Bin Yao, Feifei Li, Piyush Kumar Presenter: Lian Liu

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

 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)

 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)

 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 PFC  PFC 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 C P.

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

 CHFC (Convex Hull Furthest Cell)  Given a set of points P and a query point p: Compute CP ∪ {p} Compute fvc(p, P) using CP ∪ {p} Perform a range query with fvc(p, P)

 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.

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

 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.

 Updating Convex Hull  Inserting new points:  Lemma: P is a point set.  If point q is contained by C P,C P ∪ {q} =C P  Otherwise,  C P ∪ {q} =C Cp ∪ {q}

 Updating Convex Hull  Deleting points:  Points or MBRs with the largest perpendicular distance to p l p r are added into C P first, until there is no points outside the current convex hull.

 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.

 CPU time & number of IOs

 Thank You!  Questions?