FVR(q)

11. A solution: maintain Voronoi diagrams
Obtain the Voronoi diagram of the red points in FVR(q)
Determine the Voronoi region that contains q q
FVR(q)

12. The new problem: answering Voronoi diagram queries
For a set of points in the plane,
query: a triangle
return (implicitly) the Voronoi diagram of the points in the triangle
Our approach: Using the simplex range searching data structures (Matousek 92’, 93’, Chan 12’)
For each canonical subset
of points, maintain its
Voronoi diagram explicitly

13. The L1 version For any red point p
easy to determine the farthest point of Q to p
one of the four extreme points
The farthest Voronoi diagram of Q is determined by the four extreme points p

14. The key problem: determine the nearest neighbor of q in FVR(q)?
The solution for L2: maintaining Voronoi diagrams
A better solution for L1: using segment dragging queries
FVR(q) q

15. A key observation on the shape of FVR(q): only three types
Type-A
bounded by two vertical half-lines and a segment of slope -1 in between
On the right of the vertical line through q
b is on the horizontal line through q
assume q is the extreme point
along the southwest direction
FVR(q) a b q

16. The shape of FVR(q): three types
Type-B
bounded by a vertical half-line, a horizontal half-line, and a segment of slope -1 in between
in the first quadrant with respect to q
FVR(q) a b q

17. The shape of FVR(q): three types
Type-C
symmetric to the Type-A a q
FVR(q) b

18. The shape of FVR(q): three typesb q b b q q
FVR(q) A B C

19. Determining nearest neighbor in type-A regions
Partition FVR(q) into three subregions
Find the nearest neighbor in each subregion a b q

20. The shape of FVR(q): three typesb q b b q q A B C

21. Segment dragging queries
Preprocessing: O(nlog n) time and O(n) space
Query: O(log n) time
out-of-corner queries: Mitchell 92’
parallel-track queries: Chazelle 88’
out-of-corner query
parallel-track query

22. The query algorithm –a summary
Given a query set Q:
Compute the farthest Voronoi diagram
O(m) time
For each extreme point q,
determine the nearest point of P in FVR(q)
O(log n) time by segment dragging queries
Total query time: O(m + log n)

23. Extended to the top-k queries
After the nearest point is found in a subregion, keep searching the next nearest point by segment dragging queries
a tempting approach: keep using another out-of-corner query
does not work!!
the interior of
the purple triangle
should not contain
any point a q

24. Extended to the top-k queries
Our approach: partition into three smaller regions:
an out-of-corner query for the green region
parallel-track queries for the blue regions
Total query time: O(m+klog n) a q

25. Thank You