1. Nearest Neighbor Queries using R-trees
Based on notes from NTUA

2. Nearest Neighbor Search
Find the object nearest to a query point q
E.g., find the gas station nearest to the red point.
k nearest neighbors: Find the k objects nearest to q
E.g., 1 NN = {h}, 2NN = {h, a}, 3NN = {h, a, i}

3. R-trees - NN search A B C D E F G H I J P1 P2 P3 P4 q

4. R-trees - NN search P1 P3 I C A G F H B J E P4 q P2 D

5. K-NN search
Keep the sorted buffer of at most k current nearest neighbors
Pruning is done using the k-th distance

6. NN search: Best-First Global order [HjaltasonSamet99]
Maintain distance to all entries in a common Priority Queue
Use only MINDIST
Repeat
Inspect the next MBR in the list
Add the children to the list and reorder
Until all remaining MBRs can be pruned

7. Nearest Neighbor Search (NN) with R-Trees
Best-first (BF) algorihm:
y axis
Root 10 E 7 E E E E 1 2 3 1 e f E 2 1 2 8 8 E 8 E d E g E 2 5 1 6 h i E E E E E E E E 6 9 4 5 6 7 8 9
query point
contents 5 5 9 13 2 17 4 E
omitted b 4 a
search
region 2 c a b c d e f h g i E 3 5 13 18 13 13 10 2 13 10
x axis E 2 4 6 8 10 E E 4 5 8
Action
Heap
Result
Visit Root E E E
{empty} 1 1 2 2 3 8
follow E E E E E 1 E
{empty} 2 2 4 5 5 5 3 8 6 9
follow E E E E E E E E 2 8 2 4 5 5 5 3 8 6 9 7 13 9 17
{empty}
follow E E E E E E E 8 4 5 5 5 3 8 6 9 7 13 9 17
{(h, 2 )} g E E E i E 4 5 5 5 3 8 E 6 9 10 7 13 13
Report h and terminate

8. HS algorithm Initialize PQ (priority queue) InsertQueue(PQ, Root)
While not IsEmpty(PQ)
R= Dequeue(PQ)
If R is an object
Report R and exit (done!)
If R is a leaf page node
For each O in R, compute the Actual-Dists, InsertQueue(PQ, O)
If R is an index node
For each MBR C, compute MINDIST, insert into PQ

9. Analysis - Overview
Analyze the situation after finding the k nearest neighbors
Let o be the kth nearest neighbor of q, let r be the distance of o from q.
The region within distance r from q is called the search region.
Since we assume that q is a point, the search region is a circle (or a hypersphere in higher dimensions) with radius r.
All objects inside the search region have already been reported by the algorithm (as the next nearest object), while all nodes intersecting the search region have been examined and their contents put on the priority queue.
The contents of the priority queue are contained in nodes intersecting the boundary of the search region.
The algorithm does not access any nodes or objects that lie entirely outside the search region (i.e., that are farther from q than o is).

10. R-trees - NN search: Branch and Bound
Based on: [Roussopoulos+, sigmod95]:
At each node, priority queue, with promising MBRs, and their best and worst-case distance
main idea: Every face of any MBR contains at least one point of an actual spatial object!

11. MBR face property
MBR is a d-dimensional rectangle, which is the minimal rectangle that fully encloses (bounds) an object (or a set of objects)
MBR f.p.: Every face of the MBR contains at least one point of some object in the database

12. Search improvement Visit an MBR (node) only when necessary
How to do pruning? Using MINDIST and MINMAXDIST

13. MINDIST
MINDIST(P, R) is the minimum distance between a point P and a rectangle R
If the point is inside R, then MINDIST=0
If P is outside of R, MINDIST is the distance of P to the closest point of R (one point of the perimeter)

14. MINDIST computation R u ri = li if pi < li p = ui if pi > ui
MINDIST(p,R) is the minimum distance between p and R with corner points l and u
the closest point in R is at least this distance away
u=(u1, u2, …, ud) R u
ri = li if pi < li
= ui if pi > ui
= pi otherwise p p
MINDIST = 0 l p
l=(l1, l2, …, ld)

15. MINMAXDIST
MINMAXDIST(P,R): for each dimension, find the closest face, compute the distance to the furthest point on this face and take the minimum of all these (d) distances
MINMAXDIST(P,R) is the smallest possible upper bound of distances from P to R
MINMAXDIST guarantees that there is at least one object in R with a distance to P smaller or equal to it.

16. MINDIST and MINMAXDIST
MINDIST(P, R) <= NN(P) <=MINMAXDIST(P,R)
MINMAXDIST R1 R4 R3
MINDIST
MINDIST
MINMAXDIST
MINDIST
MINMAXDIST R2

17. Pruning in NN search
Downward pruning: An MBR R is discarded if there exists another R’ s.t. MINDIST(P,R)>MINMAXDIST(P,R’)
Downward pruning: An object O is discarded if there exists an R s.t. the Actual-Dist(P,O) > MINMAXDIST(P,R)
Upward pruning: An MBR R is discarded if an object O is found s.t. the MINDIST(P,R) > Actual-Dist(P,O)

18. Pruning 1 example
Downward pruning: An MBR R is discarded if there exists another R’ s.t. MINDIST(P,R)>MINMAXDIST(P,R’) R R’
MINDIST
MINMAXDIST

19. Pruning 2 example
Downward pruning: An object O is discarded if there exists an R s.t. the Actual-Dist(P,O) > MINMAXDIST(P,R) R
Actual-Dist O
MINMAXDIST

20. Pruning 3 example
Upward pruning: An MBR R is discarded if an object O is found s.t. the MINDIST(P,R) > Actual-Dist(P,O) R
MINDIST
Actual-Dist O

21. Ordering Distance
MINDIST is an optimistic distance where MINMAXDIST is a pessimistic one.
MINDIST P
MINMAXDIST

22. NN-search Algorithm
Initialize the nearest distance as infinite distance
Traverse the tree depth-first starting from the root. At each Index node, sort all MBRs using an ordering metric and put them in an Active Branch List (ABL).
Apply pruning rules 1 and 2 to ABL
Visit the MBRs from the ABL following the order until it is empty
If Leaf node, compute actual distances, compare with the best NN so far, update if necessary.
At the return from the recursion, use pruning rule 3
When the ABL is empty, the NN search returns.

23. Best-First vs Branch and Bound
Best-First is the “optimal” algorithm in the sense that it visits all the necessary nodes and nothing more!
But needs to store a large Priority Queue in main memory. If PQ becomes large, we have thrashing…
BB uses small Lists for each node. Also uses MINMAXDIST to prune some entries

24. References: Branch and Bound NN search:
N. Roussopoulos, S. Kelley, F. Vincent: Nearest Neighbor Queries. SIGMOD Conference 1995: 71-79
Best First NN search:
G.R. Hjaltason, H. Samet: Distance Browsing in Spatial Databases. ACM Trans. Database Syst. 24(2): (1999)

25. Some follow up works
D. Park, H. Kim: An Enhanced Technique for k-Nearest Neighbor Queries with Non-Spatial Selection Predicates. In Multimedia Tools and Applications archive, Volume 19 , Issue 1 (January 2003), Pages: 79 – 103
Ian De Felipe, Vagelis Hristidis, Naphtali Rishe. Keyword Search on Spatial Databases. IEEE ICDE 2008