Outline Spatial Index R Tree NN query Nave solution Nearest - - PDF document

9/3/2009 1

Nearest Neighbor Queries

Ling Hu lingh@usc.edu Nick Roussopoulos, Stephen Kelly and Frédéric Vincent

Outline

• Spatial Index – R‐Tree
• NN query – Naïve solution
• A better solution – Branch‐and‐bound
• Can we do better?
• Experiment Results

2

2 3 1 4 5

R-

• Tree

Tree

6 7 8 9 10 11 12

3

2 3 1 4 5

R-

• Tree

Tree

E F 2 3 1 4 5 6 7 8 9 10 11 12 G H

4

6 7 8 9 10 11 12 2 3 1 4 5 E F

R-

• Tree

Tree

B 2 3 1 4 5 6 7 8 9 10 11 12 C G H

5

6 7 8 9 10 11 12 6 2 3 1 4 5 A B E F

R-

• Tree

Tree

B C E F G H A: B: C:

Pointer MBR

One entry:

6 2 3 1 4 5 6 7 8 9 10 11 12 C G H

B: 4 5 6 1 2 3 10 11 12 7 8 9 E F G H 6

6 7 8 9 10 11 12

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Outline

• Spatial Index – R‐Tree
• NN query – Naïve solution
• A better solution – Branch‐and‐bound
• Can we do better?
• Experiment Results

7

Nearest Neighbor Search

• Retrieve the nearest neighbor of query point Q
• Simple Strategy:

– convert the nearest neighbor search to range search. – Guess a range around Q that contains at least one object say O

• if the current guess does not include any answers, increase range size

til bj t f d

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until an object found.

– Compute distance d’ between Q and O – re‐execute the range query with the distance d’ around Q. – Compute distance of Q from each retrieved object. The object at minimum distance is the nearest neighbor!!!

Naïve Approach

A B E F

B C E F G H A: B: C: 4 5 6 1 2 3 10 11 12 E F G

2 3 1 4 5 C G H

Query Point Q

Issues: how to guess range? The retrieval may be sub‐optimal if incorrect range guessed. Would be a problem in high dimensional spaces.

7 8 9 H

6 7 5 8 9 10 11 12

Outline

• Spatial Index – R‐Tree
• NN query – Naïve solution
• A better solution – Branch‐and‐bound
• Can we do better?
• Experiment Results

10

MINDIST Property

• MINDIST is a lower bound of any k-NN distance

(s1, s2) (t1, t2) (p1, p2) (p1, p2) (p1, p2) (p1, p2) (p1, p2)

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A Better Strategy for KNN search

• A sorted priority queue based on MINDIST;
• Nodes traversed in order;
• Stops when there is an object at the top of the

queue; (1‐NN found)

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• k‐NN can be computed incrementally;

I/O optimal

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Priority Queue

B C E F G H A: B: C: 4 5 6 1 2 3 10 11 12 7 8 9 E F G H

2 3 1 4 5 A B E F

A B C E F C

6 7 5 8 9 11 C G H

Query Point Q

C 5 6 4 F H 5 G 6 4 F 7 5 8 9 G 6 4 F

1NN

13

10 12

Outline

• Spatial Index – R‐Tree
• NN query – Naïve solution
• A better solution – Branch‐and‐bound
• Can we do better?
• Experiment Results

14

MBR Face Property

• MBR is an n‐dimensional Minimal Bounding

Rectangle used in R trees, which is the minimal bounding n‐dimensional rectangle bounds its corresponding objects corresponding objects.

• MBR face property: Every face of any MBR

contains at least one point of some object in the database.

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MBR Face Property – 2D

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MBR Face Property – 3D

Rectangle R 17

Improving the KNN Algorithm

• While the MinDist based algorithm is I/O
• ptimal, its performance may be further

improved by pruning nodes from the priority queue

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queue.

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Properties of MINMAXDIST

• MINMAXDIST(P,R) is the minimum over all dimensions

distances from P to the furthest point of the closest face of R.

• MINMAXDIST is the smallest possible upper bound of

distances from the point P to the rectangle R.

• MINMAXDIST guarantees there is an object within the R

at a distance to P less than or equal to it.

• MINMAXDIST is an upper bound of the 1-NN

distance

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MINDIST & MINMAXDIST

MINDIST(P,R) <= NN(P) <= MINMAXDIST(P,R)

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MinDist & MinMaxDist – 3D

Query Point Q Rectangle R MinDist(Q,R) MinMaxDist(Q,R) 21

Pruning 1

• Downward pruning: An MBR R is discarded

R R’

If there exists another R’ such that MINDIST(P,R)> MINMAXDIST(P,R’)

R MINDIST P MINMAXDIST

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Pruning 2

• Downward pruning: An object O is discarded

O R’

If there exists an R such that Actual_dist(P,O) > MINIMAXDIST(P,R)

O R Actual‐dist P MINMAXDIST

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Pruning 3

• Upward pruning: An MBR R is discarded

R

If an object O is found such that MINDIST(P,R) > Actual_dist(P,O)

R O MINDIST P Actual_dist

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MINDIST vs MINMAXDIST Ordering

• MINDIST: optimistic
• MINMAXDIST: pessimistic

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• Example: MINDIST ordering finds the 1‐NN first

MINDIST vs MINMAXDIST Ordering

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• Example: MINMAXDIST ordering finds the 1‐NN first

Generalize to k‐NN

• Keep a sorted buffer of at most k current nearest

neighbors

• Pruning is done according to the distance of the

furthest nearest neighbor in this buffer

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• Example:

R The k‐th object in the buffer MINDIST P Actual_dist

Outline

• Spatial Index – R‐Tree
• NN Query – Intuitive Solutions
• Optimized NN Query – branch‐and‐bound
• Experiment Results

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Key Insights

• # of pages accessed grows when k grews;
• The denser the dataset, the more page access;
• MinDist v.s. MinMaxDist: same in shape, but

i i h /O MinMaxDist has more I/O cost;

• In Dense area, MinMaxDist is bad;

Thanks & Questions ?