Reverse Furthest Neighbors in Spatial Databases

Bin Yao, Feifei Li, Piyush Kumar Presenter: Lian Liu Reverse Furthest Neighbors in Spatial Databases

OUTLINE • Introduction • Related Work • Algorithms • PFC (Progressive Furthest Cell) • CHFC (Convex Hull Furthest Cell) • Experiment • Discussion

INTRODUCTION • Assume you live at p1 (p2, p3), where would you prefer to build a chemical factory among q1~q3?

INTRODUCTION

INTRODUCTION • Let P={p1, p2, p3} • Q={q1, q2, q3} • fn(p1, Q)=q3 • fn(p2, Q)=q1 • fn(p3, Q)=q1 • BRFN(q1,Q,P)={p2, p3} • BRFN(q2,Q,P)={} • BRFN(q3,Q,P)={p1} Build the chemical factory here

INTRODUCTION • Problem: Given query point q, data set P (and Q), Compute MRFN(q, P) and BRFN(q, Q, P).

RELATED WORK • MBR • MBR (Minimum Bounding Rectangles) has 3 important distances to a point: • Min Distance • Max Distance • Minmax Distance

RELATED WORK • R-tree • R-tree is an index data structure. • In R-trees, points are grouped into MBRs, which are recursively grouped into MBRs in higher levels of the tree.

RELATED WORK • Range query • Range query: retrieves all points that locates within the query window. • R-tree based algorithms proves to be efficient to deal with range queries.

MRFN • How to compute the MRFN of a given query point? • BFS (Brute-Force Search) • PFC (Progressive Furthest Cell) • Main Idea: • Find the cell (region) in which all reverse furthest neighbors of the query point located • Perform a range query with the cell How to compute?

PFC • FVC (Furthest Voronoi Cell)

PFC • FVC Example: query point = q1 fvc(q1, P) fvc(q1, P)

PFC • PFC (Progressive Furthest Cell) Algorithm • Points and MBRs are stored in a priority queue L with their minmaxdist sorted in decreasing order. • Two vectors Vc and Vp are also maintained: • Vc: Furthest neighbor candidates • Vp: Disqualifying points

PFC • PFC – mechanism fvc(q)={} e is a point e∈fvc(q) c∩fvc(q)≠{} e∩fvc(q)={} e is an MBR e∩fvc(q)≠{} At last, we update fvc(q) using Vp and then filter points in Vc using fvc(q) c∩fvc(q)={}

PFC • Example: fvc(q) L={p1, R1} Vc={} Vp={} L={R1} Vc={p1} Vp={}

PFC • Example: fvc(q) fvc(q) L={p3} Vc={p1} Vp={p2} L={} Vc={p1, p3} Vp={p2}

PFC • Example: Finally, we use all points in Vp (i.e. p2) to update fvc(q). Then, we perform a range query using the updated fvc(q). The result is {p3}。 fvc(q) MRFN(q)={p3}

PFC • Efficiency of PFC • PFC makes fvc(q) quickly shrink. If the query point does not have any reverse furthest neighbors, Φwill quickly be reported. • However, it is still not efficient enough. • Improvement: CHFC algorithm.

CHFC • Convex Hull • The Convex Hull of a set of points P is the smallest convex polygon that fully contains P. • Denoted as CP.

CHFC • Lemma: Given a point set P and its convex hull Cp, for a point q, let p*=fn(q, P), then p*∈CP. • fvc(p, P)=fvc(p, CP)

CHFC • CHFC (Convex Hull Furthest Cell) • Given a set of points P and a query point p:

CHFC • BRFN • BRFN (Bichromatic Reverse Furthest Neighbor) can be found in the same way as MRFN. • The only one difference is, we compute fvc(q, Q, P) will Q, can perform range query in P.

CHFC • Efficiency of CHFC: • For most (but not all) cases, |CP| << |P|. That is, the number of points considered are likely to be greatly reduced. • Difficulty: How to compute and updateCP when |P| is very large and even |CP| cannot fit into memory.

CHFC • Computing Convex Hull • Convex hulls can be found in either a distance-first or a depth-first manner. • Distance-first approach is optimal in the number of page accesses, and the complexity is O(nlogn). • Depth-first algorithms can run in O(n) time for worst case, but not optimal in disk accessing.

CHFC • Updating Convex Hull • Inserting new points: • Lemma: P is a point set. • If point q is contained by CP,CP∪{q} =CP • Otherwise, • CP∪{q} =CCp∪{q}

CHFC • Updating Convex Hull • Deleting points: • Points or MBRs with the largest perpendicular distance to plpr are added into CP first, untilthere is no points outside the current convex hull.

CHFC • External Convex Hull Computing • Existing algorithms can found 2-Dimensional convex hulls with I/Os. • However, when convex hulls are still too large to fit into memory, we use Dudley’s approximate convex hull.

EXPERIMENT • CPU time & number of IOs

DISCUSSION • Thank You! • Questions?

Reverse Furthest Neighbors in Spatial Databases