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" t ` fff33` 3KI3ff` 33ff` /p` 3%*3|` Jy3fff3f` 3ff3̙` 33ff33` DDyq3f` ̙3n` w3ff` }ff>?" dd@,?nKd@ P nA@F`d n?" dd@   @@``PR"   @ ` `2p>> ? 7   (     6d#" `3 00  6Л #" ``   B*   Tĝd#" `P 20  <̣d#" `U 00  c $d"UY *  s *D"YW *  s *"UY *  s *" *  c $L#" `VS 00  c $"Y *  s *"X *  < #" `  `  ['Cliquez pour modifier le style du titre( (H  0X " `p  vCliquez pour modifier les styles du texte du masque Deuxime niveau Troisime niveau Quatrime niveau Cinquime niveau4 wH  0޽h ? }ff___PPT10i. 07+D=' = @B +  Pixeld  { s $ (  $bT  $ " $ Tw-d #"   20 $ c $lz- "9)e  00b e  $# "e  $ S ~-"ie  00 $ S -"9) 00 $ S ă-"0 00  $ S t-"?e  00  $ S h-") 00  $ S D-"?G 00  $ S -"oG 00  $ S  -"9G 00 $ S X-"iA 00 $ S -"A? 00 $ 6<- #" `` ` - >*  $ 6- #" ``  - @*  $ 6T- #" `` ` - B*  $ 0Ԭ- "P   - ['Cliquez pour modifier le style du titre( ( $ 0- " P  - l8Cliquez pour modifier le style des sous-titres du masque9 9H $ 0޽h ? }ff___PPT10i. 07+D=' = @B + 0 p.(    0Z( P   ( P*    0-    ( R*  d  c $ ?  (:  0a(  0 ( vCliquez pour modifier les styles du texte du masque Deuxime niveau Troisime niveau Quatrime niveau Cinquime niveau4 w  60f( _P  ( P*    6k( _  ( R*  H  0޽h ? 3380___PPT10.MA XPp(  x  c $-$  - x  c $$-$ h  - L  C $A lirmm( 0 f\  C 4Alogo-INRIA-RA R H  0޽h ? 33___PPT10i.9O+D=' = @B +F  %\z(  \ \ 0d^:e A Motivations (2   \ 0_: 8My schedule problem (as a temporal constraint network) :9 29H   \ Bd:jJB|%% RTalk  \ Bpi:jJ|% 4Mail \ B\m:jJ &  4Boss  \ B8q:jJ DH6 PSlidesdR  \ <G;HI;jJ+ R  \ s *jJ >&d  \ <G*H~IdjJ R  \@ s *jJ R \ s *jJ+ \ 6lv:S"`?t Smeets(2 \ 6z:S"`?T Tmet-by(2 \ 6w:S"`?}iW  Tequals(2 \ 6:S"`?`P  Vfinishes (2  \ 6:S"`?0O  Tstarts(2 \ 6D:S"`?V   Smeets(2 \ 68:S"`? q  Smeets(2 \ 6ܑ:S"`?J H  Vcontains (2  !\ <:  P : " "\ C x:))?  @H@ ,$D 0 3 QUESTION Is there a possible schedule solution ? 64 )>    #\ 0$: ( 2" %\ C x:))? ( (,$D 0  This problem is NP-complete *> H \ 0޽h ?_\ \ \\ \ \\ \ \\\ \\\\ }ff\T___PPT104.L9+* MD' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%\%(+p+0+"\: ++0+%\: +/  ? 7  ` (  `  ` 0\:e n Related Work (2  $O8 g `g ` 6 :&g t Generate all instantiations of relations. Test networks consistency using path-consistency [Allen:83]. Use optimizations: constraints and relations ordering [Van Beek and Manchak:96]&       ` 0+: yInstantiate constraints : 2$  "  ` Z/:))?,$D 0 [To handle mixed qualitative/quantitative relations : instantiate variables during backtrack\\b&   "  ` ZT7:))? ,$D 0 LUse backtracking optimizations such as forward checking [Schwalb et.al.:97]MM  l  -  ` - ,$D 0 ` 6(F: -   Finitely partition the infinite domains of interval values e.g., using an interval end-points encoding [Thornton et.al.:05] Backtrack on these finite partition.*<E&(      ` 0t;:   eInstantiate variables : 2  H ` 0޽h ? }ff___PPT10p.O++YD' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* `%(+p+0+ `: ++0+ `: +=     #di (  d d 0L:e WOutline(2  d 0l:P] -Instantiate variables with abstract intervals. 2.6    d  `A ? "`Gq :) d 0H:@,$D 0  Backtrack with abstract intervals Forward checking with abstract intervals Forward checking (version 2) with abstract intervals  nA     8 q4  !d4q  d 0:P]  'Instantiate variables with bi-intervals( 2(6    d 0d:   < Conclusion 2  d"  `A ? "`q4  : d"  `A ? "`q   :s  d 0t:8   A Backtrack with bi-intervals Forward checking with bi-intervals  B nAB   B "d s *DoX,$@ 0 #d 0@:x,$ 0 r(in the paper) 2,H d 0޽h ? }ff  ___PPT10 .`g +D ' = @B D; ' = @BA?%,( < +O%,( < +DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"d%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*dN%(D' =%(D' =%(Da' =4@BB BB%(D' =1:B 0.5*3>%Bstyle.opacity= B PPT<*!dD' =-6B image0BIE*v3=2Bopacity: 0.5 BIE<*!d+8+0+#d: +<%  UM@,c@(    0:] k Abstract intervals(2  |  0" An abstract interval [x,y] is a pair of elements from a totally ordered list. They encode an infinite number of intervals of real numbers.2 2ub<  o   0) b Definition : 2    0 / _ Example : 2  1 Bx2jJ   RTalk  2 B6jJ   4MailR 3 s *jJ QQ  4 69S"`? C  Smeets(2l @ <  a @ < ,$D 0ZB 6 s *DԔ ZB 8 s *DԔ   : 0>@  917H10 2 ZB ; s *DԔ  ZB > s *DԔ ZB ? s *DԔ    C 0$C\ <  918H30 2  D 0F x  8Talk 2 E 0tJ   8Mail 2 J 0&   917H30 2 ^B  6DԔ  l H   b H  ,$D 0ZB N s *DԔ x x ZB O s *DԔ x x  P 0@TH ( u  917H25 2 ZB Q s *DԔ P  ZB R s *DԔL  ZB S s *DԔP   T 0Y q  719H 2  U 0L\   8Talk 2 V 0_ p  8Mail 2 W 0@c u  917H40 2  X 0f( d,$ 0 9. . . 2 /l    c   ,$D 0`B  0DԔ p    6kC"?  7A 2   64oC"?   7B 2`B K 0DԔ t  `B L 0DԔPp P  M 6sC"?   7C 2ZB \ s *DԔ  ZB ] s *DԔ P  _ 0v   <Talk 2 ` 0|   <Mail 2H  0޽h ? 123 }ff  ___PPT10g ."Pj"+wVD ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(+8+0+X +2 B~B]e,.@(  ,LB [, c $D  LB Z, c $DLB Y, c $D>>XB , 0DԔl8Yl , <Ԙ,$@ 0 S[a,b]  , <8 ( , <$el  (X , 0  ^R  , 6GHI46 L  , c $  ^  , 6G*H~IdA6 L  ,@ c $AeL  , c $AA , 0P\C [ possibilities 2`  , 0L|c Xinstantiations 2 , <C"?,$@  0 O[a,b]`  , <0C"?" ,$@  0 O[b,c]`  , <C"?" ,$@z 0 O[d,a]`  , <C"?M ,$@E 0 O[e,c]`  , <C"? P ,$@Q 0 O[a,c]`  , <C"?~ ,$@[ 0 O[f,c]`  , <C"?,$@g 0 O[b,g]` z ,@ S  ",$@ 0z , S ",$@y 0z ,@ S  ,$@D 0z ,@ S  ,$@P 0z , S  4 ,$@Z 0z , S  ,$@d 0  , <C"? 0 ,$J 0 KX` z , S    ,$@I 0  , <C"? m ,$W 0 KX`    , <\C"?  ` ,$@b 0 KX`   !, <HC"?  ,$p 0 KX` z ", S   ,$@V 0z #,@ S 3 4  ,$@a 0z $, S  ,$@o 0 %, 03\ Smeets(2 &, 08 Tmet-by(2 ', 0, Tequals(2 (, 0d Vfinishes (2  ), 0y9 Tstarts(2 *, 0< Smeets(2 +, 0Z 0  Smeets(2 ,, 0P ]  Vcontains (2 B -, s *Do ,$D 0B .,@ s *DԔ8& & ,$@ 0B /,@ s *DԔ6 ,$@ 0 0, 0~ p e,$ 0 1a 2 1, 0~$  e,$ 0 1b 2B 2, s *DoX $ X ,$D 0B 3, s *Do ,$@ 0B 4,@ s *DԔ8 ,$@ 0 5, 0(| j c,$ 0 1c 2B 6,@ s *DԔ6,$@ 0 7, 0|!~b@ e,$ 0 1d 2B 8, s *Do0 ,$@ 0B 9, s *Do ,$@! 0 :, 0%,$ 0 S[b,c] 2  ;, 0,* ,$ 0 T,[d,a] 2 B <, s *Do ,$@) 0B =,@ s *DԔ8 ,$@( 0 >, 0.~h  e,$* 0 1e 2B ?, s *Do* ,$@- 0B @,@ s *DԔ6p p ,$@2 0 A, 0h3~$ e,$3 0 1f 2B B, s *Dot ,$@5 0B C,@ s *DԔ6 ,$@: 0 D, 08b| ZI,$; 0 1g 2B E, s *Do ,$@< 0 F, 0=.,$+ 0 S[e,c] 2  G, 0$A,$. 0 T,[a,c] 2  H, 0DX? ,$6 0 S[f,c] 2  I, 0:X? ,$= 0 T,[b,g] 2 B J, s *Do ,$@B 0 K, 04M&  ,$K 0 K 2` B L, s *Do& ,$@R 0B M, s *Dox ,$@\ 0B N, s *Do ,$@j 0B O, s *Do  ,$@{ 0 T, 0dQe Backtrack on abstract intervals (2  $   U, 0Xx0 6Talk 2 V, 0D8 6Mail 2 W, 0V 6Boss 2 X, 0T" n  RSlides 2 ], 04g 7 b ,$  0 8talk 2 ^, 0$[ ,$' 0 8mail 2 _, 04ko A  /,$G 0 8boss 2 a, 0q F ,$ 0 8mail 2 b, 0fa> !,$ 0 8talk 2 c, 0dxY ,$ 0 8mail 2 d, 0l[E ,$ 0 8mail 2 e, 0<cb 2 #,$, 0 8boss 2H , 0޽h ?,, ,,, ,,, ,,, ,,, ,,,,,,,,,, ,,, ,,, ,,, ,,, ,,",, ,#,,!,$, }ff___PPT10. c+7D' = @B DB' = @BA?%,( < +O%,( < +D ' =%(DK ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*b,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-,%(D+ ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*],%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*b,%(D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*,D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*-,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2,%(D' =%(D3' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*c,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:,%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*4,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*5,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*3,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*c,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d,%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*6,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*7,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*8,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5,%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*:,D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*d,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^,%(D' =%(D3' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*e,%(Dx' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*=,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*>,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*<,%(D|' =%(D$' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*?,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H,%(D ' =%(DE ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*@,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*B,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*A,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I,%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*C,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*D,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*E,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*F,D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*e,%(Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K,%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*K,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*J,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*=,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*>,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L,%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*G,D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*F,Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*",%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*K,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*L,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*M,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A,%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*G,D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*H,DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K,%(D"' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*H,D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*I,D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*@,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*M,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*N,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*A,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*K,%(Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K,%(D!' =%(D!' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*K,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*N,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*9,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*C,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*D,%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*4,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*5,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*O,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*F,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*I,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*H,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*G,%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*;,D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*:,D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*a,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*^,%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*_,%(++0+, ++0+, ++0+, ++0+, ++0+, ++0+, ++0+, ++0+, ++0+, ++0+!, ++0+0, ++0+1, ++0+5, ++0+5, ++0+5, ++0+5, ++0+7, ++0+7, ++0+7, ++0+:, ++0+:, ++0+:, ++0+;, ++0+;, ++0+>, ++0+>, ++0+>, ++0+>, ++0+A, ++0+A, ++0+A, ++0+A, ++0+D, ++0+D, ++0+D, ++0+D, ++0+F, ++0+F, ++0+F, ++0+F, ++0+G, ++0+G, ++0+G, ++0+G, ++0+H, ++0+H, ++0+H, ++0+H, ++0+I, ++0+I, ++0+I, ++0+K, ++0+K, ++0+K, ++0+K, ++0+K, ++0+K, ++0+K, ++0+K, ++0+], ++0+^, ++0+^, ++0+_, ++0+_, ++0+a, ++0+b, ++0+b, ++0+c, ++0+c, ++0+d, ++0+d, ++0+e, ++0+e, + QzQ~DM(  DLB D c $D  LB D c $DLB D c $D>>B lD s *Do ,$D 0B iD s *Do0 ,$@ 0 D <) W[a,b]  D <. ( D < 2el  (X D 0  ^R  D 6GHI46 L  D c $  ^  D 6G*H~IdA6 L  D@ c $AeL  D c $AA D <d6C"? O[a,b]`  D <;C"?"  O[b,c]`  D <>C"?"  O[d,a]`  D <BC"?M  O[e,c]`  D <@C"? P  O[a,c]`  D <4IC"?~  O[f,c]`  D <LC"? O[b,g]`  D <@KC"?F,$@ 0 O[h,a]`  D <SC"?s,$@5 0 O[d,i]`  D <HWC"? w,$@F 0 O[d,b]`  D <$[C"?,$@O 0 O[d,j]` F D@ S  "F D S "F D@ S  F D@ S  F D S  4 F  D S  z !D@ S ,$@ 0z "D@ S ),$@4 0z #D S ,$@E 0z $D S Z,$@N 0 %D <`aC"? 0  KX` F &D S     'D <eC"? m  KX`  (D <iC"?  `  KX`  )D <0kC"?   KX`   *D <DqC"? V ,$? 0 KX`   +D <tC"?  ,$L 0 KX` F ,D S   F -D@ S 3 4  F .D S  z /D S )) ,$@> 0z 0D S  ,$@K 0z S   1D 0 K,$@, 0zB 2D s *DoC"?  zB 3D s *DoC"?Y^ ^ zB 4D s *DoC"?  zB 5D s *DoC"?  zB 6D s *DoC"?  zB 7D s *DoC"?0  zB 8D s *DoC"?0  zB 9D s *DoC"?S0 S 6z  D  :D  d3,$@^ 0ZB ;D s *Do  ZB D s *DoC"?  zB ?D s *DoC"? C zB @D s *DoC"?X  zB AD s *DoC"? ; zB BD s *DoC"?X  zB CD s *DoC"? C zB DD s *DoC"?D D  ED 0(3\ Smeets(2 FD 0 Tmet-by(2RB MD@ s *DԔ4. . 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=1:Bhidden*o3>+B#style.visibility<*40%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*90%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*:0%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,0%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*J0%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0%(+ +0+0 ++0+0 ++0+0 ++0+ 0 ++0+!0 ++0+%0 ++0+'0 ++0+*0 ++0+*0 ++0+*0 ++0++0 ++0++0 ++0+,0 ++0+,0 ++0+,0 ++0+-0 ++0+-0 ++0+.0 ++0+.0 ++0+/0 ++0+/0 ++0+00 ++0+00 ++0+10 ++0+10 ++0+20 ++0+20 ++0+40 ++0+40 ++0+60 ++0+60 ++0+80 ++0+80 ++0+:0 ++0+:0 ++0+A0 ++0+A0 ++0+B0 ++0+B0 ++0+I0 ++0+J0 ++0+J0 ++0+K0 ++0+L0 ++0+L0 ++0+M0 ++0+M0 ++0+N0 ++0+N0 ++0+O0 ++0+O0 ++0+~0 ++0+~0 ++0+0 ++0+0 + SUKUPQ(  P P 0 ,$) 0 Tslides 2 P 0" n  RSlides 2 P 0 6Boss 2 P 04] &Forward Checking on abstract intervals'(2' 6 LB P c $D  LB P c $DLB P c $D>> P <C"? O[a,b]`  P <4C"?   O[b,c]`  P <C"?  O[d,a]` F P@ S   F P S  RB P@ s *DԔ>& & RB P@ s *DԔ<  P 0 t k 1a 2  P 08 k 1b 2RB !P s *Doa $ a RB "P@ s *DԔ>  #P 0t j m 1c 2RB $P@ s *DԔ< %P 0D b@ k 1d 2 &P 0  S[b,c] 2  'P 0  X,[d,a] 2  (P 0`  n ,$7 0 S[a,b] 2  )P 0 J`1,$ 0 S[i,a] 2  *P 0 Nr5,$ 0 T,[d,j] 2  +P 0x .,$  0 S[d,b] 2  ,P 0T p,$ 0 T,[d,k] 2  -P 0: ,$ 0 S[d,c] 2  .P 0 v ,$ 0 T,[d,l] 2 B /P@ s *DԔ> ,$@  0 0P 00# n L U,$  0 1j 2B 1P@ s *DԔ@,$@ 0 2P 0D' n,$ 0 1i 2B 3P@ s *DԔ> ,$@ 0 4P 0X+  vm,$ 0 1l 2 5P <. C"?g  KX` F 6P S  RB 7P@ s *Do & B 8P@ s *DԔ<$ $ ,$@ 0 9P 0(3  k,$ 0 1k 2B :P s *Do& ,$@ 0B ;P s *Do ,$@ 0B P s *Do ,$D 0B ?P s *Do ,$@ 0B @P s *Do& ,$@ 0 AP <9 C"?,$@- 0 O[d,j]`  BP <4> C"?",$@: 0 O[d,b]`  CP <A C"?V,$@C 0 O[d,k]`  DP <XE C"?,$DZ 0 O[d,c]`  EP <H C"?)_,$@h 0 O[d,l]`  FP <L C"?L,$@  0 O[i,a]` z GP@ S ,$@ 0z HP@ S ,$@, 0z IP@ S ,$@9 0z JP@ S ,$DJ 0z KP S \,$@Y 0z LP S ,$@g 0 MP <Q C"? J ,$@$ 0 O[a,b]` z NP@ S  ,$@# 0z S   OP  Q  ,$@' 0zB PP s *DoC"?  zB QP s *DoC"?Y^ ^ zB RP s *DoC"?  zB SP s *DoC"?  zB TP s *DoC"?  zB UP s *DoC"?0  zB VP s *DoC"?0  zB WP s *DoC"?S0 S z XP@ S   ,$@& 0  YP <(Y C"? F ,$6 0 KX` z ZP@ S  ,$@5 0  [P <X^ C"? W ,$B 0 KX` z \P@ S  ,$@A 0 ]P <lb C"? V ,$@L 0 O[a,b]`  ^P <f C"?  ,$@^ 0 O[a,b]`  _P <i C"? (^ ,$@l 0 O[a,b]` z `P S  ,$@K 0z aP S \] ,$@] 0z bP@ S  ,$@k 06z  D  cP  B ,$@P 0ZB dP s *Do  ZB eP s *Do  ZB fP s *Do 8 zB gP s *DoC"?  zB hP s *DoC"? C zB iP s *DoC"?X  zB jP s *DoC"? ; zB kP s *DoC"?X  zB lP s *DoC"? C zB mP s *DoC"?D D z nP S  . ,$@O 0z oP@ S  X] ,$@_ 0z pP@ S   ,$@m 0B qP s *Dow& w,$@" 0B rP s *Do ,$@1 0 sP 0t ,  ,$3 0 K 2` B tP s *Do| ,$D> 0B uP s *Do$ ,$@I 0B vP s *Do ,$@X 0B wP s *Do ,$@f 0 |P 0y  (  8talk 2 }P 0~  8  8mail 2 ~P 0 \@  8boss 2 P 0 e < %,$! 0 8boss 2 P <  W[a,b]  P <  ( P <0 el  (X P 0  ^R P 6GHI46 L P c $  ^ P 6G*H~IdA6 L P@ c $AeL P c $AA P 0\ 3\ Smeets(2 P 0  Tmet-by(2 P 0  Tequals(2 P 0  Vfinishes (2  P 0 y9 Tstarts(2 P 00 < Smeets(2 P 0` Z 0  Smeets(2 P 0d  ]  Vcontains (2  P 0p x0 6Talk 2 P 0x 8 6Mail 2XB P 0DԔl8Yl P 0| \C [ possibilities 2`  P 0 |c Xinstantiations 2H P 0޽h ?PPPPPPP5P6P PFPGP PAPHP PBPIP PCPJP PDPKPPEPLPFPMPNPMPXPAPYPZPBP[P\PCP]P`PDP^PaPEP_PbP]PnP^PoP_PpPPPPPPPPPPPPPPPP }ff___PPT10.æ*߄+WrD' = @B Dһ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2P%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*:P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0P%(DW' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*;P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+B#style.visibility<*,P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*+B#style.visibility<*=P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9P%(DW' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*=P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>P%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*>P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4P%(D ' =%(Dl ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*?P%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*~P%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*)PD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*GP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*FP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(DG ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*qP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*NP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*MP%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*XP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*OP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*qP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*qP%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*@P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*AP%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<**PD' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*)PD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*rP%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*sP%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*(P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ZP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*YP%(Du ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(P%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*sP%(D' =4@BBBB%(D' 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=%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*]P%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*qP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*nP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*cP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*uP%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*,PD' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*-PD' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*qP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*vP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*KP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*DP%(D/ ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*qP%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*aP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*oP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*vP%(D' =A@BBBB0B %(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*-PD' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*.PD' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*qP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*wP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*LP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*EP%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D/ ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*qP%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*bP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*pP%(D' =A@BBBB0B %(D' =,54*3>!Bstyle.color='`B@BPB<*(PD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(+x +0+P  ++0+P  ++0+P  ++0+P  ++0+P  ++0+P  ++0+P  ++0+(P  ++0+(P  ++0+(P  ++0+(P  ++0+(P  ++0+(P  ++0+(P  ++0+(P  ++0+(P  ++0+ (P  ++0+ (P  ++0+)P  ++0+)P  ++0+)P  ++0+*P  ++0+*P  ++0+*P  ++0++P  ++0++P  ++0++P  ++0+,P  ++0+,P  ++0+,P  ++0+-P  ++0+-P  ++0+-P  ++0+.P  ++0+.P  ++0+0P  ++0+2P  ++0+4P  ++0+9P  ++0+AP  ++0+BP  ++0+CP  ++0+DP  ++0+EP  ++0+FP  ++0+MP  ++0+YP  ++0+[P  ++0+]P  ++0+^P  ++0+_P  ++0+sP  ++0+sP  ++0+sP  ++0+sP  ++0+~P  ++0+P  +L  a Y  (    0tP e WOutline(2   6T P]  'Instantiate variables with bi-intervals( 2(6     <A ?"`4q   8 q  q    0X  -Instantiate variables with abstract intervals. 2.6     "  `A ? "`q2    0|c  M Backtrack with abstract intervals Forward checking with abstract intervals  N nAN     6g    < Conclusion 2   <A ?"`q    q  6 -8   A Backtrack with bi-intervals Forward checking with bi-intervals  B nAB   H  0޽h ? }ff___PPT10k.`g +D?' = @B D' = @BA?%,( < +O%,( < +D1' =%(%(D' =%(Da' =4@BB BB%(D' =1:B 0.5*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*v3=2Bopacity: 0.5 BIE<*+X  S"K"05!(    0\r ] \ Bi-intervals (2   j  0v P A bi-interval <[A,B],[C,D]> on a list of points encodes a set of abstract intervals [u,v] where u[A,B] and v[C,D].fu 2bL         0$  b Definition : 2    0  _ Example : 2 XB  0DԔ x RB  s *DԔX hh RB  s *DԔX  RB  s *DԔX `` RB  s *DԔX  RB  s *DԔX   0  w  3A 2  0  x`w  3B 2  0t    3C 2  0T  p  3D 2  0З  H 0   3E 2  0L    0PThe interpretation of the bi-interval <[A,B],[C,E]> into abstract intervals is :Q(2Q      0    g <[A,B],[C,E]> 2  B  s *DԔ\  ,$@ 0  0  p ,$ 0 3Z 2B  s *DԔZ  ,$@ 0B  s *DԔZ  ,$@ 0  0X   ,$ 0 3X 2  08  [C ~ ,$ 0 3Y 2A l N  / N,$D 0  0  v}  S[A,C] 2   0    U, [A,X] 2    0    U, [A,D] 2  ! 0 6 F   U, [A,Y] 2  " 0  }  U, [A,E] 2  # 0X T z; S[Z,C] 2  $ 0H X  ? U, [Z,X] 2  % 0(  V = U, [Z,D] 2  & 0 : V J = U, [Z,Y] 2  ' 0  T ; U, [Z,E] 2  ( 0 z S[B,C] 2  ) 0    U, [B,X] 2  * 0l    U, [B,D] 2  + 0 : J  U, [B,Y] 2  , 0D   U, [B,E] 2  - 0h N \w  4{ 2 . 0  4} 2l 0 h  40 h ,$@ 0 0 0 8 X  Q[ 2(  2 0`0 h  Q] 2( l 8 @  58 @ ,$@  0 1 0@8 P  Q[ 2(  3 0p 8 @  Q] 2( H  0޽h ? }ff66___PPT105."Pj"+RD1' = @B D1' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D ' =%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D/' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =%(D>' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =%(D' =%(D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D/' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =%(Dp' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +  FzFdn*D(    0|-d  $,$ 0 8mail 2 , 00d  $,$" 0 8boss 2  0,5I.0 ( 2  07e {Backtrack on bi-intervals(2 $  Lz DP P  D ,$@! 0ZB  B s *Do P P`B  B 0DoDP P  0X<d@ $,$ 0 8talk 2  0=dX $,$ 0 8mail 2LB  c $D  LB  c $DLB  c $D>>  <DC"?,$@ 0 O[a,b]`   <LC"?  ,$@ 0 O[b,c]`   <PC"? ,$@] 0 O[d,a]` z @ S   ,$@ 0z  S  ,$@\ 0B @ s *DԔ>& & ,$@  0B @ s *DԔ< ,$@  0  08V| t c,$  0 1a 2  0Z| c,$  0 1b 2B  s *DoZ $ Z ,$@  0B @ s *DԔ: ,$@ 0  0^~ j e,$ 0 1c 2B @ s *DԔ<,$@` 0  0c|jH c,$a 0 1d 2B  s *Do ,$@ 0,  0gx,$ 0 <[b,b],]b,[>,B B ! 0tnQ,$ 0 <]-,a[,[a,a]>X 2 B # 0`wYq,$# 0 <]-,b[,[c,c]>X 2 ( $ 0J6 ,$$ 0 <[b,b],]c,[>@ 2   % 0&  ,$2 0 K 2` B (@ s *Do & ,$@^ 0 ) 0| ) w ,$ 0 8talk 2 * 0Hj  ,$ 0 8mail 2 + 0 T ,$_ 0 8mail 2f - <,$@ 0 <]-,[,]-,[>n  . << ( / <\el  (X 0 0  ^R 1 6GYNH`IYN46 L 2 c $  ^ 3 6G*H~Id<A6 L 4@ c $<AeL 5@ c $AF 6 03\ Smeets(2 7 0< Tmet-by(2 8 0$ Tequals(2 9 0ܵ Vfinishes (2  : 0y9 Tstarts(2 ; 0T< Smeets(2 < 0<Z 0  Smeets(2 = 0@ ]  Vcontains (2  > 0x0 6Talk 2 ? 08 6Mail 2 @ 0 6Boss 2 A 0X" n  RSlides 2XB B 0DԔl8Yl C 0\C [ possibilities 2`  D 0||c Xinstantiations 2 F 0d p-2 2  G 0@5c b& 2 B H 0DoLd,$@ 0Lz D&$ & I D$ ,$@ 0ZB J s *Do &$ &`B K 0DoD& &Lz  &t& L  p,$@ 0ZB M s *Do & &`B N 0Do &t&Lz  xbx Q  b,$@% 0ZB RB s *Do x x`B SB 0Do xbx T <lC"?M ,$D/ 0 O[e,c]`  U <C"? P ,$@8 0 O[a,c]`  V <|C"?~ ,$@@ 0 O[f,c]`  W <C"?,$@L 0 O[b,g]` z X@ S  ,$@. 0z Y@ S  ,$@7 0z Z S  4 ,$@? 0z [ S  ,$@K 0  \ <@C"? 0 ,$1 0 KX` z ] S    ,$@0 0  ^ <p C"? m ,$; 0 KX`   _ <C"?  ` ,$E 0 KX`   ` <C"?  ,$S 0 KX` z a S   ,$@: 0z b@ S 3 4  ,$@D 0z c S  ,$@R 0B d s *Do ,$@) 0B e s *Do& ,$D9 0B f s *Dox ,$@A 0B g s *Do ,$@M 0 h 0o A  /,$* 0 8boss 2B i@ s *DԔ8 ,$@+ 0 j 04~h  e,$, 0 1e 2B k@ s *DԔ6p p ,$@B 0 l 0~$ e,$C 0 1f 2B m@ s *DԔ6 ,$@N 0 n 0"b| ZI,$O 0 1g 2H  0޽h ?-01/02.03./4-.5 TX UY VZ W[ T\]U^aV_bW`c }ffƹ___PPT10.4rC+hD2' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*-Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D/' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*L%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*I%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<* Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(DW' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*Q%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*j%(D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*#D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D ' =%(DC ' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*i%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*j%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D ' =%(DG ' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*e%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*k%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*l%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*f%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*n%(D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*#D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*$Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =%(D]' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*m%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*n%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*g%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*#%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<* D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*!D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<**%(+@ +0+  ++0+ : ++0+,- ++0+,- ++0+ - ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+! ++0+! ++0+# ++0+# ++0+$ ++0+$ ++0+$ ++0+% ++0+% ++0+% ++0+% ++0+% ++0+% ++0+% ++0+% ++0+) ++0+* ++0+* ++0++ ++0+- ++0+- ++0+T ++0+U ++0+V ++0+W ++0+\ ++0+^ ++0+_ ++0+` ++0+h ++0+h ++0+j ++0+j ++0+l ++0+l ++0+n ++0+n + PRHR0~N(    0e {Backtrack on bi-intervals(2 $  LB  c $D  LB  c $DLB  c $D>>  <C"? 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"`q2 '  0' M Backtrack with abstract intervals Forward checking with abstract intervals  N nAN     6'P]  'Instantiate variables with bi-intervals( 2(6     <A ?"`q4  'y  6'8   A Backtrack with bi-intervals Forward checking with bi-intervals  B nAB   H  0޽h ? }ff___PPT10k.`g +D?' = @B D' = @BA?%,( < +O%,( < +D1' =%(%(D' =%(Da' =4@BB BB%(D' =1:B 0.5*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*v3=2Bopacity: 0.5 BIE<*+Z.  q-i-gyy'(  LB  c $D  0(] @ Conclusion (2  LB  c $D& 6 R  s *R  s * R  s * R  s *kaR  s *aR  s *aR  s *aR  s *( aR  s * aR  s * aR  s *aR  s * R  s * ( L @ c $o L @ c $o L  c $o )L  c $o DL @ c $oOL  c $oOL @ c $o dL @ c $o L  c $o L  c $o  L  c $oade L  c $oa   0' r Backtrack on abstract intervals (2 $   ! 0(m r &Forward checking on abstract intervals'(2'6  " 0(  yBacktrack on bi-intervals(2$   # 0((m   Forward checking on bi-intervals!(2!6 R $ s *R % s *-  !R & s *!R + s *7bR , s *.bR - s *bR . s *bR / s * R 0 s * 7L 5@ c $o ^L 6 c $o^&L 7@ c $o!s&L 8@ c $o!&L 9@ c $o!&L : c $o!&L ; c $obst L < c $ob R = s *R7aR > s *eJ`R ? s * Q6R @ s *eJL A c $o!&L B c $o!&L C@ c $oa L D c $o`R E s *C R F s *  ^ R G s * ^ R I s *( F+ R J s *( 4 R K s *( ! R L s *I " L N@ c $oC I L O c $oC I0 L P@ c $o^ 0( L Q@ c $o^ 0( L R c $o^ 0( L T c $o I R V s *0 \A R X s *G [@ L Z c $o^ 00 L \@ c $o G R ^ s *B R _ s * ] R ` s * x] R a s *' ~c R b s *'  R c s *'  R d s *'  R e s *'  R f s *'  R g s *'  R h s *'  R i s *H  R j s *G  L k@ c $o] ' L l@ c $o] ' L m c $o] !' L n c $o] <' L o@ c $oB G L p c $oB G L q@ c $o] \' L r@ c $o] x' L s c $o] ' L t c $o]  ' L u c $o \]H L v c $o G  w 05(  rDomains are smaller (2$ x 0:(  ? Future Works: 2^ y 04>( RExplore other optimizations techniques and use bi-intervals in a hybrid algorithm S 2St    H  0޽h ?           $%5$&6&.7 &-8!&,9"&+:#./;$+0<&&=A(&>B*=?C,>@D-EFN.EGO/GKP0GJQ1GIR3KLT5GVZ7VX\9_ak:_bl;_cm<_dn=^_o>^`p?`hq@`grA`fsB`etChiuDejv }ff___PPT10i._3+D=' = @B +  S(    0`F(! Thank you for your attention(2 6 #  00N(X  -baget@lirmm.fr sebastien.laborie@inrialpes.fr.(2.U 0U 0-H  0޽h ? }ff___PPT10i.&ПI+D=' = @B +g K0 f^P(    0O(] vExperimental Results(2 $   `A  ALL50"`x QB  (   nA   ALL100"`xQ J  (  `A  opt50"`P `9  (  bA  opt100"`p p I (H  0޽h ? }ff___PPT10i.Xk+D=' = @B +H  K0 G ? ` (    0&-] xRefining a bi-interval(2 $ P  0*- JAlgorithm: Refining a bi-interval constrained by the finished-by relation.K 2Kl     5  02- EData: A bi-interval I=<(a,b),(c,d)> and an abstract interval J=[x,y].0F 2=<  !   0=-D 6Result: Let R be the refinement of I following J and the finished-by relation. We return FALSE if R is empty or a bi-interval whose interpretation in pointsList is R otherwise. 2D         0G-  ,: if y (c,d) then return FALSE; if b d" x then return FALSE; if a e" y then return FALSE; if (a d" x and b e" y) then return <]x,y[,[y,y]>; if (a d" x and b < y) then return <]x,b),[y,y]>; if (a > x and b e" y) then return <(a,y[,[y,y]>; if (a > x and b < y) then return <(a,b),[y,y]>;x 2 4 V              H  0޽h ? }ff___PPT10i.Xk+D=' = @B +rx@ 5v $/Mh8Rh.7spG9G2'? 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