Faster Con struction of Plan ar 2-Cen ters David Eppstein Dept. In - - PDF document

Faster Con struction of Plan ar 2-Cen ters

David Eppstein

• Dept. In form ation an d Com puter Scien ce

Un iv. of Californ ia, Irvin e http:/ / www.ics.uci.edu/ ∼eppstein /

1

The problem

Cover n poin ts w/ two m in im um -radius circles

2

It’s safe to assum e:

• Both circles are the sam e size
• On e circle has three tan gen t poin ts,
• r is diam eter circle of two poin ts
• (Other circle is less con strain ed)

3

History

Agarwal an d Sharir, SODA 1991: O(n2 log3 n) Eppstein , FOCS 1991: O(n2 log2 n log log n) ran dom ized Katz an d Sharir, SCG 1993: O(n2 log3 n) Jarom czyk an d Kowaluk, SCG 1994: O(n2 log n) Sharir, SCG 1996: O(n log9 n) New result: O(n log2 n) ran dom ized

4

Two Cases (based on circle separation )

d < (2-ε)r d > εr

5

Overlappin g Case → Matrix

Fin d poin t in in tersection of disks (by testin g O(1) can didates) Look for partition by two rays Form m atrix represen tin g possible partition s: Row in dex = position of upper ray Colum n in dex = position of lower ray

6

Overlappin g Case: Quadtree Search

(based on m atrix selection of Frederickson an d John son )

Represen t poten tial partition s as set of n

k ×n k squares

(in itially, on e square for whole m atrix) For O(log n) stages: Subdivide each square in to four Prun e back down to O(k) squares

7

Overlappin g Case: Prun in g

Too m any squares → m any in terior corn ers Pick a corn er x ran dom ly an d evaluate correspon din g circum radii Com pare x again st all other in terior corn ers If x better than y, elim in ate on e of y’s squares (above an d to right of y, left circle on ly gets larger; below an d to left, right circle on ly gets larger.) Expect 50% of in terior corn ers to becom e exterior Repeat O(1) tim es un til few in terior corn ers left

8

Overlappin g Case: An alysis

O(log n) stages. On ly slow part: com pare corn ers to ran dom choice (= test circum radii from correspon din g partition s) Con n ect in to path, use Hershberger-Suri offlin e circum radius decision algorithm : O(n log n) per stage Use exact circum radius data structure: O(k logc n) per stage Com bin e both m ethods: Exact circum radius when k = O(n/logcn) Offlin e alg for rem ain in g O(log log n) stages Total: O(n log n log log n)

9

Separated Case: Cut Lin e

Fin d halfspace con tain in g on ly poin ts of con strain ed disk, in cludin g at least on e tan gen t poin t (by testin g O(1) can didates)

10

Separated Case: Main Idea

Param etric search Let A1 be decision algorithm (com pare given radius again st optim um ) Let A2 be any algorithm that is discon tin uous at the

• ptim al value (e.g. the decision algorithm again )

Sim ulate A2(r∗) by replacin g each com parison in A2 with a call to A1. Because of discon tin uity, calls m ust in clude A1(r∗)

11

Separated Case: Efficien cy Con sideration s

To m ake param etric search efficien t:

• Sim ulate a parallel algorithm
• Batch calls to decision alg usin g bin ary search
• Rem ove as m uch as possible from sim ulation

– preproc. n ot depen din g on param eter – postproc. after already discon tin uous

12

Separated Case: Decision Algorithm

Swin g circle aroun d circular hull of pts in half- plan e, testin g circum radius of rem ain in g poin ts Total: O(n log n)

13

Separated Case: Sim ulated Algorithm

• Com pute circular hull of poin ts
• Sweep circle aroun d hull
• Fin d sequen ce of poin t sets swept by circle

Already discon tin uous – don ’t have to apply offlin e decision algorithm to sequen ce

(Proof: 4 cases. Is optim al circle supported by two or three tan gen t poin ts, an d are on e or two of them on circular hull?)

14

Separated Case: Fast Circular Hull

Circular hull arcs correspon d to certain edges of the farthest poin t Voron oi diagram Com pute Voron oi diagram (preproc.) Test which Voron oi edges give hull arcs (O(n) processors, O(1) tim e) Con n ect the dots (in depen den t of param .)

15

Separated Case: Swin gin g the Circle

For each poin t p n ot in the hull fin d hull vertex v the circle is pivotin g on when it crosses p (bin ary search) For each hull pivot v sort the associated poin ts by sweep tim e O(n) processors, O(log n) tim e but suitable for Cole’s speed-up

16

Separated Case: An alysis

Preprocessin g: Voron oi diagram O(n log n) Sim ulate: Fin din g hull arcs O(n) bin ary searches On e sortin g algorithm Total: O(n log2 n)

17

Open Problem s

Deran dom ize

(on ly uses O(log n log log n) ran dom bits!)

Im prove tim e boun d

(on ly slow case: n early tan gen t circles)

Make sim ple en ough to be practical

(m ost com plicated part: param etric sort)

18