Computational Geometry. Kirkpatrick-Seidel s Prune-and-Search Convex Hull Algorithm

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1 COSC 6114 Coutational Geoety Kikatick-Seidel Pune-and-Seach Convex Hull Algoith Intoduction Thi note concen the coutation of the convex hull of a given et P ={ 1, 2,..., n }of n oint in the lane. et h denote the ize of the convex hull, ie the nube of it vetice. The value h i not known befoehand, and it can ange anywhee fo a all contant to n. Wehave aleady een that any convex hull algoith euie at leat Ω(n lgn)tie in the wot cae, and have tudied a nube of algoith, uch a Gaha can algoith, whoe wot cae tie colexity i O(n lgn). If n wa the only eaue of oble ize, then thee algoith ae otial. Howeve, we alo know that the Javi ach algoith euie O(nh) tie. The latte can ange anywhee fo O(n) to O(n 2 )deending on the value of h. Ithee an algoith which i aytotically ueio to both Gaha can and Javi ach, fo all oible value of h? Below, we will decibe Kikatick and Seidel [KiS86] algoith that euie O(n lgh)tie. Kikatick-Seidel algoith alie a deign techniue known a the une-andeach ethod o Megiddo techniue. Niod Megiddo howed, eg, how thi techniue can be ued to olve fixed dienional linea oga in linea tie [Meg83, Meg84], and how tocoute the allet cicle that encloe a finite nube of given oint in the lane in linea tie [Meg89]. Dye [Dye84] indeendently dicoveed the ae techniue. Many othe alication of thi oweful algoith deign techniue aea in the liteatue. Edelbunne book [Ede87] alo give a bief decition of the ethod in ection 15.6 and how it alication, eg, to linea ogaing in chate 10, and to ha-andwich cut in ection Fance Yao in ection 6, chate 7 of van eeuwen book [van90] alo dicue thi techniue. The une-and-each techniue can be taced back to the fit linea tie edian finding algoith of Blu-Floyd-Patt-ivet- Tajan [BFP73]. The latte algoith find the edian (and in geneal, the k-th allet eleent) of a finite et of given nube in linea tie and i alo decibed in ection 10.3 of Coen-eieon-ivet [C91]. Kikatick-Seidel Algoith Conide the iniu and axiu x-coodinate of oint in P, denoted x in and x ax. Convex Hull of P can be be viewed a a ai of convex chain called the ue hull and the lowe hull of P (excluding the oible vetical edge at x in o x ax ). (See Fig. 1(a).) The algoith that coute the ue-hull of P i given below. The lowehull can be couted in a iila anne and i oitted fo futhe dicuion.

2 -2- ue hull vetical t edge lowe hull x in x ax (a) Fig. 1 (a) The ue and lowe hull, (b) The ue bidge. Algoith UeHull(P) 0. if P 2 then etun the obviou anwe 1. ele begin 2. Coute the edian x ed of x-coodinate of oint in P. 3. Patition P into two et and each of ize about n/2 aound the edian x ed. 4. Find the ue bidge of and,, and 5. { x() x() } 6. { x() x() } 7. UH UeHall( ) 8. UH UeHall( ) 9. etun the concatenated lit UH,, UH a the ue hull of P. 10. end (b) Analyi Thi i a divide-&-conue algoith. The key te i the coutation of the ue bidge in te 4 which i baed on the une-&-each techniue. (See Fig. 1(b).) In the next ection we will how that thi te can be done in O(n) tie. We alo know that te 2can be done in O(n) tie by the linea tie edian finding algoith. Hence, te 3-6 can be done in O(n) tie. Fo the uoe of analyzing algoith UeHall(P), let u aue the ue hull of P conit of h edge. Ou analyi will ue both aaete n (inut ize) and h (outut ize). et T (n, h) denote the wot-cae tie colexity of the algoith. Suoe UH and UH in te 7 and 8 conit of h 1 and h 2 edge, eectively. Since and, the two ecuive call in te 7 and 8 take tie T (n/2, h 1 ) and T (n/2, h 2 ) tie. (Note that h =1+ h 1 + h 2. Hence, h 2 = h 1 h 1.) Theefoe, the ecuence that decibe the wot-cae tie colexity of the algoith i T (n, h) = O(n) + ax { T ( n h1 2, h 1) + T ( n 2, h 1 h 1)} if h >2 O(n) if h 2

3 -3- Theoe: T (n, h) =O(n lg h). Poof: Suoe the two occuence of O(n) inthe above ecuence ae at ot cn, whee c i a uitably lage contant. We will how by induction on h that T (n, h) cn lg h fo all n and h 2. Fo the bae cae whee h = 2, T (n, h) cn cnlg2 =cn lg h. Fo the inductive cae, T (n, h) cn + ax { c n h1 2 lg h 1 + c n 2 lg (h 1 h 1)} = cn + c n 2 ax lg ( h 1 (h 1 h 1 )) h 1 cn + c n 2 lg ( h 2 h 2 ) = cn + c n 2 2 lg h 2 = cn lgh. Finding the Ue Bidge in inea Tie The oble i thi: we ae given acollection P of n oint in the lane, which i eaated into two non-ety ubet and by a known vetical line, with on the left and on the ight. We wih to find a line t aing though one oint fo each ubet, uch that none of the given oint lie above t. See Fig 1(b). In othe wod, we want the ue exteio coon tangent (the ue bidge) of the convex hull of and. If the convex hull of and wee known, the coon tangent could eaily be found in linea tie (ee the ege te in the divide-&-conue convex hull algoith dicued ealie in the coue). Howeve, couting the convex hull of Θ(n) oint cot Θ(n lgn)tie in the wot cae. Coutation of the ue bidge of and can be foulated a a 2-vaiable linea oga with n linea containt, and hence, can be olved in O(n) tie by Megiddo linea-ogaing algoith. The linea oga foulation i a follow. Suoe the euation of the (non-vetical) bidge line t i y = x + β. The two coefficient (the loe) and β (the y-inteet) ae the two unknown that we have to coute. Suoe the x-coodinate of the vetical eaato line between and i x = a. (See Fig. 1(b).) Then, the y-coodinate of the inteection of t and i y o = a + β. Clealy any line that i at o above evey oint of cannot inteect at a y-coodinate lowe than y o.thi give uthe deied 2-vaiable linea oga; find and β to: iniize ubject to: a + β x( i ) + β y( i ) fo all i. Intead of dicuing Megiddo olution of thi linea oga, we will dicu Kiktick-Seidel diect ethod. The key to thei algoith i a ile une-&-each citeion that in linea tie allow u to eliinate a good any of the oint that do not define the ue bidge.

4 -4- et u fix fo the oent ou attention on line of a aticula loe. Wecan coute in linea tie a uoting line of of loe. Suoe thi line i tangent to at oe oint. Wecan do the ae fo and obtain a uoting line of of loe tangent to at oe oint. Now if the line ha loe le than,then o ut the coon tangent t; iilaly, if ha loe geate than then o doe t; and if ha loe then t =. See Fig 2(a,b). (a) (b) Fig 2. (c) Now uoe the fit cae hold, o loe of t i le than. et, be any two oint of, uch that i to the left of and the line ha loe geate than. Then we can conclude that t cannot a though, becaue a line of loe le than though ut a below. See Fig 2(c). The econd cae, whee the loe of t i geate than, i entiely yetical: we can eliinate the econd ebe of any ai,, with to the left of, ifthe line ha loe le than. Thee eak ugget the following une-&-each ethod. Pai u the n given oint in an abitay way, and find the edian loe of the n/2 line defined by thoe ai. Now coute the uoting line of, eectively, and of loe and aue thee line, eectively, ae tangent to and at oint and. Ithould be obviou why the edian i a good choice: ince half the ai have loe le than,and half have loe geate than. Theefoe, half the ai will atify the citeion, no atte whethe the loe of i geate o le than. So, in eithe cae we eliinate one oint fo half the ai, fo a total of at leat n/4 oint. Of coue if ha loe then we ae done. It i oible to find the edian loe in tie linea in n uing the ae edian finding algoith entioned in the eviou ection. The othe oeation clealy take O(n) tie. Theefoe, in linea tie we eithe to, o eliinate at leat n/4 of the oiginal oint. If we eeatedly aly thi eliination (o uning) oce on the eaining oint, we ae guaanteed to find the coon tangent, at a total cot of O(n + (3/4)n + (3/4) 2 n +...)=O(n) tie. Note that we ay eliinate a diffeent nube of oint fo and fo, but thi doe not affect the analyi. Now wecan tate the following eult. Theoe: The ue bidge of two vetically eaated oint et can be couted in linea tie.

5 -5- Coollay: Kikatick-Seidel convex hull algoith take O(n lgh) tie. Kikatick-Seidel alo howed that in te of the two aaete n and h, Ω(n lg h) i a wot-cae lowe bound to coute the convex hull (uing a geneal coutational odel known a the algebaic deciion tee odel). Theefoe, thei algoith i wotcae otial. 3D Convex Hull The convex hull of n oint in 3 can alo be couted in O(n lgn)tie by a divide-&-conue algoith. ecently, Edelbunne and Shi [EdS91] have hown that 3D convex hull can be couted in O(n lg 2 h)tie, whee h i the nube of hull vetice (extee oint). Futheoe, Kenneth Clakon and Pete Sho [ClS89] give a andoized 3D convex hull algoith with O(n lgh) exected tie. Chazelle and Matouek [ChM93] have eoted that deandoizing an algoith of [ClS89] give an O(n lgh)tie deteinitic algoith. See alo [CSY95]. efeence [BFP73] Blu, M., W. Floyd, V. Patt,. ivet,.e. Tajan, Tie bound fo election, J.ofCoute and Syte Science, vol. 7, , [CSY95] Chan, T.M.Y., J. Snoyink, C.K. Ya, Outut-enitive contuction of olytoe in fou dienion and clied Voonoi diaga in thee Poc. SODA 95, , [ChM93] Chazelle, B., and J. Matouek, Deandoizing an outut-enitive convex hull algoith in thee dienion, Manucit, [ClS89] [Dye84] [EdS91] [KiS86] Clakon, K., and P.W. Sho, Alication of ando aling in coutational geoety, II, Dicete & Coutational Geoety, vol. 4, no. 5, , Dye, M.E., inea tie algoith fo two- and thee-vaiable linea oga, SIAM J. Couting, vol. 13, , Edelbunne, H., and W. Shi, An O(n log 2 h) tie algoith fo the theedienional convex hull oble, SIAM J. Couting, vol. 20, no. 2, , Kikatick, D.G., and. Seidel, The ultiate lana convex hull algoith?, SIAM J. Couting, vol. 15, no. 1, , [Meg83] Megiddo, N., inea-tie algoith fo linea ogaing in 3 and elated oble, SIAM J. Co. 12, , [Meg84] [Meg89] Megiddo, N., inea ogaing in linea tie when the dienion i fixed, J. Aociation fo Couting Machiney (JACM) 31, , Megiddo, N., On the ball anned by ball, Dicete & Coutational Geoety, vol. 4, no. 6, , 1989.