Lecture Note 08 EECS 4101/5101 Instructor: Andy Mirzaian. All Nearest Neighbors: The Lifting Method

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1 Lecture Note 08 EECS 4101/5101 Instructor: Andy Mrzaan Introducton All Nearest Neghbors: The Lftng Method Suose we are gven aset P ={ 1, 2,..., n }of n onts n the lane. The gven coordnates of the -th ont n P are =(x, y ), =1..n. A nearest neghbor of P s a ont P { }such that the Eucldean dstance dst(, )smnmum ossble. We denote ths nearest neghbor as = NN( ). In case has more than one nearest neghbor (wth the same mnmum dstance from ), we ck one of them arbtrarly. The All Nearest Neghbors Problem (ANNP): Inut: Aset P ={ 1, 2,..., n }of n onts n the lane. Outut: The nearest neghbor NN( ), for each ont P, =1..n. Fgure 1(a) shows a set of onts, for each ont an arrow ontng to ts nearest neghbor. Fgure 1(b) shows the roerty of nearest neghbor, namely, that = NN( )fnone of the onts n P { } are n the nteror of the crcle wth center and radus dst(, ). (a) The Nearest Neghbor Grah (b) = NN ( ) Fgure 1. (c) Emty crcle A related roblem s the Closest Par Problem (CPP), whch asks to comute a dstnct ar of onts n P so that the Eucldean dstance between them s mnmum among all dstnct ars of onts n P. Chater 33 of [CLRS] descrbes an O(n lgn) tme algorthm to comute the closest ar. Ths dvde-and-conquer algorthm s due to Shamos and Hoey [ShH75]. It should be obvous that a ar (, )n P s the closest ar only f and are mutually nearest neghbors of each other. Note that the latter condton s necessary, but not suffcent. However, ths shows that f we already have the soluton to ANNP, wecan solve CPP n O(n) addtonal tme. (Just fnd the ar (, NN( )), for =1..n that mnmzes dst(, NN( )).) In ths lecture note we want to descrbe a method to solve ANNP n O(n lgn)tme. The man ngredents of the soluton are: () The Sarsfcaton Method: We can vew ANNP as a grah roblem: consder the comlete grah wth n vertces n one-to-one corresondance wth the n onts n P, and an edge between each ar (, )ofvertces wth weght dst(, ). The

2 -2- roblem s now tocomute, for each vertex n ths grah, ts nearest adacent vertex. But snce the grah s comlete, t has Θ(n 2 )edges. The sarsfcaton method wll elmnate many rrelevant edges to obtan a sarse subgrah that stll contans the edges we are seekng. The sarse grah n our soluton turns out to be a lanar grah (known as the Delaunay trangulaton), wth n vertces and O(n) edges. Once we have such a sarse grah, we can nsect all ts edges n O(n) tme to comlete the soluton to ANNP. () The Lnearzaton Method: Ths method, tres to turn non-lnear equatons nvolved n the roblem formulaton to lnear equatons as much as ossble. () The Lftng Method. Due to the lnearzaton method, ths method lfts the 2D onts of P nto 3D. (v) The use of an O(n lgn)tme algorthm to comute 3D convex hull (e.g., the dvdeand-conquer 3D convex hull algorthm). Fgure 1(b) shows the condton for = NN( ). Fgure 1(c) also shows the crcle wth dameter (, )whch s nsde the crcle as shown n Fgure 1(b). The roerty of the smaller crcle s that none of the onts of P are n ts nteror. Any such crcle s called an emty crcle. Thus, we arrve at our frst necessary condton for the nearest neghbor roerty: Fact 1: If s nearest neghbor of,then the crcle wth dameter (, )san emty crcle. As we shall see later, there are a total of O(n) ars (, )that are dameter of an emty crcle. Ths would accomlsh ngredent (), sarsfcaton. Now, let us concentrate on the emty crcle roerty. The equaton of a crcle C wth center coordnates (a, b) and radus r s (x a) 2 + (y b) 2 = r 2. Ths s the quadratc equaton: x 2 + y 2 =2ax + 2by + (r 2 a 2 b 2 ). (1) Usng ngredent (), the lnearzaton method, we name the quadratc quantty x 2 + y 2 on the left hand sde of eq. (1) as a new varable z. Now, eq. (1) can be exressed as a ar of smultanous equatons n 3D (wth coordnate system (x, y, z)) as: z =2ax + 2by + (r 2 a 2 b 2 ) z = x 2 + y 2 (2) From eq. (2) we see that the crcle arameters (a, b, r) all show uonly n the lnear equaton z =2ax + 2by + (r 2 a 2 b 2 ), whch s now the equaton of a lane n 3D. Let s denote ths lane Π(C). The second, and quadratc equaton z = x 2 + y 2 s fxed, ndeendent of the crcle arameters. The latter s the equaton of the so called arabolod of revoluton, denoted Λ. See Fgure 2. Thus, we can nterrete the crcle equaton (1) as follows: n 3D, take the ntersecton λ(c) = Π(C) Λ of the lane Π(C) wth the arabolod of revoluton Λ and roect t back down to the ground lane,.e., the xy-lane, to get crcle C. Remark 1: Assumng coordnate z s the vertcal drecton n 3D, the ntersecton of any non-vertcal lane wth Λ s an ellse n 3D, but ts roecton on the xy-lane s always

3 -3- z Λ 2 2 : z = x + y λ(c ) Π(C ) y x Fgure 2. a crcle. The ntersecton of a vertcal lane wth Λ n 3D s a arabola whose roecton on the xy-lane s a lne.) From the above dscusson and the convexty of Λ, weobtan our next observaton: Fact 2: Let Π be any non-vertcal lane n the xyz-sace. The roecton of the ntersecton of Π and the arabolod of revoluton Λ down on the xy-lane s some crcle C. Conversely, any crcle C n the xy-lane s the roecton of the ntersecton λ(c) = Π(C) Λof the non-vertcal lane Π(C) wth Λ n the xyz-sace. Furthermore, any ont on Λ below the lane Π(C) roects down to a ont n the nteror of crcle C, and any ont on Λ above the lane Π(C) roects to a ont n the exteror of crcle C. Consder a ont =(x, y) onthe ground lane, and vertcally "lft" t to the ont λ( ) =(x, y, x 2 + y 2 )onthe arabolod of revoluton n the xyz-sace. Smlarly, let λ(p) ={λ( ) P }denote the n onts n P lfted vertcally on to Λ. Now, the next observaton: Fact 3: Let C be a crcle on the xy-lane, let Π(C) denote ts corresondng lane n xyz-sace. Then, C s an emty crcle (.e., none of the onts of P are n ts nteror) f and only f none of the 3D onts λ(p) are below the lane Π(C). Remark 2: Thus, by ths lftng method, we have converted the 2D non-lnear "nsde/outsde crcle test" to the 3D lnear "below/above lane test". Now consder a ar of onts (, )n P. Accordng to Fact 3, the crcle C wth dameter (, )sanemty crcle f and only f all lfted onts λ(p) are on or above the lane Π(C). The latter lane asses through λ( )and λ( ). Ths, and Fact 1 mly that f s nearest neghbor of,then n the xyz-sace there s a lane (namely, Π(C)) that asses through λ( )and λ( ), and all n onts λ(p) are on or above t. But such a lane s a suortng lane of λ(p), one that touches the 3D convex hull of λ(p) along the edge (λ( ),λ( )) and the remanng onts of λ(p) are above that suortng lane. Furthermore, snce ths suortng lane touches convex hull of λ(p) from below, then the lne segment (λ( ),λ( )) s anedge of one of the "lower" faces of the convex hull of λ(p) C

4 -4- (.e., a face whose normal outsde the convex hull s ontng downward). Hence, we arrve at our next observaton: Fact 4: If s a nearest neghbor of,then (λ( ),λ( )) s one of the edges of the "lower" convex hull of λ(p). Let G =(P, T )bethe straght-lne grah on the xy-lane wth n vertces P, and the edge set T,where each edge of G s the roecton of a lower-convex-hull edge of λ(p). Remark 3: Snce the arabolod Λ s convex, all onts λ(p) are convexly ostoned and they wll all be vertces of the lower-convex hull of λ(p). Thus the vertex set of G s ndeed the entre set P of the nut onts. From Fact 4, we see that all the nearest neghbor edges aear n ths grah G. Furthermore, G s sarse, snce t s lanar and hence t has at most 3n edges. If on the contrary, G were nonlanar,.e., had a ar of crossng edges, that ar of edges would corresond (n the lfted verson) to a ar of edges of the lower hull of λ(p) such that one s vertcally "above" the other. Ths s mossble for a convex hull. Remark 4: In fact, ths straght lne lanar grah G =(P, T )sthe well known Delaunay Trangulaton of P. Atrangulaton of P s a maxmal set of non-crossng edges T that artton the nteror of the convex hull of P nto trangles. A ont set P has many trangulatons. The Delaunay trangulaton s the unque trangulaton wth the roerty that the crcumcrcle of each of ts trangles s an emty crcle. In other words, each trangle n T s the roecton of a (trangular) face of the lower convex hull of λ(p). See Fgure 3. z y x G Fgure 3. From the above develoment we conclude the followng algorthm:

5 -5- Algorthm All-Nearest-Neghbors (P) Ste 1. Construct the 3D ont set λ(p) Ste 2. Comute the 3D convex hull CH(λ(P)) Ste 3. T edges of the lower hull of CH(λ(P)) comuted n ste 2. Ste 4. Ste 5. end T roecton of T down on the xy-lane Search the edges T to fnd all NN s. That s, for each, =1..n, look at all ts ncdent edges (, ) T and ck the shortest one. Comlexty Analyss: Ste 2 takes O(n lgn) tme, by usng any worst-case effcent 3D convex hull algorthm (e.g., the dvde-and-conquer algorthm). The other stes take O(n) tme. Sace comlexty s O(n). Bblograhy The toc of ths note s covered n many standard books on comutatonal geometry such as [BKOS97], [ORo94] or [Ede87] ust to name a few. Delaunay Trangulaton, and ts dual, the Vorono Dagram are well known. There are lterally hundreds of ublcatons on these geometrc structures. You may also fnd more detals about them n the books mentoned above. References: [BKOS97] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkof, "Comutatonal Geometry: Algorthms and Alcatons," Srnger, Second Edton, [Ede87] [ORo94] [ShH75] H. Edelsbrunner, "Algorthms n Combnatoral Geometry," Srnger-Verlag, J. O Rourke, "Comutatonal Geometry n C," Cambrdge Unv. Press, 2nd edton, M.I. Shamos and D. Hoey, "Closest-ont roblems," In Proceedngs of the 16th Annual IEEE Symosum on Foundatons of Comuter Scence (FOCS), 1975,

Solving Optimization Problems on Orthogonal Ray Graphs

Solving Optimization Problems on Orthogonal Ray Graphs Solvng Otmzaton Problems on Orthogonal Ray Grahs Steven Chalck 1, Phl Kndermann 2, Faban L 2, Alexander Wolff 2 1 Insttut für Mathematk, TU Berln, Germany chalck@math.tu-berln.de 2 Lehrstuhl für Informatk