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

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

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.) 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; CHFC(Query q; R-tree T (on P)) // compute only once

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: insertion

Dynamically Maintaining CH: deletion The qhull algorithm

Dynamically Maintaining CH Adapt qhull to R-tree

Outline MRFN  Progressive Furthest Cell Algorithm  Convex Hull Furthest Cell Algorithm  Dynamically updating to dataset 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

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