Reverse Furthest Neighbors in Spatial Databases

Bin Yao, Feifei Li, Piyush Kumar

Florida State University, USA

A Novel Query Type Reverse Furthest Neighbors (RFN)

Given a point q and a data set P, find the set of points in P that take q as their furthest neighbor

Two versions: Monochromatic Reverse Furthest Neighbors (MRFN) Bichromatic Reverse Furthest Neighbors (BRFN)

Motivation and Related works

Motivation: inspired by RNN Reverse Nearest Neighbor

Set of points taking query point as their NN.Monochromatic & Bichromatic RNN

Many applications that are behind the studies of the RNN have the corresponding “furthest” versions.

MRFN Application P: a set of sites of interest in a region For any site, it could find the sites that take itself

as their furthest neighbors This has an implication that visitors to the RFN of

a site are unlikely to visit this site because of the long distance.

Ideally, it should put more efforts in advertising itself in those sites.

BRFN Application P: a set of customers Q: a set of business competitors offering similar

products A distance measure reflecting the rating of

customer(p) to competitor(q)’s product. A larger distance indicates a lower preference. For any competitor in Q, an interesting query is to

discover the customers that dislike his product the most among all competing products in the market.

BRFN Example : customer : product

876531 ,,,,: of RFN pppppq

1p

2p

1q

4p

3p

6p

5p8p

2q

3q

7p

4213 ,,: of RFN pppq : of RFN 2q

MRFN and BRFN

MRFN for q and P:

BRFN for a point q in Q and P are:

q),fn(),,( QpPppPQqBRFN

q)}{,fn(),( qPpPppPqMRFN

Outline

MRFNProgressive Furthest Cell AlgorithmConvex Hull Furthest Cell AlgorithmDynamically updating to dataset

BRFN

MRFN: Progressive Furthest Cell Algorithm (first algorithm) Lemma: Any point from the furthest Voronoi cell(fvc) of p

takes p as its furthest neighbor among all points in P.

1p

3p2p

)( 1pfvc

5p4p

Progressive Furthest Cell Algorithm (PFC)PFC(Query q; R-tree T)

Initialize two empty vectors and ; priority queue L with T’s root node; fvc(q)=S;

While L is not empty do Pop the head entry e of L If e is a point then, update the fvc(q)

If fvc(q) is empty, return; If e is in fvc(q), then Push e into ;

else If e fvc(q) is empty then push e to ; Else for every child u of node e

If u fvc(q) is empty, insert u into ; Else insert u into L ;

CV PV

CV

PV

PV Update fvc(q) using points contained by entries in ; Filter points in using fvc(q);CV

PV

1p

3p2p

)( 1pfvc

4p

)( 1pfvc

Outline

MRFNProgressive Furthest Cell AlgorithmConvex Hull Furthest Cell AlgorithmDynamically updating to dataset

BRFN

MRFN: Convex Hull Furthest Cell Algorithm(second algorithm)

Lemma: the furthest point for p from P is always a vertex of the convex hull of P. (i.e., only vertices of CH have RFN.)

Find the convex hull of P; if , then return empty; else

Compute using ; Set fvc(q,P*) equal to fvc(q, ); Execute a range query using fvc(q,P*) on T;

PC

PCq

*PC }{qCP

*PC

CHFC(Query q; R-tree T (on P))

// compute only once

Outline

MRFNProgressive Furthest Cell AlgorithmConvex Hull Furthest Cell AlgorithmDynamically updating to dataset

BRFN

Dynamically updating to dataset

PFC: update R-tree CHFC:

update R-tree& re-compute CH (expensive)Qhull algorithm

Dynamically Maintaining CH: insertion

1p4p

3p2p

6p

5p

7p}{}{ 77 pCpP P

CC

Dynamically Maintaining CH: deletion

2p

8p

1p9p

3p

4p5p

6p

7p

The qhull algorithm

Dynamically Maintaining CH

2p

3p

2e

3e

1e

1p

minVdist

maxVdist

Adapt qhull to R-tree

Outline

MRFNProgressive Furthest Cell AlgorithmConvex Hull Furthest Cell AlgorithmDynamically updating to dataset

BRFN

BRFN

After resolving all the difficulties for the MRFN problem, solving the BRFN problem becomes almost immediate.

Observations: all points in P that are contained by fvc(q,Q) will have

q as their furthest neighbor. Only the vertexes of the convex hull have fvc.

BRFN algorithm

BRFN(Query q, Q; R-tree T) Compute the convex hull of Q; If then return empty; Else

Compute fvc(q, );Execute a range query using fvc(q, ) on T;

QC

QCq

QC

QC

BRFN: Disk-Resident Query Group

Limitation: query group size may not fit in memory

Solution: Approximate convex hull of Q (Dudley’s approximation)

Experiment Setup

Dataset: Real dataset (Map: USA, CA, SF)Synthetic dataset (UN, CB, R-Cluster)

MeasurementComputation time Number of IOsAverage of 1000 queries

MRFN algorithm

CPU computation Number of IOs

BRFN algorithms

CPU: vary A, Q=1000 IOs: vary A, Q=1000

Scalability of various algorithms

MRFN number of IOs BRFN number of IOs

Conclusion

Introduced a novel query (RFN) for spatial databases.

Presented R-tree based algorithms for both versions of RFN that feature excellent pruning capability.

Conducted a comprehensive experimental evaluation.

Thank you!Questions?

Datasets: San Francisco

Datasets: California

Datasets: North America

Datasets : uncorrelated uniform

Datasets : correlated bivariate

Datasets : random clusters