Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37
Prliminris CGCS p.2/37
Gomtri onstrint prolm C 2 D L BC L AC h α A L AB B 1 A gomtri onstrint prolm onsists o st o gomtri lmnts, {A, B, C, D, L AB, L AC, L BC }, st o gomtri onstrints in twn thm, n st o prmtrs, { 1, 2, α, h}. CGCS p.3/37
Gomtri onstrint solving A gomtri onstrint prolm n rprsnt y prit ϕ in irst orr logi. ϕ(a, B, C, D, L AB, L AC, L BC ) (A, B) = 1 on(a, L AB ) on(b, L AB ) on(a, L AC ) on(c, L AC ) on(d, L AC ) on(b, L BC ) on(c, L BC ) h(c, L AB ) = h (L AB, L BC ) = α (C, D) = 2 Gomtri onstrint solving onsists in proving th truth o th xistntilly quntii prit ϕ tht rprsnts th gomtri onstrint prolm. A B C D L AB L AC L BC ϕ(a, B, C, D, L AB, L AC, L BC ) CGCS p.4/37
Gomtri Constrint Grph 2 C 2 C A D L L AC AC h L AB 1 L BC D α B A h L AB 1 L BC α B A gomtri onstrint prolm n lso rprsnt y mns o gomtri onstrint grph G = (V, E) whr th nos in V r gomtri lmnts with two grs o rom n th gs in E V V r gomtri onstrints suh tht h o thm nls on gr o rom. CGCS p.5/37
Wll-onstrin grphs Thorm 1 (Lmn, 1970) Lt G = (P, D) gomtri onstrint grph suh tht th vrtis in P r points in th two-imnsionl Eulin sp n th gs in D P P r istn onstrints. G is gnrilly wll-onstrin i n only i or ll G = (P, D ), inu sugrph o G y th st o vrtis P P, 1. D 2 P 3, n 2. D = 2 P 3. p p p p r p p p p r r p p p p CGCS p.6/37
Struturlly wll-onstrin grphs A nssry onition or gomtri onstrint prolm to solvl is tht th ssoit onstrint grph must struturlly wll-onstrin. Lt G = (V, E) gomtri onstrint grph. 1. G is struturlly ovr-onstrin i thr is n inu sugrph with m V nos n mor thn 2m 3 gs. 2. G is struturlly unr-onstrin i it is not struturlly ovr-onstrin n E < 2 V 3. 3. G is struturlly wll-onstrin i it is not struturlly ovr-onstrin n E = 2 V 3. CGCS p.7/37
Construtiv Gomtri Constrint Solvrs CGCS p.8/37
Arhittur or Construtiv Gomtri Constrint Solvrs Astrt prolm Prmtrs ss. Anlyzr Inx sltor Astrt pln Inx ssign. Construtor Rliztion CGCS p.9/37
Arhittur or Construtiv Gomtri Constrint Solvrs p 3 2 h 1 l 1 p 1 1 p 2 Prmtrs ss. Anlyzr Inx sltor Astrt pln Inx ssign. Construtor Rliztion CGCS p.9/37
Arhittur or Construtiv Gomtri Constrint Solvrs p 3 2 h 1 l 1 p 1 1 p 2 Prmtrs ss. Anlyzr Inx sltor p 3 Inx ssign. l 1 p 1 p 2 p 3 Construtor Rliztion CGCS p.9/37
Arhittur or Construtiv Gomtri Constrint Solvrs p 3 2 h 1 l 1 p 1 1 p 2 Prmtrs ss. Anlyzr Inx sltor p 3 s 3 = +1 s 1 = 1 l 1 p 1 p 2 p 3 Construtor Rliztion CGCS p.9/37
Arhittur or Construtiv Gomtri Constrint Solvrs p 3 2 h 1 p 1 l 1 1 p 2 1 = 34 2 = 16 h 1 = 14 Anlyzr Inx sltor p 3 s 3 = +1 s 1 = 1 l 1 p 1 p 2 p 3 Construtor CGCS p.9/37
A lustr is st o two imnsionl gomtri lmnts with known positions with rspt to lol oorint systm. Clustrs P 1 P P 2 h L L 2 α L 1 P 3 L 2 3 1 P 1 P 2 h 1 h 2 1 P 1 P 2 L 2 α h 2 P h 1 L 1 CGCS p.10/37
Tr omposition CGCS p.11/37
Thr r grphs tht n tr ompos {,,,,,} CGCS p.12/37
Thr r grphs tht n tr ompos {,,,,,} {,} {,,,} {,,} CGCS p.12/37
Thr r grphs tht n tr ompos {,,,,,} {,} {,,,} {,,} {,} {,} {,} CGCS p.12/37
Thr r grphs tht n tr ompos {,,,,,} {,} {,,,} {,,} {,} {,} {,,} {,} {,} {,} {,} {,} {,} CGCS p.12/37
St ompositions C 1 C2 V 1 V2 C 3 Lt C st with, t lst, thr irnt mmrs, sy,,. Lt {C 1, C 2, C 3 } thr susts o C. W sy tht {C 1, C 2, C 3 } is st omposition o C i 1. C 1 C 2 C 3 = C, 2. C 1 C 2 = {}, 3. C 2 C 3 = {} n 4. C 1 C 3 = {} V 3 Lt G = (V, E) grph n lt {V 1, V 2, V 3 } thr susts o V. {V 1, V 2, V 3 } is st omposition o G i it is st omposition o V n or vry g in E, V () V i or som i, 1 i 3. CGCS p.13/37
Tr omposition Lt G = (V, E) grph. A 3-ry tr T is tr omposition o G i 1. V is th root o T, 2. Eh intrnl no V V o T is th thr o xtly thr nos, sy {V 1, V 2, V 3 }, whih r st omposition o th sugrph inu y V, n 3. Eh l no ontins xtly two vrtis o V. A grph G is tr omposl i thr is tr omposition o G. CGCS p.14/37
Rution nlysis CGCS p.15/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Thr r grphs tht n ru CGCS p.16/37
Lt G = (V, E) gomtri onstrint grph. W in th initil st o lustrs S G = {{u, v} (u, v) E}. Rution nlysis Lt S st o lustrs in whih thr r thr lustrs C 1, C 2, C 3 suh tht {C 1, C 2, C 3 } is st omposition o C. S r S is rution rul whr S = (S {C 1, C 2, C 3 }) C. Th gomtri onstrint prolm rprsnt y th gomtri onstrint grph G is solvl y rution nlysis i S G rus to th singlton {V }. I G is not struturlly ovr-onstrin, th strt rution systm inu y th rution rul r is trminting n onlunt whih implis th uniqu norml orm proprty n noniity. CGCS p.17/37
Th omin o solvl grphs y rution nlysis Lt G = (V, E) wll-onstrin gomtri onstrint grph. Th ollowing ssrtions r quivlnt: 1. G is tr omposl. 2. G is solvl y rution nlysis. CGCS p.18/37
Th omin o solvl grphs y rution nlysis {,,,,, } {, } {,,, } {,, } {, } {,, } {, } {, } {, } {, } {, } {, } {, } CGCS p.19/37
Th omin o solvl grphs y rution nlysis {,,,,, } {, } {,,, } {,, } {, } {,, } {, } {, } {, } {, } CGCS p.19/37
Th omin o solvl grphs y rution nlysis {,,,,, } {, } {,,, } {,, } {, } {, } {, } CGCS p.19/37
Th omin o solvl grphs y rution nlysis {,,,,, } {, } {,,, } {,, } CGCS p.19/37
Th omin o solvl grphs y rution nlysis {,,,,, } CGCS p.19/37
Domposition nlysis CGCS p.20/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Thr r grphs tht n ompos CGCS p.21/37
Lt G = (V, E) gomtri onstrint grph. W in th initil st o lustrs O G = {V }. Domposition nlysis Lt O st o lustrs in whih thr is lustr C suh tht {C 1, C 2, C 3 } is st omposition o th sugrph o G inu y C. O o O is rution rul whr O = (O C) {C 1, C 2, C 3 }. Th gomtri onstrint prolm rprsnt y th gomtri onstrint grph G is solvl y omposition nlysis i O G rus to S G. Th rution rltion o inus n strt rution systm. CGCS p.22/37
Th omin o solvl grphs y omposition nlysis Lt G = (V, E) wll-onstrin gomtri onstrint grph. Th ollowing ssrtions r quivlnt: 1. G is tr omposl. 2. G is solvl y omposition nlysis. CGCS p.23/37
Th omin o solvl grphs y omposition nlysis {,,,,, } {, } {,,, } {,, } {, } {,, } {, } {, } {, } {, } {, } {, } {, } CGCS p.24/37
Th omin o solvl grphs y omposition nlysis {, } {,,, } {,, } {, } {,, } {, } {, } {, } {, } {, } {, } {, } CGCS p.24/37
Th omin o solvl grphs y omposition nlysis {, } {, } {,, } {, } {,, } {, } {, } {, } {, } {, } {, } CGCS p.24/37
Th omin o solvl grphs y omposition nlysis {, } {, } {,, } {, } {, } {, } {, } {, } {, } {, } CGCS p.24/37
Th omin o solvl grphs y omposition nlysis {, } {, } {, } {, } {, } {, } {, } {, } {, } CGCS p.24/37
Rormulting Own s lgorithm CGCS p.25/37
Own s lgorithm rlis on omputing trionnt omponnts... SPLIT REDUCE SPLIT REDUCE SPLIT... ut tr h split som wll hosn gs shoul rmov to ontinu th pross. It is iiult to unrstn whih gs shoul rmov n th rson why thy shoul rmov. CGCS p.26/37
Whih gs n why shoul thy rmov? Th trionnt omponnts lgorithm suivis th grph n s virtul gs to prsrv onntivity proprtis. To urthr suivi, Own s lgorithm rmovs virtul gs t ny rtiultion pir with no singl g n xtly on mor omplx sugrph. CGCS p.27/37
Th proprty to prsrv in omposition lgorithms is th iit Wht is ssntil to prsrv in th grph suivision pross is rigiity proprtis, not onntivity proprtis. Diit = 0 Diit untion o grph G = (V, E) is in s Diit(G) = (2 V 3) E Diit = 1 Diit = 0 At vry grph split, iit vlu shoul mintin. Thus nw gs must to ulill this rquirmnt. CGCS p.28/37
Two rsults show how iit n mintin Lt G wll-onstrin onstrint grph n G n G sprting grphs o G. Thn Diit(G) = Diit(G ) + Diit(G ) 1 I Diit(G ) > Diit(G ), G is unr-onstrin n G is wll-onstrin. Thror To mintin wll-onstrintnss on virtul g must to th sprting grph G. Th virtul g susums th rigiity proprtis u to th sprting grph G CGCS p.29/37
Exmpl o iit ompnstion Diit = 0 Compnstion Diit = 0 Diit = 1 Diit = 0 CGCS p.30/37
Exmpl o iit ompnstion Diit = 0 Compnstion Diit = 0 Diit = 1 Diit = 0 CGCS p.30/37
A nw ormultion o Own s omposition lgorithm A lr n simpl pplition o ivi-n-onqur. Uss sprting pirs to suivi th grph. Applis iit ompnstion to mintin rigiity strutur. un Anlysis(G) i Trionnt(G) thn S := BinryTr(G, nulltr, nulltr) ls G 1,G 2 := SprtingGrphs(G) i Diit(G 1 ) > Diit(G 2 ) thn G 1 := AVirtulEg(G 1 ) ls G 2 := AVirtulEg(G 2 ) i S := BinryTr(G, Anlysis(G 1 ), Anlysis(G 2 )) i rturn S n CGCS p.31/37
Th rsult o th nw ormultion is n s-tr Th nw lgorithm yils inry orm o th Own s tr. W nm it s-tr. SPLIT REDUCE REDUCE SPLIT SPLIT CGCS p.32/37
Th omin o Own s mtho Lt G = (V, E) wll-onstrin gomtri onstrint grph. Th ollowing ssrtions r quivlnt: 1. G is tr omposl. 2. G is s-tr omposl. CGCS p.33/37
Th omin o Own s mtho {,,,,,} {,} {,,,} {,,} {,} {,} {,,} {,} {,} {,} {,} {,} {,} CGCS p.34/37
Th omin o Own s mtho {,,,,,} {,} {,,,} {,,} {,} {,} {,,} {,} {,} {,} {,} {,} {,} CGCS p.34/37
Th omin o Own s mtho {,,,,,} {,} {,,,} {,,} {,} {,} {,,} {,} {,} {,} {,} {,} {,} CGCS p.34/37
Domin quivln o onstrutiv mthos CGCS p.35/37
Construtiv mthos hv th sm omin Lt G = (V, E) wll-onstrin gomtri onstrint grph. Th ollowing ssrtions r quivlnt: 1. G is tr omposl. 2. G is s-tr omposl. 3. G is solvl y rution nlysis. 4. G is solvl y omposition nlysis. Th lss o grphs ulliling th ov proprtis is nm th onstrutivly solvl grphs lss. CGCS p.36/37
Summry W hv introu th tr omposition o grph. Tr omposl grphs hrtriz th omin o rution nlysis, omposition nlysis n Own s mtho. Th omins o onstrutiv mthos r th sm. W hv lrii n rormult Own s lgorithm. Th rormult lgorithm pplis ivi-n-onqur shm n it is onptully simplr. Th output o this lgorithm is n s-tr. CGCS p.37/37