Convex Hull Algorithms. Convex hull: basic facts

Transcription

1 CG Leture D Conve Hull Algorithms Bsi fts Algorithms: Nïve, Gift wrpping, Grhm sn, Quik hull, Divide-nd-onquer Lower ound 3D Bsi fts Algorithms: Gift wrpping, Divide nd onquer, inrementl Conve hulls in higher dimensions Conve hull: si fts Prolem: give set of n points P in the plne, ompute its onve hull CH(P). Bsi fts: CH(P) is onve polygon with ompleity O(n). Verties of CH(P) re suset of the input points P. p 3 p 9 p p4 p 8 p 3 p 5 p 7 p0 p 6 p p p Input: p,, p 3 CH verties: p,p,p,p,p 3,p 9,p 3

2 Algorithm Nive lgorithm For eh pir of points onstrut its onneting segment nd supporting line. Find ll the segments whose supporting lines divide the plne into two hlves, suh tht one hlf plne ontins ll the other points. Construt the onve hull out of these segments. Time ompleity All pirs: n nn ( ) O( ) = O( ) = O( n ) Chek ll points for eh pir: O(n) eh, O(n 3 ) totl. No Yes 3 Tringle Are Are = ( P P ) ( P P ) 3 = P P P P sinα 3 y y = y y = 3 3 y y y 3 3 α P (,y ) P 3 ( 3,y 3 ) P (,y ) The determinnt is twie the re of the tringle whose verties re the rows of the mtri. 4

3 Orienttion nd point lssifition y (,y ) Are = y + 3 y3 (,y ) ( 3,y 3 ) The re sign indites the orienttion of the points. Positive re ounterlokwise orienttion left turn. Negtive re lokwise orienttion right turn. This test n e used to determine whether given point is ove or elow given line Tringle Are Are = ( P P) ( P3 P ) = P P P3 P sinα y y = 3 y3 y y = y y P (,y ) P 3 ( 3,y 3 ) P (,y ) The determinnt is twie the re of the tringle whose verties re the rows of the mtri. α 6 3

4 Orienttion nd point lssifition y (,y ) Are = y + 3 y3 (,y ) ( 3,y 3 ) The re sign indites the orienttion of the points. Positive re ounterlokwise orienttion left turn. Negtive re lokwise orienttion right turn. This test n e used to determine whether given point is ove or elow given line 7 Possile prolems Degenerte ses, e.g., 3 olliner points, my hrm the orretness of the lgorithm. Segments AB, BC nd AC will ll e inluded in the onve hull. A B C Numeril prolems We might onlude tht none of the three segments (or wrong pir of them) elongs to the onve hull. Question: How is olinerity deteted? 8 4

5 Generl position ssumption When designing geometri lgorithm, we first mke some simplifying ssumptions, e.g.: No three olliner points; No two points with the sme or y oordinte; Other onfigurtions: no three points on irle, Lter, we onsider the generl se: Behvior of lgorithm to degenerte ses? Will the orretness e preserved? Will the running time remin the sme? Modify/etend lgorithm to hndle degeneries 9 Algorithm: Gift wrpping lgorithm. Find the lowest point p nd its hull edge e. For eh remining point p i (i > ) do Compute the CCW ngle α i from the previous hull edge Let p j e the point with the smllest α i Mke (p p i ) the new hull edge Rotte ounterlokwise line through p until it touhes one of the other points Time ompleity: O(n ) In ft, the ompleity is O(nh), where n is the input size nd h is the hull size. p p 3 p 8 p4 p 5 p 7 p0 p 6 p p p p 3 0 5

6 Line eqution nd ngle Let (,y ) nd (,y ) e two points. The epliit line eqution is: y = m + y = y tnθ = y y + y y Singulrity t = (vertil line) y y = m+ m = tn θ Grhm s s sn lgorithm Algorithm: Find point p 0 in the interior of the hull. Compute the CCW ngle α i from p 0 to ll other points. Sort the points y ngle α i. Construt the oundry y snning the points in the sorted order nd performing only right turns (trim off left turns ). p 0 Use stk to proess sorted points Time Compleity: O(n log n) Question: How do we hek for right or left turn? p 0 Right turn p 0 Left turn 6

7 Grhm s sn: ompleity nlysis Sorting O(n log n) D i = numer of points popped on proessing p i, n time ( ) = D + = n+ D i i= i= Eh point is pushed on the stk only one. One point is popped it nnot e popped gin. Hene n D n i= i n i 3 Quik hull lgorithm Algorithm: Find four etreme points of P: highest, lowest, leftmost, rightmost d. Disrd ll points in the qudrilterl interior Find the hulls of the four tringulr regions eterior to the qudrilterl. To proess tringulr regions, find the etreme point in liner time. Two new eterior regions A, B will e formed, eh proessed seprtely. Reurse until no points re left in the eterior: the onve hull is the union of the eteriors. Time Compleity: T(n) = O(n)+T(α)+T(β) where α+β=n = T(n ) + O(n) = O(n ) worst se! etreme point A = α d B = β 4 7

8 Algorithm: Divide-nd nd-conquer Find point with medin oordinte in O(n) time Prtition point set in two hlves Compute the onve hull of eh hlf (reursive eeution) Comine the two onve hulls y finding their upper nd lower tngents in O(n). left hull medin tngents right hull Time Compleity: O(n log n) n T ( n) = T + O( n) 5 Finding tngents () Two disjoint onve polygons hve four tngents tht lssify them s either eing entirely to the left (+) or to the right ( ) of the line: (+,+) (+, ) (,+) (, ) 6 8

9 Finding tngents () Lower tngent: Connet rightmost point of left hull to leftmost point in right hull nd wlk round oth oundries until the lower tngent is rehed. Compleity: O(n). : (-,+)( : (-,*)( 3: (*,*) 5: (*,-) 7 Lower ound for onve hull in D Clim: Conve hull omputtion tkes Θ(n log n) Proof: redution from Sorting to Conve Hull: Given n rel vlues i, generte n points on the grph of onve funtion, e.g. ( i, i ). Compute the (ordered) onve hull of the points. The order of the onve hull points is the order of the i. Compleity(CH)=Ω(n log n) Sine there is O(n log n)-time lgorithm, Compleity(CH)= Θ(n log n) 8 9

10 z Conve hulls in 3D Input: Points in 3D y Output: Conve hull Representtion: Plnr sudivision 9 Conve hull in 3D: properties Theorem: A onve polyhedron with V = n verties hst t most E = 3n 6 edges nd F = n 4 fes. Proof: from Euler s formul for plnr grphs: V E + F = Every fe hs t lest 3 edges (tringle) Every edge is inident to t lest fes 3 F E E 3n 6 nd F n 4 The ompleity of the onve hull is O(n). The equlity holds when ll fes re tringles. When no three points re on line nd no four on plne, the fes of the onve hull re ll tringles. 0 0

11 Gift wrpping in 3D () Ide: generlize the D proedure. The wrpping element is plne insted of line. Pivot plne round the edges of the hull; find the smllest ngle of the plnes Π i ontining segment nd points p i Gift wrpping in 3D () Form tringulr fe ontining,, nd repet the opertion for edges nd. Strt from lowest edge in the onve hull nd work round nd upwrds until the wrp is over. e d f d

12 Gift wrpping in 3D (3) Algorithm sketh Mintin queue of fets nd emine their edges in turn, omputing for eh the smllest ngle with it. Compleity O(n) opertions re required t eh edge to find the minimum ngle. Eh edge is visited t most one. Sine there re O(n) edges, the ompleity is O(n ). In ft, it is O(n F ), where F is the numer of fets in the finl hull (s in D se). 3 Divide-nd nd-conquer in 3D Ide: generlize the D proedure. Reursively split the point set into two disjoint sets, ompute their hulls nd merge them in liner time. Reursion ends when 4 points re left (tetrededron). A B Key step: merging two disjoint onve polyhedr in O(n). 4

13 Merging two disjoint onve polyhedr Ide:. Identify the merge oundries of A nd B.. Crete new tringulr fes with two verties from the merge oundry of A nd one verte from the merge oundry of B (nd vie-vers). 3. Delete hidden fes of A nd B. Remrks: The hin of hidden fes strts nd ends t the merge oundry edges. The nd of new fes hs the topology of ylinder with no ps (it wrps round). 5 Merging two disjoint onve polyhedr Algorithm sketh. Find the lowest new edge of the hull formed y one verte of A nd one verte of B.. Pivot plne Π round edge to find the first p. 3. Form tringulr fe (p) nd repet with the new pivot edge formed y p nd its opposite (either or ) 4. Repet step 3 until the wrpping is done. 5. Delete the hidden fes y following the fes round A nd B whose edges re on the merge oundry. 6 3

14 Find the hull strting edge Strting edge How do we find the net point p? How do we know tht the fes will not self-interset? 7 Plne rottion Lemm: when the plne Π is rotted round segment, the first point enountered, must e djent to either or. Proof: y onveity rguments (omitted here) Only the neighors of edge need to e tested t eh time. Sine there re t most n neighors, the overll ompleity is O(n ). 8 4

15 Merge oundry tringultion. Strt from lowest hull edge. Alternte etween left nd right merge oundry points, reting tringles. No serh or testing neessry. Compleity: O(n). Overll ompleity: O(n ) Cn we do etter? O(n log n) Look gin t rottion Plne rottion improvement () Improvement: the testing of ll neigoring points is wsteful nd repets work. Only O(n) verties should e emined overll (mortized ost). Keep trk of A-winners nd B-winners: A-winner: the verte α djent to with smllest ngle. B-winner: the verte β djent to with smllest ngle. Lemm: If α i is winner t itertion i, the B-winner t the net itertion β i+ is ounterlokwise of β i round (sme for α nd β reversed proof omitted). 30 5

16 Divide nd Conquer in 3D (end) The numer of ndidte verties in eh loop itertion dereses monotonilly. Eh edge is emined t most twie opertion tkes O(n). Divide-nd onquer reurrene eqution: n T ( n) = T + O( n) Theorem: the onve hull of set of points in spe n e omputed in optiml O(n log n) ) time nd O(n) spe. Sme ompleity s D prolem! 3 Inrementl onve hull in 3D Ide: inrementlly dd point to onve polyhedr P Two ses:. The new point is inside P do nothing. The new point is outside of P fi P! Memership test is done in O(n) ) time. Wht needs to e done to dd point? 3 6

17 Inrementl onve hull in 3D How to updte the onve polyhedron:. Add fes from edges of the horizon to the new point.. Remove hidden fes strting from edges in the horizon. 33 Inrementl onve hull in 3D One we know wht the horizon is, oth opertions n e performed in O(n) ) time. 34 7

18 Fe visiility The visiility of fe F from point p n e determined y the signed volume of the tetrhedron formed y three points on the fe nd p. p p F F Visile V < 0 V = 6 p p y y y y p z z z z Not visile V > 0 35 Inrementl 3D onve hull lgorithm For ll fes, ompute the signed volume. Keep hidden fes nd disrd visile fes. Edges shring visile nd hidden fes form the horizon. Algorithm sketh Initilize CH to the tetrhedron (p,p,p 3,p 4 ) For eh remining point, do For eh fe F of CH, ompute the signed volume V of the tetrhedron formed y p nd F. Mrk F visile if V<0 If no fes re visile, disrd p (p is inside CH) Else for eh order fe F onstrut one fe. for eh visile fe F, delete F. 36 8

19 Compleity nd degeneries Overll ompleity is O(n ). Cn we do etter? Degenery: if points n e oplnr, oplnr tringles will e reted. This n e fied y deleting the shred digonl, therey reting lrger fe. 37 Improvement () Look-hed omputtion to mke it heper to find visile fets. Mintin for eh fet f of the urrent onve hull CH(P i ) onflit set: P onflit (f) {p i+, p n } ontining the points tht f n see. For eh point p j, j>i, mintin the set of fets of CH(P i ) visile from p j, lled F onflit (p j ) Point p j is in onflit with fe f euse one p j is dded, the fe must e deleted from the onve hull. 38 9

20 Improvement () We otin iprtite grph, lled the onflit grph G. Updte the onflit grph when dding p i : Disrd neighors of p i Add nodes to G for the newly reted fets nd disrd p i Find onflit list of new fets. All others remin unffeted! Algorithm sketh: dd points p, p n sequentilly, using the onflit grph G to determine visile fes. Compleity: O(n log n) epeted rndomized

21 Conve hulls in higher dimensions Prolem: given n points in R d, find their onve hull (lso lled onve polytope). Fes eome hyperfes of dimension,3,,d. Hyperfes form grph struture where djenies etween fetures of dimension i nd i re stored. Some of the previous lgorithms sle up (pplile in priniple) with proper etensions. Theorem : the onve hull of n points in d-dimensionl spe hs t most d / Ω( n ) hyperfes. Theorem : the onve hull n e omputed with the gift-wrpping lgorithm in d / + d / Ω( n ) + Ω( n log n) 4