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### Reverse Furthest Neighbors in Spatial Databases

Outline

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.

MRFN and BRFN

- MRFN for q and P:
- BRFN for a point q in Q and P are:

Outline

- MRFN
- Progressive Furthest Cell Algorithm
- Convex Hull Furthest Cell Algorithm
- Dynamically 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.

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

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

- 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 ;

Outline

- MRFN
- Progressive Furthest Cell Algorithm
- Convex Hull Furthest Cell Algorithm
- Dynamically 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.)

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

// compute only once

- 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;

Outline

- MRFN
- Progressive Furthest Cell Algorithm
- Convex Hull Furthest Cell Algorithm
- Dynamically 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: deletion

The qhull algorithm

Dynamically Maintaining CH

Adapt qhull to R-tree

- MRFN
- Progressive Furthest Cell Algorithm
- Convex Hull Furthest Cell Algorithm
- Dynamically 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;

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)
- Measurement
- Computation time
- Number of IOs
- Average of 1000 queries

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.

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