RELAXED REVERSE NEAREST NEIGHBORS QUERIES

Arif HidayatMuhammad Aamir CheemaDavid Taniar

Outline

Motivation Problem Definition Technique Experiment Conclusion

RkNN Query

R2NN of is

Nearest Neighbor Query (NN)– Find the object

closest to

Reverse Nearest Neighbor Query (RNN)– Find object which

consider as its NN

2 Nearest Neighbors (2NN) of are and

However, and are not Reverse 2 Nearest Neighbors (R2NN) of

Motivation

In R2NN query, is influenced by and

However, it is believed that is also influenced by

Normally, user will not mind to travel slightly farther to the next closest facility

In this case, RNN may miss influenced objects or retrieve non-influenced ones

u1f4

u2

f3

f2 f1

u330 Km

31 Km

Contribution

Complement RNN query with relative distance

New pruning techniques Extensive experimental

study

RRNN-Problem Definition

Given a set of users , a set of facilities , a query facility and a value of

an RRNN query returns every user for which where denotes the distance between and its nearest facility in

u1f4

u2

qf2 f1

u3

1 km

1.5 km

𝑥=1.5

RRNN-Pruning

Compute regions on which users cannot be RRNN of q

q

a

ce

P1P2

P3

P4 P5 P6

60ob

d

fu1

u2g

q

a

ce

b

d

fu1

u2g

New pruning rule

Six-regions and half-space pruning not applicable in RRNN problem

r

Point-based Pruning

𝒓=𝒙 .𝒅𝒊𝒔𝒕 (𝒒 ,𝒑)

𝒙𝟐−𝟏

Given a query , a value of and a point , the pruning circle of is a circle centered at with radius where

q p

is on the line passing through and

c

𝒅𝒊𝒔𝒕 (𝒒 ,𝒄 )= 𝒙𝟐 .𝒅𝒊𝒔𝒕 (𝒒 ,𝒑)𝒙𝟐−𝟏

Cp

and Proof:

u

Ɵ

Point-based Pruning

qc

b

p

u

Ɵ

Cpr

u’ The pruning rule is tight (proof is in the paper)

Given a query point , a user (outside ) and its nearest facility

–

– cannot be pruned by

MBR-based Pruning

Given a query , a value of , and a line representing a side of an MBR,

q

a ba' b'

CbCa

u

a user cannot be the RRNN of if it lies inside both of the pruning circles and ,

can be pruned if lies in

q

a b

Cb

Ca

cd

CcCd

Filtering

Prune users using defined pruning

regions

Straightforward approach: Store pruning regions in a

list Check user against entries

in the list O(n)

Our approach: Define interval for

each pruning region Build interval tree for

each partition Check users against

overlapped interval O(log n + k)

RRNN Candidates

Verification

Verify candidates: Circular boolean range query

on facility R*-Tree A user candidate is RRNN of if

no facility ,

More techniques: Computing

interval Trimming Cb

Ca

R

Rb

Ra

Rt

Ca

R

Ra

q

P1P2

P6

P3

P4P5

Ai

a

b

Ai.max

Ai.min

e1

e2

e3

Experiment Design

Implemented in C++

Run on Intel Core I5 2.3GHzx4 PC with 8GB memory running on Debian Linux

Users and facilities are indexed with R*-Tree

Each experiment runs 100 queriesParameter Values

Data size 2K, 200K, 2M, 20M

x factor 1.1, 1.3, 1.5, 2, 4

Real data set NA, LA, CA

13

Experiment Design

Synthetic and real data sets

175,812 points from North America (NA), 2.6 m points from Los Angeles (LA) and 25.6 m points from California (CA)

Data set: divided into 2 almost equal user and facility size

Improved range query

– For user and facility R*-Tree entry, , is immediately pruned if

– is not opened if

14

Experiment Results

No previous method for RRNN problem

We compare with naïve range query and improved algorithms

Experiment Results

Our algorithm is several orders of magnitude better than improved algorithm

Conclusion

An RRNN query relaxes the definition of influence using the relative distances between the users and the facilities

Our algorithm based on proposed effective pruning technique is several magnitude better than the competitors

Future works:

— Continuous RRNN

— Relaxed Reverse Top-