ETREE — A DATABASE-ORIENTED METHOD FOR
GENERATING LARGE OCTREE MESHES
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Tiankai Tu David R. O’Hallaron Julio C. Lopez´ £
Computer Science Department, [email protected] ¤
Computer Science Department and Electrical and Computer Engineering Department, [email protected] ¥ Electrical and Computer Engineering Department, [email protected] Carnegie Mellon University, Pittsburgh, PA, U.S.A.
ABSTRACT
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H Etree API n g e o r application-specific input n l i e i t r a o a c t t element a e c n d u g n o i a i database a l A v L L u a a q b n construct unbalanced balance balanced transform octree octree etreelibrary etreelibrary etreelibrary B-Tree node database Etree library Figure 2: Etree method of generating octree meshes. Figure 3: Etree architecture. ¨=./!E 3$* *tR %f 3!E:§` . E mEB$h?6©#() !* () B QVU©t § 3! ©#S 3 )bfR8EBO? [ ,/ ©t ©#.$*6 - ,$? NR! N (FH> 4.1 The etree API $ © "$ B G .hB$E,§X. E ,$ 6 §- U E.+ ! ¨ §!K* . EB © .$>$ ¦¨§ G ( $* *aKk'h.5K §!G F* ©#?,R4© ) § .BVUL© J¬© "$@ © §>.$5$°. E: ,$ BH ¦¨§ L § F. E: ,$ )EB$°?[ B 3!ho!HD¦¨§ )`kma© ©#! ©tB w? Rw©K ©t_> *#h$I $ wEB $ G;7¿$;º F$ 0 N .$B 4© O9A·fô#7G § DE.$ .$ ! h . ?EB- "/.©#§B F© + .h §!QE* H ¢ § a. E $H © EU § U N 3!E 3!,*©#5.( © ./ © ¨* / R !E.+ !B >© > § bfR>2$$* 3!/% § © + :$*j. 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E ,+ U.$> !© bH /* Initialization and cleanup */ e‘sleft-lower corner (2, 3) etree_t *etree_open(const char *path, int flag, ...); int etree_close(etree_t *ep); binary form (010, 011) /* Octant-level operations */ int etree_insert(etree_t *ep, location_t loc, const void *value); interleave the bits to obtain Morton code int etree_search(etree_t *ep, location_t loc, location_t *hitloc, void *value); int etree_update(etree_t *ep, location_t loc, const void *value); 010 011 int etree_delete(etree_t *ep, location_t loc); int etree_append(etree_t *ep, location_t loc, void *value); 00 11 01 /* Octree-level operations */ append the level of e typedefint decom_t(location_t loc, const void *value, void *childvalue[8]); 001101_11 int etree_construct(etree_t *ep, location_t rootloc, const void *rootvalue, decom_t *autodecom); int etree_balance(etree_t *ep,decom_t *baldecom); int etree_sprout(etree_t *ep, location_t loc, const char *childvalue[8]); int etree_initcursor(etree_t *ep, location_t loc); Figure 6: Interleaving bits and appending level to int etree_getcursor(etree_t *ep, location_t *loc, void *value); obtain the locational code. int etree_advcursor(etree_t *ep); ¦j. © ./*# ¦ § ª ! ,©#* .5 §!ª?!©t @W© * BB2+© " $ * 2f*w !© "%wV¨ ! $? 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E,bw( $ Q .[[ 3 ? . . B% §+3 Q. E , ¤h(3! N ? E. ./ !© "w .M©t D©#"$§ )E:§!© * +&H $K.$ © EX §- D./! O ! E !$ .$5 §!©# 3!?! F. . 1n .$5&©t 6E:§!© * + (3! N =§ -2/a §!K $()a* 5C @A* .BV¨UE.$ :U$ V¨., ! %. E ,+ ¤`(h3 N ]?Q E:§!©#*# .$5& ./()`. E ,$ ¨ § ¨© ©t *t5dHm¦¨§!5./%/ §!].$*# © §¨2$$* 3!¨ ¨¢ $£ VU© * *!?].-2f@ dVU© Eh$ `*#"/h$ §¤$Ha«K.BV=2fB% ©#!Eh$w. E © Q) !© N@ V6©t &Hc\]R) ./© " .%fV¨a$*tV¨BR ]bf §!6*# N =. E ,$ #N.$© + 6 !E.$() ./ ©t © ./ .$5i § ` !.$( $© &%f© ¨©# ]© () ./ © ? * `5C.$ © !5C.$( © .$>©#F §! ? * . E:b+© " -,R HUW a § .$3 *# F? .$ . E ,+ ¤m .Q§B2f6D© *!!© "/§+?.$' §- '.-2f*#$ ! ' §! §- ' 2f,*+© :, © ./! m( R ?] B B .` !.$ $"/ = §! "$©#.$>E.-2fB X? R ¤ a B%!$ a § .BVUF© [¬©#"$3!)u ¥ H © () E U.$5' ©# RX. E ,+ BH Impossible for an octant to overlap a5 > . E © "°$* * § 8? * . E:b+ %Q §!M: 1.$?! ,$© ! w the region covered by b‘sparent E.$() * F* © N [.5. E ,+ ) § X2+©#.$*# F § >y-@^ .$@u>E.$!@ N ,$© w* . EB$* *tR .°?M53! §!8 !E./() .$ B iH ¦¨§ 1 § X$E: 3$*= - ,$? ./ ,- ©#.$ h.$5a !*#: ©#!"w$ © © !"D. E ,$ `EB-© B [./3 a©#G?- E:§G().+ !/H b b `* .% = §!6 ©#()U.5j . E © "B$E,§ ? * . E,bg%/ § U. E ,$ ./ § ]? * . E,b ?.$3 R=E.$, ! © ha ,- ¨* ©# N BH Figure 15: Case(3) is an impossible scenario. Ts©t §4EB$ >nNu-xh snAy/xh$ §!G .$ ©#?!* F© ©t ©#$*6E.$ !©t@ & -~ fCjj T1 . EB .¥© :, © ./85.$ §!h N ©t ,- © ./0.5¨?./3! !Rw?*#$ E©#!"!%V= §!.$3 "$§F § ?./3! -R[. E: ,$ Q* © N B%i$ FE:§!E:bXBE:§w. E@ EB1 !.-2fh? Rw©# !3!E © ./w § §!© *'. E ,$ VU© *#* ,$ 6 ©#"$§+?.$ 6 .D ()© aVU§ §!6 § yB@^ .$@uQE.$!@ 2fm?algE D?+R § 6© ,E © ./ ]? dV=)$ #N$E N ,$© ]©# ]2+©#.$*# B g%+$ D© 5i .% 5.( E.$() ./ ©t © ./&H ? * . E,b+ H d8 § © !. E %cF V¦* © N h5C.$h V¦?./3! !R8. E ,$ ¬© "/3 uBvh §!.BVU U § 6VU©t §[E$ )nNu-x¨ ¨ § Q© !© ©#*&E.$!@ © "$ ,- B i%mVU§ © E,§4©# h3 B 4 § © !3! .w §!G!_+ !©t © ./i% § h !E./() ./ ©t © ./w.5¤QVU© * *c 3 *t Q© >5.$3!KE:§!©#*t@ ©t , © .$&H ¢ ). E ,$ =.5jS+3 $*! ©#§` .¤ =© $*!© "/§+?.$ H . $K. § U §!Q .$ ,$*jE./ N `.$5' B-E:§!© " © "$§?. `© [ §! § :`© ¨!. 5C3! §!6© () E =.$X §!`© $*. E , H¨IK - ,$? $ =5C.$'?./3! !R? $*#$!E© "K© (h3 E,§E:§!B$ : § § . § §$ i%c §! WV¨.wE,§ © *# O.5¤)© 0 § X?.$ .$( § >E$ 8.$5 "/* .$?$*K?$*# E© "4?E$3 > §!>3!(? .5 §*t5j?E./()U § a V°?./3 -R). E ,$ mVU§ ./ K©# $* ?./3! !R0. E: ,$ © h5)* h §$0 §! 3!(?.5a §! ©#"$§+?.$ '] § ©t' © ? * © "$ c.$5 §!= $()] © §/Hm¦¨§+3 E$ . E ,+ U© [ §!Q ©t !./( © &H nAy/x¨© 6 § Q!V © ©t ©#$*jE./ ©t ©#.$&H © § ª()./ N © () .$ , © !:_ N 3 E 3 © Internal octants Internal octants \]@^ ¥%muBv$£^HG¬© "/3!Fuz in the same block not affected by the - ,$? $ X$ 8* ) NR N () )¢tuBq%]uo %]u as b decomposition Decompose § .BVU §!¨ N 3 E 3=.5K\m@A /Hc¦¨§!6] dV¨.a dR .5 .+ ¨© F\]@^ ¦c §!´#7·¨K¶¿Bº 7 Q ) § µ¶º 7 ¶¿Bº 7 H octant b ¡ b An array of records in ascending key order Unused space A tiny octant on key data key data … key data … the boundary of 1 1 2 2 n n another block Pointer to next leaf node Figure 16: Case (1): internal neighbors not affected by the decomposition of the boundary octant. Figure 19: B-tree leaf node. ¦¨§ =*#5!.+ ! E./ ,$© D ! ,U .K?= E,§ B gH'¦¨§ ] N 3 E@ ¬© "/3u U § .BVU §- VU©t §hE$ QnAy$xj$ §!=© ©t ©#$*+E.$ !©t@ 3!U.5jQ* B-5i .+ !U© ¨ ,RD.5gE.$, ]VU©t §D § 65C.$( © ./i%f § 6 E.$() ./ ©t © ./).$5 ¤]VU©#* * 3 *t m©#h5./3=E,§ © *# £¢ ¤¥¤ £§¦©¨ ¦ ":%B § .BVU © ¬©#"$3!1u#¢H'¦¨§!' ©# m N .$ . E ,+ a§*t5'$ a*#"/ §¤ U© *j!© "/§+?.$BH]\=3 a §! © F E © "Dbf:R .$, !:a$ G$* *g § Qbf:R! =©#G* B5c!.+ !Q© ¢ y-@^ .@,u E./ N ,© + K© K N © * *( $© ,$© iH .D § © () $E: `© ( $* * U §F+RXbf:RG N .B [© [ §!Q _+ U* B-5' .+ !$H . K ./ $"f- B F© . §!? * . E,b[©t § :BHa`"f© &% §!!V ?./3! !R). E , 6§B2fK© $*g © "$§?. 6.$5& § K () An array of Internal octant Internal octant not key1 key2 ¼ keym ¼ in the same affected by the block as b decomposition Decompose octant b Figure 20: B-tree index node. b ¦¨§ `© !:_ .+ ! 6E./ ,$© ./3 ©#!"© !5C.$( - ©#.$X .h"/3!© § B-E:§>5. "/© 2fwbfR[2/* 3 /HQ¦¨§ N 3 E 3h.5¨ £¢ ¤¥¤ © £¨ ¤ ":%j §!.BVU A tiny octant on © _w .+ © $1,R8.$5¨ $©t ¬©#"$3! y¡H ¦¨§ M ©# FM$* .L N .$B s© $ N@ the boundary of © another block E !© "h.$, BHm¦¨§ Q S+3! EK.$5bfR ¨© [[© _ .+ § n£ ppp x) !© 2+©# ! ) §!> BE,§J E § E.-2f M? R> §- .+ !$H ®]E:§MbfRw2$$* 3 U§ $ $1$ N@ . E©# 8 .$©# %VU§ © E,§8 .$© + a .X 3!EE . .+ Figure 17: Case (2): internal neighbors not affected by the decomposition of the boundary octant. §- )E.$+ ,© 5C3! § :)© !5.( © ./°$?.$3! *#*Ubf:R! 3 E,§G § ! " $# H § ¦¨§ :5./%& §!© () E K5./( $ #N$E ?!* . E:b+ Q© $*tV¨BR \]@^ .+ a ( $ ! B G .) © bG $"/ HU\]@^ ¿$µ¶!9^½ $?! .$?B D? R § n^ +R! $()© EB*#*tRE,§$!"/© "fx'?.$3 Rh. E@ 7:;¡ [-[E 3* * RO !© b0 "/X+3 (?: H8 \m@A G© _ ,$ H .+ EB$G§B2f *-"/ +3 (h?6.$5' .$© + : % © [E./ ,$ N .w>? © -R1 )VU§!G$E,§L© !_M .+ X§ $ h()./ N 4.5 B-tree dV=.F 3 EE .$ H>` G 3!* B%c\]@^ D .F?DVU©# ! $ G §!.$ a© F N 3 E 3/H IK ,- ©#.$ 6.$[\]@^ Q- !: © X 3 E,§[V6R) § … Index § ¨\]@^ 6 N 3 E 3¨© m$*tV6R ]? $*#$!EB gH'¦¨§- ]© % 2fR nodes §!G. .$ .8 § F* B-5 $*tV6R § D § [ () … - §L5./( … … … … * " §&H4¦¨§!5C.$/%] § G !./()© $ :5.$( EX5<E .$ ©# bM "/G$EE %m©# $*tV6R h §! $())5.)$* *= §! Leaf Y nodes B-E:§F.$ , © .$ H ¢ E,§ © "65.'6bfR%¢Q N , j5./( § ]\]@^ c. .$ B%VU§ © E,§ © $1© !_8 .+ /HXª .$© + :& © 5.$* *#.BV=B 0 3 E,§M § '¢ HKW5( * B$ ! K .X$w© _>!.+ !/% $# Figure 18: B-tree structure. § § U . EB !3/H]I` § :VU©# $%+ BE,§) § U* B5& .+ a5C.$= §! _ © () B%iV¨) BS 3 + ©#*#*tRM EB$0 WV¨.1u$H ¥>rQ\ * . R VU©t §>( E,§ © !"bfR 2$$* 3!/H £3! §G §!Q./ ,- © ") NR N ( 6? 3lgUEB$E,§ $H d! © "O8 V E.$, °© .1 § F\]@^ F( RLEB$3! w * ©t `.$5] \]@^ .+ ! © .D dV=.)© 5' §!!.+ ! © a5C3 * *tR>. E@ 5.2 Is the etree method feasible? ¢ E3! © B iH © ()© *-*tR %K * © "40E., J( R°EB3 w dV=. ¦j.`$! NV¨c §!©# jS 3 N © ./i%$V=="$ ,- B () § &.$5 !©tlg@ \]@^ U .+ an^ m § 6 $()6* 2/*xc .?U()"$B ©t5g./!a.5 ` ©#§ $ F()B$ 3B F § © () S+3!©tB F .X"$ :, § .+ ! . 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B2+© "f- ©#.$¥£Mn {[ § ®'* () \J © § ¦¨© () ¦¨§ 3 nA{F\¨x n< ,x n<*#(X ,x ¢ ¬=uB¡ +% ¢$o/¡ y+H ¥ q¨¢H ¢ u¢2¢ ¢ ¥ v% q/q$¡ y-o uz/vH ¡ o!u¡ 5.1 Methodology ¬ ¢ ¬'y u/% z$q/z% ¥$yo ¥z/q u$% v$q/v+H u/% u-yq d ., !J . 2$$* 3 ± §!±: () §!.+ © B* @ ¢ uBq% ¥ ¢ +% uByo o!% q$¡/¡ ¢+% o$ofz+H z u/% ofq2¢ ¬=u V¨.*# E © .!%V¨O 2f* .$ B @A? $ B . E () §"/ ,- .$w §- > !.+ 3 E FL5<()© * R.5 ©t 1*t@ ¢ $²¬! !. B § S+3 $bfV6B2f@ () 4() §! 05. Figure 22: Etree-based mesh generator running !.$ "f © .$> ©#(h3 *# © .$ ¢tuBz$£^H time and throughput. ¬© "/3Uy+u] 3 ()( -©#§ & §!=E,§,E © N © E '.$5!B$E,§() §&% IK3!= N 6.$? 2$ © ./X© ¨ §- ¨ § a .$ ,$*3! © !" © ()a . VU§ © E,§ E./() !© J$ © © EB$*>2f.$* 3 () .$5 ¥¡¦b+(¨§ ¢ "/ ,- )./3!Q*-"/ N () §i% ¬=u/%&© $ .B_!© ( - *tRMy+H v ¥$¡Jb+(©§ uBy+p ¥Jb+(GH a./3!"/§!* R± Bb© !"!%D() § N5 °© § .$3! Hwa*t § ./3!"/§1V= §B2f) .>? 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