Computational Geometry
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1 Problem efto omputatoal eometry hapter 6 Pot Locato Preprocess a plaar map S. ve a query pot p, report the face of S cotag p. oal: O()-sze data structure that eables O(log ) query tme. pplcato: Whch state s osto located? Trval Soluto: O() query tme, where s the complexty of the map. Why? p S F Naïve Soluto The Trapezodal Map raw vertcal les through all the vertces of the subdvso. Store the x-coordates of the vertces a ordered bary tree. Wth each slab, sort the segmets separately alog y. Query tme: O(log ). Problem: Too delcate subdvso, of sze Θ( 2 ) the worst case. (ve such a example!) ostruct a boudg box. ssume geeral posto: uque x coordates. xted upward ad dowward the vertcal le from each vertex utl t touches aother segmet. Ths works also for ocrossg le segmets Propertes Notato very trapezod (tragle) s defed by Left( ): a segmet edpot (rght or left); Rght( ): a segmet edpot (rght or left); Top( ): a segmet; ottom( ): a segmet. otas tragles ad trapezods. ach trapezod or tragle s determed: y two vertces that defe vertcal sdes; ad y two segmets that defe overtcal sdes. refemet of the orgal map
2 omplexty Theorem (lear complexty): trapezodal map of segmets cotas at most 6+4 vertces ad at most 3+ faces. Proof:. Vertces: orgal extesos box 2. Faces: out Left( ) left e.p. rght e.p. box 2 V 6 F Possbly by L. Map ata Structure alteratve: For each trapezod store: The vertces that defe ts rght ad left sdes; The top ad bottom segmets; The (up to two) eghborg trapezods o rght ad left; (Optoal) The eghborg trapezods from above ad below. Ths umber mght be lear, so oly the leftmost of these trapezods s stored. Note: : omputg ay trapezod from the trapezodal structure ca be doe costat tme. 86. The Search Structure Q P Q 2 S P 2 S Q P F P S 2 2 Q 2 rachg Rules Query pot q, search-structure ode s. s s a segmet edpot: q s to the rght of s: go rght; q s to the left of s: go left; s s a segmet: q s below s: go rght; q s above s: go left; S 2 S 2 F Q P Q 2 S P 2 S 2 Usg the S 2 S Q P F P S 2 2 Q 2 ostructo Fd a oudg ox. Radomly permute the segmets. Isert the segmets oe by oe to the map. Update the map ad search structure each serto. The map s depedet of the order of serto ad ts sze s Θ(). The ad ts sze depeds o the order of serto. F
3 Updatg the Structures Fd the exstg structure the face that cotas the left edpot of the ew segmet. (*) Fd all other trapezods tersected by ths segmet by movg to the rght. (I each move choose betwee two optos: Up or ow.) Update the map M ad the. (*) Note: : Sce edpots may be shared by segmets, we eed to cosder ts segmet whle searchg. Update : : Smple ase The segmet s cotaed etrely oe trapezod. I M : Splt the trapezod to four trapezods. I : The leaf s replaced by a T subtree. P O() tme. Q P Q M S Update M: : eeral ase Updatg : : Splt The th segmet tersects k > trapezods. ach er trapezod s replaced by: S Splt trapezods. Merge trapezods that ca be uted. O(k ) tme. ach outer trapezod s replaced by: Q S S Updatg : : Merge Leaves are elmated ad replaced by oe commo leaf. O(k ) tme. L S S F L S F 76. Sze of : : Worst-ase alyss ach segmet adds trees of depth at most 3, so the depth of s 3. Query tme (depth of ): O(), Θ() the worst case. The th segmet s - may tersect wth k O() trapezods! The sze of ad ts costructo tme s the worst case: Θ( ) Θ( 2 ) 86. 3
4 Segmet/Trapezod Iteracto Oe segmet may affect may trapezods s verage-ase alyss ompute the expected depth of : q: pot, to be searched for. p : The probablty that a ew ode was created the path leadg to q the th terato. Oe trapezod may affect at most four segmets ompute p by backward aalyss: q (M - ): The trapezod cotag q M -. Sce a ew ode was created, q (M ) q (M - ). elete s from M. Prob[ q (M ) q (M - )] 4/ xpected epth of x : The umber of odes created the th terato the path leadg to the leaf q. The expected legth of the path leadg to q: x 2 [ x ] (3 p ) O(log ). q 26. efe a dcator xpected Sze of dsappears from M f s s removed δ (, s) 0 otherwse k : Number of leaves created the th terato. S : The set of the frst segmets. verage o s: [ k ] (, s) (, s) s S M M s S M O( ) δ δ 4 (4 M ) O(). s 226. xpected Sze of (cot.) k -: Number of teral odes created the th step. Total sze: O( ) + ( k ) O( ) + k leaves teral O( ). xpected ostructo Tme of ( O(log) + O([ k ])) O( log ) Fdg the frst trapezod The rest of the work the th step
5 adlg egeeraces What happes f two segmet edpots have the same x coordate? Use a shear trasformato: x x + ε y ϕ y y gher pots wll move to the rght. ε should be small eough so that ths trasform wll ot chage the order of two pots wth dfferet x coordates. I fact, o eed to explctly shear the plae. omparso rules ca mmc the shearg. Prove: The etre algorthm remas correct
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