Geometric Algorithms. Geometric Algorithms. Warning: Intuition May Mislead. Geometric Primitives

Transcription

1 Geometri Algorithms Geometri Algorithms Convex hull Geometri primitives Closest pir of points Voronoi Applitions. Dt mining. VLSI design. Computer vision. Mthemtil models. Astronomil simultion. Geogrphi informtion systems. Computer grphis (movies, gmes, virtul relity. Models of physil world (mps, rhiteture, medil imging. Referene: Referene: Chpters 24-25, Algorithms in C, 2 nd Edition, Roert Sedgewik. History. Anient mthemtil foundtions. Most geometri lgorithms less thn 25 yers old. Prineton University COS 226 Algorithms nd Dt Strutures Spring 2004 Kevin Wyne 2 Geometri Primitives Wrning: Intuition My Misled Point: two numers (x, y. Line: two numers nd [x + y = 1] Line segment: four numers (x 1, y 1, (x 2, y 2. Polygon: sequene of points. ny line not through origin Wrning: intuition my e misleding. Humns hve sptil intuition in 2D nd 3D. Is given polygon onvex? Primitive opertions. Distne etween two points. Compre slopes of two lines. Given three points p 1, p 2, p 3, is p 1 -p 2 -p 3 ounterlokwise turn? Do two line segments interset? Is point inside polygon? Other geometri shpes. Tringle, retngle, irle, sphere,... 3D nd higher dimensions sometimes more omplited we think of this lgorithm sees this 3 4

2 Wrning: Intuition My Misled Wrning: Intuition My Misled Wrning: intuition my e misleding. Humns hve sptil intuition in 2D nd 3D. Intersetions mong set of retngles. Wrning: intuition my e misleding. Humns hve sptil intuition in 2D nd 3D. Computers do not. Neither hs good intuition in higher dimensions! we think of this lgorithm sees this 5 6 Convex Hull Pkge Wrp Convex hull. Shortest fene surrounding the points. Smllest (onvex polygon enlosing the points. Intersetion of hlfspes defined y point pirs. Pkge wrp. Strt with point with smllest y-oordinte. Rotte sweep line round urrent point in w diretion. First point hit is on the hull. Repet. onvex not onvex Prmeters. N = # points. M = # points on the hull. 7 8

3 Pkge Wrp How Mny Points on the Hull? Implementtion. Compute ngle etween urrent point nd ll remining points. Pik smllest ngle lrger thn urrent ngle. 2D nlog of seletion sort: (MN time. Prmeters. N = # points. M = # points on the hull. How mny points on hull? Worst se: N. Averge se: diffiult prolems in stohsti geometry. Uniform on irle: N. Uniform in onvex polygon with O(1 edges: log N. Uniform in dis: N 1/ Grhm Sn: Exmple Grhm Sn: Exmple Grhm sn. Choose point p with smllest y-oordinte. Sort points y polr ngle with p to get simple polygon. Consider points in order, nd disrd those tht use lokwise turn. Implementtion. Input: p[1], p[2],..., p[n] re points. Output: M nd rerrngement so tht p[1],..., p[m] is onvex hull. Given three points,, nd, is -- ounterlokwise turn? Totl ost: O( for sort nd O(N for rest. p // preproess so tht p[1] hs smllest y-oordinte // sort y ngle with p[1] points[0] = points[n]; // sentinel int M = 3; for (int i = 4; i <= N; i++ { while (Point.w(p[M], p[m-1], p[i] >= 0 { M--; // k up to inlude i on hull } M++; swp(points, M, i; // dd i to puttive hull } 11 12

4 Implementing CCW Implementing CCW CCW: Given three point,, nd, is -- ounterlokwise turn? Ide: ompre slopes. Diffiulty: degenery. CCW: Given three point,, nd, is -- ounterlokwise turn? Plys sme role s omprisons in sorting. Determinnt gives twie re of tringle. x x x y y y x y yx yx xy xy x y Yes No Yes ( slope??? (olliner??? (olliner??? (olliner If re > 0 then -- is ounterlokwise. If re < 0, then -- is lokwise. If re = 0, then -- re olliner. Lesson. Geometri primitives re triky to implement. Need to hndle ll degenerte ses. ( x, y > 0 ( x, y < 0 ( x, y ( x, y ( x, y ( x, y Quik Elimintion Convex Hull Algorithms Costs Summry Quik elimintion. Choose qudrilterl Q or retngle R with 4 points s orners. If point is inside, n eliminte. 4 CCW tests for qudrilterl 4 omprisons for retngle Three-phse lgorithm Pss through ll points to ompute R. Eliminte points inside R. Find onvex hull of remining points. Impt. Almost ll points re eliminted if points re rndom: O(N. Improve performne of ny onvex hull lgorithm. Q these points eliminted R Gurnteed symptoti ost to find M-point hull in N-point set. Algorithm Pkge Wrp Grhm Sn Quikhull Mergehull Sweep Line Quik Elimintion Best in Theory Running Time N M N * N log M * ssumes "resonle" point distriution 15 16

5 Closest Pir of Points Closest Pir of Points Given N points in the plne, find pir tht is losest together. For onreteness, we ssume Euliden distnes. Foundtion of then-fledgling field of omputtionl geometry. Grphis, omputer vision, geogrphi informtion systems, moleulr modeling, ir trffi ontrol. Algorithm. Divide: drw vertil line so tht roughly N / 2 points on eh side. Conquer: find losest pir in eh side reursively. Comine: find losest pir with one point in eh side. Return: est of 3 solutions. Brute fore solution. Chek ll pirs of points p nd q. (N 2 omprisons. 1-D version (points on line. O( esy Assumption. No two points hve sme x oordinte. 12 solely to mke presenttion lener Closest Pir of Points Closest Pir of Points Key step: find losest pir with one point in eh side. Extr informtion: losest pir entirely in one side hd distne. Oservtion: only need to onsider points within of line. Sort points in 2-strip y their y oordinte. Only hek distnes of those within 6 positions in sorted list! s[] = rry of points in the 2-strip, sorted y their y-oordinte. Ft: if i j 12, then the distne etween s[i] nd s[j] is t lest. No two points lie in sme /2 y /2 ox. Two points t lest 2 rows prt hve distne 2 / 2. Ft: still true if we reple 12 with 7. 2 rows i j / 2 / 2 / = min(12,

6 Closest Pir of Points Closest Pir of Points Closest pir lgorithm. Running time. Compute seprtion line x = x med suh tht hlf the points hve x oordinte less thn x med, nd hlf re greter. O( T( N 2T N /2 O ( N logn T( N O ( N log 2 N 1 = ClosestPir(left hlf 2 = ClosestPir(right hlf = min ( 1, 2 Delete ll points further thn from seprtion line. Sort remining points in strip y y oordinte. Sn in y order, nd ompute distne etween eh point nd next 6 neighors. If ny of these distnes is less thn, updte. Return. = ClosestPir (p 1, p 2,..., p N 2T(N / 2 O(N O( O(N Cn we hieve O(? Yes. Don't sort points in strip from srth eh time. Eh reursive ll should return two lists: ll points sorted y y oordinte, nd ll points sorted y x oordinte. Sorting is omplished y merging two lredy sorted lists. N /2 O ( N T( N O ( N log T ( N 2T N Nerest Neighor Voronoi Digrm / Dirihlet Tesseltion Input: N Euliden points. Nerest neighor prolem: given query point p, whih one of originl N points is losest to p? Input: N Euliden points. Voronoi region: set of ll points losest to given point. Voronoi digrm: plnr sudivision delineting Voronoi regions. Ft: Voronoi edges re perpendiulr isetor segments. Brute fore: O(N time per query. Gol: O( preproessing, O(log N per query. Quintessentil nerest neighor dt struture

7 Applitions of Voronoi Digrms Adding Point to Voronoi Digrm Anthropology. Identify influene of lns nd hiefdoms on geogrphi regions. Astronomy. Identify lusters of strs nd lusters of glxies. Biology, Eology, Forestry. Model nd nlyze plnt ompetition. Crtogrphy. Piee together stellite photogrphs into lrge "mosi" mps. Crystllogrphy. Study Wigner-Setiz regions of metlli sodium. Dt visuliztion. Nerest neighor interpoltion of 2D dt. Finite elements. Generting finite element meshes whih void smll ngles. Fluid dynmis. Vortex methods for invisid inompressile 2D fluid flow. Geology. Estimtion of ore reserves in deposit using info from ore holes. Geo-sientifi modeling. Reonstrut 3D geometri figures from points. Mrketing. Model mrket of US metro t individul retil store level. Metllurgy. Modeling "grin growth" in metl films. Physiology. Anlysis of pillry distriution in ross-setions of musle tissue. Typogrphy. Chrter reognition, eveled nd rved lettering. Zoology. Model nd nlyze the territories of nimls. Chllenge: ompute Voronoi. Bsis for inrementl lgorithms: region ontining point gives points to hek to ompute new Voronoi region oundries. How to represent the Voronoi digrm? Use multilist ssoiting eh point with its Voronoi neighors. Referenes: Rndomized Inrementl Voronoi Algorithm Disretized Voronoi Digrm Add points (in rndom order. Find region ontining point. use Voronoi itself Updte neighor regions, rete region for new point. Use grid pproh to nswer ner-neighor queries in onstnt time. Approh 1: provide pproximte nswer (to within grid size. Approh 2: keep list of points to hek in grid squres. Computtion not diffiult (move outwrd from points. Running time: O( on verge

8 Deluny Tringultion Some Geometri Algorithms Input: N Euliden points. Deluny tringultion: tringultion suh tht no point is inside irumirle of ny other tringle. Running time to solve 2D prolem with N points. Ft 1: Dul of Voronoi (onnet djent points in Voronoi digrm. Ft 2: No edges ross O(N edges. Ft 3: Mximizes the minimum ngle for ll tringulr elements. Ft 4: Boundry of Deluny tringultion is onvex hull. Ft 5: Closest pir of of Deluny grph is losest pir. Prolem onvex hull losest pir nerest neighor Brute Fore Cleverness N 2 N 2 N log N polygon tringultion N 2 furthest pir N 2 Deluny Voronoi 35 36