Bin Yao, Feifei Li, Piyush KumarPresenter: Lian Liu

IntroductionRelated WorkAlgorithms

- PFC (Progressive Furthest Cell)- CHFC (Convex Hull Furthest Cell)

ExperimentDiscussion

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

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

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

MBR MBR (Minimum

Bounding Rectangles) has 3 important distances to a point:

Min Distance Max Distance Minmax Distance

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.

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.

How to compute the MRFN of a given query point?

BFS (Brute-Force Search)PFC (Progressive Furthest Cell)Main Idea:

1. Find the cell (region) in which all reverse furthest neighbors of the query point located

2. Perform a range query with the cell

How to compute?

FVC (Furthest Voronoi Cell)

FVC Example: query point = q1

fvc(q1, P)

fvc(q1, P)

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 – mechanism

e is a point

e is an MBR

fvc(q)={}

e∈fvc(q)

e∩ fvc(q)={}

e∩ fvc(q)≠{}

c∩ fvc(q)≠ {}

c∩ fvc(q)={}At last, we update fvc(q) using Vp and then filter points in Vc using fvc(q)

Example:

L={p1, R1}Vc={}Vp={}

L={R1}Vc={p1}Vp={}

fvc(q)

Example:

L={p3}Vc={p1}Vp={p2}

L={}Vc={p1, p3}Vp={p2}

fvc(q)

fvc(q)

Example:

MRFN(q)={p3}

fvc(q)

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}。

Efficiency of PFCPFC makes fvc(q) quickly shrink. If

the query point does not have any reverse furthest neighbors, Φ will quickly be reported.

However, it is still not efficient enough.

Improvement: CHFC algorithm.

Convex Hull

The Convex Hull of a set of points P is the smallest convex polygon that fully contains P.

Denoted as CP.

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 (Convex Hull Furthest Cell) Given a set of points P and a query

point p:

BRFNBRFN (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.

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 update CP when |P| is very large and even |CP| cannot fit into memory.

Computing Convex HullConvex 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.

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}

Updating Convex Hull Deleting points: Points or MBRs with

the largest perpendicular distance to plpr are added into CP first, until there is no points outside the current convex hull.

External Convex Hull ComputingExisting 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.

( log )M

m mOB MB

CPU time & number of IOs

Thank You!Questions?