Technion - Computer Science Department - Technical Report CIS9609 - 1996
ikj\lnmVoqp%orslnotvuwVp%oryxKzqzS{|oj9o#}~{tqjP%wp%o
OtM#j9mVwlm*jlmV
1[cV [¡;n¢c(£V¤A¥,¦§ #¤¨©¡¥q¦fA[¦fSq©[(¥¥
ª«¦§ ;A¤*¬¯®°k±*²¨³*³¨³
´%µY,
¦.¶¦·µ¹¸º¥¦( »¤*\¸¼*¦*¸½»[µY [¡A¾¶¦·µ¹¸º¥¦ A¤¨\¸G¨¦*¸½A
¿MÀ[Á%Â(ÃĨŧÂ
Ç
~÷
~÷=øÙå?âiÙàlÜnÚêBà,ç¢Úî!Þ4áã¢ÙÚßåáSåß#à'ë
à,ä?æ4ÞÜnå?ã¢Ùßý÷4éçnÚãnÙÚçià,ßÚ;Ûäà'èå?áæ
ç¢ãnÜià,ãnÚ6æè!à,æà,å?áçnã;à
Æ
Ùå?äÚeÙéßàlá#Ûäà'èÚÜ¢ç
à,î héçnãã¢ÙÚå?ÜÛäà'èåáSæçnã¢ÜUàlãnÚæ4è¯à,ââÞ4Ünîå?áæ#ã¢Þ!ãnÙÚ6åÜÞÛÛ1ÞáÚ6á`ã â6ÞßÛSéãnÚÜ
ÛÜnÞ4æÜUàlßç
áÞ
åâ6ÚÒàlç;à,æà,å?áçnãà,á#ÚëÛ1ÚÜnã'ìØÙå?çå?çîéÚãnÞgã¢ÙÚ
àlçnçnéSßÛã¢åÞ4áÞlݲßå?áå?ßà,ëã¢ÙBà,ãã¢ÙÚÞÛSÛBÞ4áÚáãéçnÚ6ç
øÞ4ÜÉç¢ãnéîSåÚ6ç;ãnÙSÚÛÜnÞ4êä?ÚßÞ,ÝÞÛÛ1ÞáSÚáãßÞîÚä?å?áæå?á#æ`àlßÚ
ãnÙÚçUàlßÚçnã¢ÜUàlãnÚæ4èÈàlçãnÙÚÛäà'èÚÜ'ì
ØÙå?ç
7
Ûäà'è`å?áæ1ì
F
à,áBàläè Ú6îì
Æ
÷
±
ü 1à,á!à,ä?æ4ÞÜnå?ã¢ÙßóÝ2Þ4ÜçnÚàlÜnâiÙå?áææà,ßÚ kã¢ÜnÚÚ6çéçnå?áæàláÞ4ÛÛ1ÞáÚáãrßÞîÚäå?çîÚç¢âÜnå?ê1Úî£à,áî
²
ãnÙÚÛ1Þã¢Úáãnåà,äLÙBà,Ü¢ßâà,éSçnÚî.êèãnÙSÚ
éç¢ÚÞ,Ýà,á#å?áBà,ââ6éÜUàlãnÚeßÞîÚä/ì
°
ø
d÷
Ú
îSÚßÞ4áçnã¢ÜUà,ã¢ÚÚëÛ1ÚÜnå?ßÚáãUà,ä?ä?è#ãnÙÚêBÚ6áÚ BãgÞlÝ~éç¢å?áæà,á£Þ4ÛÛ1ÞáÚáãeßÞîÚäàláî£çnÙÞ
÷
Æ
ÚãnÙÚáËîÚçnâ6Ünå?êBÚàlá!à,ä?æÞ4Ünå?ãnÙß
]øÙÚá¯àêBÞ4éáî£Þ4áã¢ÙÚßÞ`îSÚä
ÚÜnÜ¢ÞÜå?ç,ÉáÞøáì
ü 1ÝDÞÜéç¢å?áæ#éáâÚÜ¢ãUàlåá£ßÞîÚä?ç
åçeå?ßÛ1Þç¢çnå?êä?Ú
å?ácãnÙÚ#æÚ6áÚÜUàläâà,ç¢Úì
Æ
ÚÛÜ¢Þ
«È
´
ÜnéáSåáSæå?á
ü
ÚàËçné âå?ÚáãgâÞáîSåã¢å?Þá.Ý2Þ4Ü
ÛÜ¢éáå?áæ£à,áîcÛÜnÚ6çnÚáãà
ý.÷ #% ü ÷ ãnÙBàlãÜ¢ÚãnéÜ¢áçãnÙSÚG ü àläéSÚÞ,ÝqàãnÜnÚ6Ú=÷ ç¢Úà,Ü¢âUÙSåáSæ
Þ4áä?ègáÚâÚç¢çUàlÜnè#çnéêSãnÜnÚ6Úçì
.
E
÷
ø
e
Ú
Û
n
Ü
6
Ú
n
ç
Ú
áãàgßÚãnÙÞîÝ2Þ4ܨà,⯮`éSåÜ¢å?áæàgßÞ`îSÚäÝ2Þ4Ü;à,á!éSáÉáÞøá!Ûäà'èÚ6Üì É å?Ünç¢ã÷øÚeîÚç¢âÜnå?ê1Ú
É
àeäÚ'à,ÜnáSåáSæà,ä?æÞ4Ünå?ãnÙß
ãnÙBàlã;à,âd®éå?ÜnÚ6çÒàeßÞîÚäÞ,Ýã¢ÙÚÞÛSÛBÞ4áÚáã ç;îSÚÛãnÙËÞ,ÝçnÚàlÜnâiÙ.ê`èéçnå?áægå?ã¢ç;ÛBàlçnã
ßÞ
Ú6çÒàlç
Ú6ëàlßÛä?Úç'ì#ØÙÚác
÷ àlá¯à,ä?æÞ4Ünå?ãnÙSßÝ2Þ4Ü^àlâd®éå?Ünå?áæ!à!ßÞîÚäÞlÝ;ãnÙÚÛäà'èÚÜ~ ççnãnÜià,ã¢Úæè.
÷ êBÞ4ãnÙ
îÚÛã¢ÙGà,áSî£Ý2éáSâãnå?ÞáS
÷ åçeîÚç¢âÜnå?ê1Úîfà,áîcÚ à,ä?éBàlãnÚîLì ¶ ë`Û1ÚÜ¢åßÚáãnç3
ø å?ãnÙ£ãnÙSåçgà,ä?æ4ÞÜnå?ã¢Ùß¾ç¢ÙÞÅ
ø ãnÙ1à,ã
øÙÚáàeç¢éÛBÚ6ÜnçnÚ6ãÞ,ÝãnÙSÚçnÚãÞlÝqÝDÚà,ã¢éÜnÚçéç¢Úî!êè!à²BëÚî!Þ4ÛÛBÞ4áÚáã;å?çà à,å?äà,êä?ÚãnÞãnÙÚäÚ'à,ÜnáSÚÜS÷ Ý2Ú!ø
Úëà,ßÛSäÚ6ç~àlÜnÚçnéÈâ6åÚ6á`ã¦ÝDÞÜä?Úà,Ü¢áå?áægàeßÞîÚä1ãnÙBàlã;à,æ4ÜnÚÚçà,ä?ßÞ4çnãÛ1ÚܪÝ2Úâã¢äè
ø å?ãnÙãnÙÚÞÛÛ1ÞáÚ6á`ã'ì
ÛÜnéáSåáSæ
à,ä?æÞ4Ünå?ãnÙSß
å?áBàlä?äè
Ê
Ë
L5ék_!Ê
Ê
)1"2%6 03¯L+%,!+!%60+e**
Æg7h(_Sa¯a1+1"7!+,°
vgÉg
Êñ\7B!+7;a
-k+'0*
%'1±qa1+,1*e+g)a1+,7i*1}aø*+.%1"+ß&
Ë '
*;*+(+77e%'1-k+,6+,%'+ W "*
¡¨S q¦+ga1#"**6+a!**;+ea!6+,_"+,¡+a£S)*±-/*U3#-og;;&Ê·
·+e-
*+
ËB7i**10+g%640?0+,S+7;*6*;*+
*h"±q%6"240m¢B*+'0?0?"+,%'+6+7i+%6%'!#)1"*i3#7L+,+,
*3S"b*c-ª%'+£a1)i"f*+Ë027i*±_+£a1+%,a1+7"27#*+£%'+*h"S$-gG%'S!()*+,#('**È%,$L+*
*+e¡i02a%6+77%6!(1";]cB7i*%'*i"_S"*i3#".*+g+-B!+,°/(10243S"Ë(7¦7r+,+,¯%'%'+,6+a
¡"*¹+'ýï%6"+,B*£¡43}7-7h+%61"¸02+cB!+£*6+,+7,ó¡+,_+, W
aaS"*i"40*i31(+7-"1*+'0?0"+,1*.(}%'+767i+7,b+
1}a
(1043S"(+,i-/%'+."1_0_+7
S)*+£L`_S+1"10?"1*7#+.*i3B(L+-g7i)%c(%'+77
**r"27(+,i-/!+aøB3.+'(+,*1)c(1023S+,7,% S)1"h"¯È!}a1+'0¦-*+'"e((+,1*ñ\77i**6+,3
dó1"?0+1)£(10243+,7Sa4ùU)7i**+'"(1023S"ï7h**+,S3%,%'aS"!*6!*+'"((L+,B* W
%'!()*+,(°
7(10243#*+7!+B4"7i*;e_S"2%'+
7¦B4"7i*;#+'(+,*4~]cB7h*(10243"(7)7h+*+#"1"4
¡
5
ÓÞß×kÚXÒ$â&ã~ÚXÙ5×XÖEâ&Ò$ÚX×XÒ$Ø=Ò$×8î_1ÌfÍYÎÏÐ
40h"*6
-/7i+%¹n£+Ë4"ÇS77i)!(*i"&-*1"27#40i"*
"27*6**+Ë((+,1*#)7i+7*+
7Ë+7i**6+,37*6+g(1043+,4
Ò+,+
+7i+,_S+,40²7U"*6)*i"7¡+,+*+Ë}a1+'0?"((1%Ë7a1_`B*+_+,*+S°/!}a1+'0?"
Technion - Computer Science Department - Technical Report CIS9609 - 1996
((1%¸-¨*+.7i*a'ab#"1"4$(%'+a1)+£Â7h+,ú\Ä4 W
a1+7%'i"+7*i¡¯7¢"*)*i"7Ê"f¡1"2%6f"*Ê"7
"!(S*1*
*6ï%'7U"a1+,*+S((+,1*ñ\7e(10243"¯1"?0?"*i3!·+"27gc@ºÎ<DJ
t =?Ke(17¢"*i"
W
¡+,+#*+#(10243+,
76+7i£*#L+'0"+,_+#***+g((L+,B*¡"?00)a1+,+7i*h".*+#1a.!_+ W a.¡"?0?0~*+,6+'-k6+g(1043£
(1+,e!_+e"7i*+Sa
*+,g7¢"*)*i"c"27
cy/ziE'A£(B7¢"*i"
¡+,+#*+(10243+,e+'}(L+%'*7e*+S((+,1*
W
*g_+,+7h*i"*+a#*6+,+'-kS+(10243ÈÊa#!_+;1B7¢"#*(#;7i¡Ò"aS0+g(LB7¢"*i"7"271}aË7i**6+,3
¡+,&*+.(1043+,#7+S7i&*¯L+'0"+,_+c**Ê"*'7#((S+,B*!7i+'%+7*b7i40?0`¡+,!a1+,(**b"*7i+'0?-h
*+,;7¢"*)*h"S
W ¡+,+
((S+,B*r!}a1+'0%,!L++,+'±q%6"240 W "270B7¢"(LB7¢"*i"£\[nÁm/-40?0(B77U"10+
!_+7Ò0+ac*¯#0B767 W
#"1"4$%1B7i+7+#-*+,
1"*i"?03 W
"$%'SB*'7i**.1)b(1023S+,7e**
%,£)*i"?0?"À,+!*+'"((L+,1*
!a1+'0"¯Sa1+,*7h+'0+%'*g#7i¡"aS0+#!_+
0?0~*6+È_+%,S7i+7
S77i)!+**e*+((L+,1*g +7+,Ò"b+,_40)*i"¯!_+7l
`¡+,_+, W *+
(S*+,B*h"`0
+,+'±*£-g!a1+'0?"&"27.*È0"#"*+a¹*&7i)%ó%,7i+7,ódó+,¹*+£((S+,B*£a*6+¯(10243+,
+,_``0)*+b(B7¢"*i"7caS"?ò+,6+,B*i03 W
(+'-/+,+,%'+7,
¿Z£+'Z!(10+ W
"*£.3L+b(B77U"10+f*6$+'(10"*£*+b B¡0+a1+fL)**6+GS((+,1*
"?-È*6+G(10243+,c(+'-k+,'7£%'¡
a1+aó1a7£aó 1¡
7Ë**£*+fS((+,1*
+'%S+7±S)+7r¡+,(LB77¢"10+ W
7i+g%,£4_"2a#(17¢"*i"7**+,10+g*6+
((L+,1*¨*6+,1-/%'+±)+
+'%S+7,
Ò+
#"1"4¯((1%.43#-n4"?0*(+aS"2%'*e*+e((+,1*ñ\7r!_+7+,_+,£¡+,£*+e(10243+,a.*+
((L+,B*e+,_``0)*+#(B7¢"*i"7Ò"2a1+,B*h"%,`003
77i)Ë+e-ke+'!(10+***6¯(10243+,7(+'-k+,e*!+'%6+
±)+7Ò¡+,£(B767¢"10+1)7*+e**`0q1)L+,-±S)+7¡"*¯+,B*h"_S+7¢"."27#%'!(S+,B*-*+
(10243+,ñ\7gac*+#((L+,1*ñ\7e+,_``0)*i"f-/)%'*i"#¡+,_+, W
*i"+7Ê*È"1"#"À,+£*+#(10243+,ñ\7-/)%'*i"
W
*6+(10243+,767i)!+7e**e*+!S((+,1*
*+,+'-/+#*£.4B"#"À,+#"*7Ê+,B*i"mn**)7e)*g*6*g*+
(10243+,¨767i)!+7*6**+e((+,1*ñ\7~-k)%'*h"Sï%'1*4"7*+**'40}1)#+,-
±)+7¡"*#A>@º<Dy/< u K7¢"
*r!+7r**r#"1"4."Ë(10"2%6"*i03£-/%'+7r*+(1043+,*#¡603!77i)!+e**r*+e((+,1*4_"2a7
±)+g+'%6+7,
+,_S+,40+7h+%6+,7e4_+(L"B*+ac)*e*+"!(S*%'+#-;((L+,1*
!a1+'0?"$¬ W
[ W
¬ W
W
W
{ W
)*e*+% S)1"27¢"*i"$aø)*i"?0?"À*i"f-;¯S((+,1*
Ë}a1+'04_+*+%'+'"_+ab#)%b**+,1*i"."b*+
%'!()*'*i"401!+g+7i+%¯%'S!#)1"*i3Æei-\4)*i0?"+ab#!+,*a£-~+,+,'40?"À'"£*+#"1"4
40h"*6
-/)*i"?0?"À'"$#)10*i"(10+,°D0+,_S+'0^!a1+'027-+,_`40)*i"&-/)%'*i"7,b+!¡6 ja1+76%'i"+a¸"*1"27
((L+,
)1"?02a7e¯Æei-ñ\7r+7i+%6¯a.+'(a7Ò"*Ò"f7i+,_S+,40¡37,
Ò+¯B`0
-*1"27.+7i+%
"27.*Ç7h*)a13*+b)*i"?0?"À*i"õa¹% )1"27¢"*i"
BË+,°/(1043S"df+ +gG**+,!(**±a¯7i¡+,7*#*+Ê-k0?0¡"
S
77h)#"!**¡+g(LB77i+77£((S+,B*e!a1+'0 W
-((S+,B*£!a1+'027Ë"
S)+7i*i"7l
.0
¡Å%,¯¡+e)*i"?0"À,+#"*
E0
[Zdú*Ò+g*+g(S*+,B*h"`0+,+'±*7-)7¢"S((+,1*
Ë}a1+'027
0
¡Åa1B+7*+%,%')'%'3-~*+gË}a1+'0+'Ã+%'*"*7+,+'±*
|0
´L¡Å%,¯¡+e)7i+£)%'+,*4"¯Ë}a1+'0
XZ¡Å%,G(S
.0
% S)1"+#Ë}a1+'0-"*7((S+,B*
M)+7h*i"oe+,_+'0(1"40Si"*7L-/)7¢"gS((+,1*
´)+7i*h"S7,
]c+7i)h".*+(*+,1*i"240+,+'±*7
+%'*h"S[-*1"27Á((+,a1+4027¡Ò"*6*+¦±7h*
!a1+'0n+%'*h"S
a1+4027¡"*b*+7i+%'SaGa£*+*1"a
-;!a1+'0?"¯a£*+7h*i"*++'Ã+%'*7-~!a1+'0?"G%,%')S%'3££"*7+,+'±*7lmnf7i+%'*h"S¯´ W
¡+a1+7%'i"+
b40i"*÷-/e)7¢"£)%'+,*4"b!a1+'07l#+%'*i"fXa1+76%'i"+7Ë!+,*aø-kS"%'(L*h"£(6)1"
"B*6*6+
40i"*.7,r+%'*i"¯Äa1+7%'h"L+740h"*67-k% S)1"h"#((L+,1*¨Ë}a1+'027,;¿Á"40?03 W 7i+%'*h"S
¬%'%60)a1+7,
[
a 8
c 8
Technion - Computer Science Department - Technical Report CIS9609 - 1996
b 6
ϕ
ϕ
e 8
d 6
3
6
2
8
C§çnÚàlÜnâiÙ!ãnÜ¢ÚÚç¢ÛBà,ááSÚî#ê`èãnÙÚ, à,ä?æÞ4Ünå?ãnÙSß øå?ã¢ÙîSÚÛãnÙ#ä?å?ßå?ã û ì[HO®éBàlÜnÚçrÜnÚÛSÜnÚç¢Úáã¨áSÞ`îSÚçøÙÚÜ¢Úå?ã
ç;ã¢éÜná!ã¢ÞÛäà'èì\I¦åÜ¢âä?Úç;Ü¢ÚÛÜ¢ÚçnÚáãáSÞ`îSÚç]øÙÚÜnÚeå?ã;å?ç;ã¢ÙÚÞ4ÛÛBÞ4áÚáã çrãnéÜ¢áã¢ÞÛäà'èìØÙÚeÞÛÛ1ÞáSÚáã
ßÞîÚ6ä^]¸å?çéç¢Úî¯ã¢Þæ4Úãã¢ÙÚßÞ
ø ÙSåä?ÚeãnÙÚçnãià,ãnå?âgÚ àlä?éBà,ã¢åÞ4á#Ý2éáSâãnå?Þá£å?çéç¢Úî¯ã¢Þ
Úç¢Úä?ÚâãnÚ6îGêè!ã¢ÙÚÞ4ÛÛ1ÞáÚáã
¿Á")+!
å?ç;ã¢ÙÚÛäà'èÚ6Ü
Ú
àläé1à,ãnÚã¢ÙÚÜnÚç¢éä?ãnå?áægê1Þ`à,Ü¢îì
_
'98$8ýʵ
È
pr+'-kS+7i*)a13S"*+g(610+,
;
`
-EC 4xB<Dz'<2J1Hg£((S+,B*e!a1+'0 W ¡¨+a1+,_S+'0S(G!+,*a103Ë-kSx@,<DJBH
"*m¢¯*1"27e7i+%'*i"£¡+g(+7i+,1*
!7i+,*-`0Si"*7r-/)7U".((L+,B*!}a1+'027Ò"Ga1_+,'73£7i+%6
M©
º_zX½k·5»}¼X¹Á °aÀ.¼/³C½¾º_³ º»¼X¹}»}º».¿·7¶8¶·5³¸c³7¹ Á¾·;¸&z
uY8¸á
K
Ì
+,*
+#*+!7i+,*e-;(B767¢"10+ËB!+#7i**+7l
Ì
+,*
R
((L+,B*4ñ\77i+'0+%'*+aø!_+-/+%6ï7h**++û
+,_``0)*i"£-/)%'*i"
Ò+È`0Si"*
W
Kcb
K
L+!¯((L+,1*g!}a1+'0**g7i(L+%6"±+7Ê*+
40i"*ö* +7r#(B7U"*h"S
¯1"*3S((+,1*!a1+'0
R
W
W !a1+,(*.0?"#"* W
7h**i"2%
a.6+,*)7!7h+'0+%'*6+a¯Ë`_S+^a£È_`40)+
7h+'0+%'*'7.!_+"f*+#-/0?0¡"¯¡43m¢*g+,+,*+7e*6+7i)%,%'+77iS7e-*+!%')+,1*
(17¢"*i"mn*e*+,G((10"+7
R
£+%6¯-*+7h)%,%'+77i7*!S+,**+S((+,1*ñ\76+7i(7h+aø+,_`40)*+7
+%6b-;*+#+7i)10*i"¯LBa713G((103S"c*+!40i"*
*n¡"*¯6+a1)%'+afa1+,(*c0?"È"*,m¢*
6+%')7¢"_+'03
*+,$7h+'0+%'*'7*6+7i)%,%'+77iS¡"*f*6+!1"+7i*_40)+£+40h"*6+,*)67g*+7h**i"2%+,_`40)*h"S$*+e"()*e(B7¢"*i"b¡+,£*6+a1+,(*.0?"#"*e"27À,+,L
¿Á")6+fG7i¡
7Ëb*+,+c-Êa1+,(*6
+'(a1+a¹B3*6+¯û
40i"*¯
77h}%6"2*+aÈ¡"*#*+((L+,B* W *+(10243+,)7i+7*6+¨Ë}a1+'0-/)%'*i"
v
*a1+
W
R
"27;%,`00+a£a+,*6)7a1+
`
R
~Ò+(1023S+,;*+,%,`0027rû
ýg¡*6GÄË-/g*+#(1023S+,!"#"?02+7i1"¯7i¡
7e*6*
a1+ G
*+g(10243+,;Ò+,+'-kS*+g(10243+,^7h+'0+%'*'7*+Ë`_S+0+aS"£*#a1+ ¿Á")6+[!7i¡
7r*+g-/.40
0"27i*h".-*+û
¼*+Ë***6+
R
uY8¸á
Ì
+,*
û
[
&
R
û
+,_`40)*+7
W
*+,+'-/g}a1+
W
¡¨S*G{Ë-/
40i"*¯
fe fdDzhg
ü ü
+*6+g+,)102#"1"4G`0Si"*¯
40h"*6¾"27û
`
*+.4B"`0;_`40)+b+,+'-/+
º_zX½k·5»}¼X¹Á °aÀ.¼/³C½d
[
0+,_S+'0^-g*+c*+,+
"27#)7i+a&7#£102%6 $Lcmn*+Ë-k00¡"f7i)7i+%'*i"7È¡¨+. S+!!+.+'(10"2%6"*
77h)!(*i"7L)**+*)+g-
M©
*!+S%
*6g+,_`40)*+ea1+
*+!*i¡£0+4_+7Ê)7¢"G*6+ï7i*'*i"2%#+,_``0)*i"f-/)%'*i"5aj+,*)7#Ä W
a1+
*6±a#*6+((S+,B*ñÖ7¦+7i(S7i+
º_Àº
Á¾·;¸&z
*)'40~%,aS"2a*+*!7i+,_+S7*+g!a1+'0
à ¡ %$%
¡"*¯"_+,¯7h**i"2%+,_`40)*i"ø-k)%'*h"S
R
d¸+%,40?0²*61"77i(+%6"240~%,S7i+g-~û
"¯*+û
W ¡+,+
Technion - Computer Science Department - Technical Report CIS9609 - 1996
9;:=<5>.?A@CBD:E? û * Vcz.YN| Ã % É |~`UG¥VAL-e7|R ,
^ `UG¥VAL-e \ Y
:=?m.B5:} F| Ã % É * Vcz.Y ,
?A> =?
THUR HUWX{SG g 9@
K ; g û z.UGEG=J * VSz&Y ,
^ <_: +%6 Y={SB
AÅK
^ `G!VAL$e \ VcWXHPG=I HUWX{SG g à % É * YE{c ,
?A> =?
z~V}V z.HI&` g R * Y={S ,
VSWXHUPG=I z.HI.`c|)VcWXHUPG=I HWX{cG. g
^ VcWXHPG=I HUWX{SG THR UHWX{cG
THR HUWX{SG g VcWXHPUGEI UHWX{cG
THR z.HUI.` g YE{c
:=?m.B5:} THUR z&HI.`c|~THUR HWX{cG.
,
¿Á")6+[ØÙÚ
R
\
[
û
"
W *+gû
40Si"*¯;1"2740h"*6
* z~V}V z.HI&`S| Ã % É |~`UG¥VAL-e &[ |R ,
û
à,ä?æÞ4Ünå?ãnÙß
¡6 }7B37U"#)102*i"#*+e((S+,B*ñÖ7;È"1"`ø7i+%.*
±a."*77i+'0+%'*+ac!_+ W 7a1+7%'i"L+a£"c*+g(+,_S"S)7
7i)7h+%'*i"
¿Á")6+
7h`¡e7!&+'ZË(10+c-7i+%6$*6ja1+,(*6
"
(L+,i-kS!+a&13f*+¯û
40Si"*¯*f©¢E «
7i¡
7Ê*+!*i¡¯%,40?07*c#"1"4$*b7¢"#)10*+*6+!((S+,B*ñÖ77i+'%£¼*+!***+!S((+,1*g"27#
[
41"#"À,+,f*¯©
"
-e*+!±S)+ï7h`¡e7g*+.*i¡¨c+%')7¢"_+£%,40?027#*bû
«
-/ò*+.B'a77h+'0+%'*6+a
13¯*+#((L+,B*4¼S*+**
¡1"?0+.*+#((+,1*ñ\7#"1"4f(%'+a1)+!L+'0"+,_+7**"fa1+£Kg*+
S
UT
(10243+,^7h+'0+%'*'7*+!_+e0+aS"£*!a1+
W *+(10243+,g%'*)40?03£(+'-/+,7}a1+
º¼S*+g***6+(10243+,
"27g103ø"1*+,+7h*+ac"f*+£w#> u K#7i+'0+%'*+ab13¯*6+7¢"#)10*i"f-*+##"1".4$7i+%6b-¨*+#((L+,1*
+
u E=?xBKe+,*)6+a.B3.*+È"1"`G7i+%6."27%'!(10+,*+'03£"S+a
M©pª
uY8¸á
df+%,ja1+'±+.*+û
s
û
°²´³ º»8¼t¹»Nº_»¿Á¾·;8¸z/¼)³½Kz/¸VCk¸&z
º zX½k·5»}¼X¹Á
x _
40Si"*
N
-/
13
W
`0Si"*
J
*£L+!*+7i(L+%6"240Ò%,7h+!-Òû
*6L+*6+Èû
`0Si"*
¡"*
-/0?0¡
7l
*.,
*U[,
û
û
ÒB)7 W (1043+,)7¢"!*6+^û
)7¢"&*+Gû
N
*+!û
ü
40i"*
40i"*
s
[
W * Vcz.Y
Ò%X ¹ f * Vcz.Y
Ò ¹f
"
,
,
\
R
\
û
gû
R
s
\
"
û
#df+%,fa1+'±+.*+
%,bL+-/.40?03¯a1+'±+a¸7
5* Vcz.Y}| Ã [ | ` ,
* VSz&YN| Ã s |`S| û s S# " ÒYX%Z Ý ¹ f #S" ,
#"1".4
\
û
40i"*
[ *nÈ"1"`_,
77h)!+7r**"*7r((L+,1*¨)7h+7û
77i)!+7Ë**È"*'7!((L+,1*)7i+7.û
**"%60)a1+7+,_+,3Gû
¡+,+
s #c"
s
"
W
W (10243+,
+,*%¹mnó*1"277h)7i+%'*i"5¡+ba1+'±+
40i"*7e.7i(+%6"240%,7i+gû
ü
+,*7e7e!('!+,*+,g
Aq=?EKze¡1"2%6&"%60)a1+7"1-/.*i"$L)*#*+!(10243+,ñ\7Ê+,_`40)*h"Sf-/)%'*i"a$)*Ê"*'7g!a1+'0¨*+g((L+,1*
Qd?×5ôm= <5
Ô
s
(10243+,#<@EAE<2z
t K/§¨JLK t
E@;¤,>=D=>SΦ@
´
a
a
Technion - Computer Science Department - Technical Report CIS9609 - 1996
b
2d
h
5
4e
i
2
j
6
9 b
c
5 g
7 f
k
4
m
9
l
7
3 c
n
8
9 e
o
5
3 f
j
9
k
2
l
1
(a)
m
3
(b)
fÍ"]
ø å?ãnÙ!îÚ6ÛãnÙ£ä?å?ßå?ã¦Þ,Ý û ì ´ àlÜnã]2à çnÙÞ
ø çãnÙSÚ
ãX¦
ø Þ#âà,ä?ä?ç;ã¢Þ
~ çâUÙSÞå?âÚì9E;Þ4ãnÚãnÙ1à,ããnÙSÚÞÛÛ1ÞáSÚáã;å?ç;àeßàlë`å?ßgåGFÚÜ'ì;Hý®`é1à,ÜnÚ6ç;ÜnÚ6ÛÜnÚç¢Úáã
áÞîÚç]øÙÚ6ÜnÚå?ã;å?çã¢ÙÚ
ÛSäà'è4ÚÜ çã¢éÜná!ã¢ÞgÛäà'è`ì9I¦å?Ünâä?ÚçÜnÚ6ÛÜnÚç¢ÚáãáÞîÚç
ø ÙÚÜ¢Ú
å?ãå?ç;ã¢ÙÚÞÛSÛBÞ4áÚáã ç;ã¢éÜnáËãnÞÛSäà'èì
§ÝDãnÚÜã¢ÙÚÛäà'èÚ6ÜîÚ6ãnÚÜ¢ßå?áÚçrãnÙBàlããnÙÚÞÛSÛBÞ4áÚáãâiÙÞÞçnÚãnÞgßÞ Ú¦Ý2ÜnÞ4ßKJrãnÞMLà,áSîÝDÜnÞßON;ã¢ÞücS÷ å?ãÚ àlä?éBà,ã¢ÚçãnÙÚ
" Ü¢ÚâéÜ¢çnå Úä?èì
à,ä?éÚrÞ,ÝPLàláîëüËêèâàlää?å?áæ"Í
¿Á")+
eØÙÚeçnÚàlÜnâiÙ£ãnÜ¢ÚÚçeçnÛBàlááÚî.êè!âàlää?å?áæ
ßå?áå?ßàlëÝ2ÞÜîSÚãnÚÜ¢ßå?áå?áæeãnÙÚÞÛÛ1ÞáÚ6á`ã
ÃMM \ *-Ã |/F , <@^E#(10243+,`
E=?x1Ey/<D>J#¤,xBJLCy/<D>J
à EJ t E#(10243+, MM \ *-à |/ ,^<@E(1043+,
OËÏ< u KJ¯ESJGK u E=?x1Ey/<D>J#¤,xBJLCy/<D>J
P ËÏ< u KJ¯ESJGK u
:BKʧ¨z6@ºyeK=?K4wK4Jye>i¤£E.Aq=?EKz.<@£C'E=2=?K t
yD:BK!(10243+,ñ\7#7i*6*+,3EJ
yD:BKø@ºK'C>SJ
t
K=?Kw#K4Jye<@£CES=2=?K t
t
yD:1K
((L+,B*!}a1+'0U
Qd?×5ôm= <5ö
ziKJLCK
\ * KM|E ,ºø:1Kg!a1+'0"ø0+,_S+'0^>i¤#E#Aq=?EKz<@
Ï< u K4JfEAq=?EKz
* ,
û
ÒB)7 W ÊÀ,+,°D0+,_+'0!a1+'0?"(10243+, W
0+,_+'0qË}a1+'0?"#(1043+, W
* K " | * K [ |/F ,!,
)7i+77h**+,S3
K "
K
N
W
û
F *
,
* K [ |/F ,
<¤
\
AF
W "27r+g**a1B+7r*!a1+'0
"*'7((S+,B*
ûü+,*'7¨(LB7¢"*i"
W
* K [ |/F ,
#a1+,(*.0?"#"* W
+,°
* K N | * K " | * K [ |/AF ,!,¥, W "27+
* K " | * K [ |EF !, ,+#((+,1*ñ\7!a1+'0
*i¡°D0+,_+'0!a1+'0?".(10243+, W
W -*+
(10243+,r++%')7U"_S+a1+'±1"*h"S£-(10243+,"7
´*U¶]b] ,B3cÉ
#31*7U"+,¡Ò"%'À
"¯*+7h(1"h"*e-*+¶+%')7U"_S+È]ca1+'0?"£]c+,*a
"*7_`40)++40Si"*
T
>yD:BKz'Î<@ºK
ac7
Ë!a1+'0;-¦"*'7e((+,1* W
a#7g!a1+'0 W
[,.yD:BK.zUK'Cx1z
W "27+**S7g!a1+'0-
"*7¦((S+,B* W )*;767i)!+7*6*¦"*7
((L+,B*Ò"7À,+,6°D0+,_+'0!a1+'0?".(10243+,
**e)7i+7e7h**+,S3
Y
F *-;, \ãÛ
t K/§JK t
W
oe)i-k+,+a£df+,+!4X¢
a£ÊAq=?EKz W a.)*()*7!_+g7i+'0+%'*+a.B3.*+e(1023S+,
a
S+,+,*+7*6+7i)%,%'+77ie1a7ac7¢"#)10*+7*6+g((+,1*ñ\77i+%6£-/
+%6b-;*+,÷"ba1+,g*£1*i"2%6"(*+"*'7g%"2%'+Ò1"7g7U"#)102*i"j"7ga1S+13G((103S"¯*+!40Si"*
+%')7U"_S+'03c¡"*b*+S((+,1*
Ë}a1+'0;7*+(1023S+,
+(1023S+,
*+,c+,_`40)*6+7e+%c-¦"*'7(*h"S40
!_+713£+,_``0)*i"£*+)*'%'!+g-¦"*7S((+,1*ñ\7+%'*i"cB3¯((103S"£*+`0Si"*÷+%')7¢"_+'03
)7¢"Ë"*'7¡¯7i**+,3
¿Á")6+´¯7h`¡e7
b+'!(10+#-/£7i+'%c*+,+!7i(+ab13Gû
*+#S**
ü
* HS| | Ã N -* Ã " | Ã [ ,!,+#B)#+,'7*
+#*+7i*'*i"2%_`40)+7e-*+g0+4_+7,g+#+%')7U"_S+È%,`0027((10?"+a¸*.+%6¯a1+È+0"27i*6+a
+'**!*+ga1++a7i+aï0"+7"aS"2%,*+¡*!_+"27
7h+'0+%'*6+a¯13.+%£6+%')7¢"_+%,`00n
!a ¯+.((S+,B*!7¢"#)102*+7*+
` a G *}a1+ ` *+g!a1+'0-*+e(10243+,
Ò+!(1023S+,È7U"#)102*+7Ê"*7((L+,B*4ñ\7#7i+%6b-/a1+7
(10243+,B3!)7U"È"*'7¡!!a1+'0-*+g(10243+,r-k6¾a1+7
X
M*(a,3,(f2,(f1,f0)))=10
a
Technion - Computer Science Department - Technical Report CIS9609 - 1996
M*(b,2,(f1,f0))=-6 b
M*(d,1,f0)=4
M*(d,1,(f2,(f1,f0)))=8
c
M*(e,1,f0)=10 e
d
M*(c,2,(f1,f0))=0
M*(f,1,f0)=4
f
g
M*(g,1,f0)=6
M*(g,1,(f2,(f1,f0)))=10
j
h
i
k
M*(h,0,(f1,f0))=-6
M*(j,0,(f1,f0))=-8
f2= 8
f2=-4
f2= 4
f2= 7
f1=-6
f1= 6
f1=-8
f1=-7
f0= 4
f0=-8
f0=10
f0= 3
¿Á")+´ØÙÚç¢Úã;ÞlÝÜnÚâéSÜnçnå
m
n
o
M*(m,0,(f1,f0))=-2
M*(o,0,(f1,f0))=0
f2=-6
f2= 1
f2=10
f2= 2
f1= 7
f1=-2
f1=-4
f1= 0
f0=-4
f0= 4
f0= 4
f0= 6
l
]
Úâà,ä?ä?çæÚ6áÚÜUàlãnÚî.êèâàlä?äå?áæ
ø øÙBà,ã¦ßÞ
ãnÙSÚ¨áSÞ`îSÚå?ãå?çâàlää?ÚîÝ2Ü¢Þß.ìØÙÚîBà,ç¢ÙÚî!ä?å?áÚçç¢ÙÞ
ü
>îï û ï/pü N 1ü
" ï$ü d[
¶
ì
øÜnå?ãnã¢Úá#áÚëã;ã¢Þ
à,âiÙ!âàlääå?ç
Ú;å?çç¢Úä?Úâã¢ÚîêèÚà,âiÙ#âà,ä?ä/ì
*-Ã [ ,;7i+'0+%'*7ra1+ e +e((+,1**+,£((10?"+7r"*7]Ã " -k)%'*h"S£*a1+ e a
%'%60)a1+7**}a1+ e W a#*+,+'-/+}a1+ ` W +¡*Ä+e((L+,B*r*+,£((10?"+7r*+(1043+,4ñ\7
!a1+'0*-Ã
, *6a1+
[ ,¦*#}a1+ G W %'%60)a1+7**;*6+
(10243+,
7i+'0+%'*a1+ P W ((10?"+7"*7r¡Ë-k)%'*i"´*$Ã " ~
{+,+'-/+ W *6+S((+,1*g!a1+'0 W
P W afa1+%6"2a1+7g**ea1+ P W aj*+,+'-/+a1+ G W +#¡* ¡+,$((10"+a&*.a1+
W 7i+'0+%'*7Ê*+!!_+#**e0+Sa7g*£a1+ ` !+#(1043+,*+,f*+7h*7e¡)%6
a1+
` "27g¡*b%,%'aS"£*."*7g%'h"*6+,i"±*-à N , Êmn*((10?"+7#ûõüÊ*a1+ ` ab%'%60)a1+7**ea1+
` W ac*+,+'-/+ga1+i W +¡¨S*¯{")102*h"Sf-*+((L+,1*e-kS }a1+ 3S"+'0a7Ê*+7h+'0+%'*h"S
-}a1+ O !Ò+(10243+,*6+,f+,_`40)*6+7 O
%,%'aS"c*."*7`¡j7h**+,S3Gac±a7Ê**e"*#¡*Y
* *+g_40)+#-~a1+ J , ¨¼*+g*6*¡+,c*+((L+,1*^7¢"#)102*+7*+(10243+,-/ö}a1+ O W "*e¡03
n
77h)!+7***+g(10243+,
7h+'0+%'*'7}a1+ z ;+,+'-/+ W *+g(10243+,
7i+'0+%'*7*+g!_+e**r0+a7* ¡"*
Ë_`40)+!-
4Yg¼*6+**e)7¢"GË+,)102gÈ"1"`j7i+'%b¡"*O
à Ne¡)102a¯4_+#+7h)10*6+a£"$7h+'0+%'*h"
*+g!_+e0+a7*#a1+ëe
¡"*¯#_40)+-¬B
)7i+aËB3Ë*+
S((+,1*
Ò+e-k640
0?"7h*i"-*+ûüg40i"*
d¸+
%6024"!+a.**û
¡+,
J
"7e7i¡."£±S)+X
"27g+,+,`0"À*h"S£-*+^û
ü
+%6+7£Y¾¡+,_+, W
s
`0Si"*¯+gû
s
40i"*Ó%,40?027#"1"4
S*+b**£ûõüca1B+7ø*¯%'1*4"Å7i)%Å%,40?0n
f
+b+7h¹"7ø**
T y J y THUR *-Ã |~`c| z.HUI.` ;, :óû ü * < *-Ã | *Ã | =*$> Ã |EÌ=Ì=Ì ,¥,!? ,$,*U¿r*+
7!++7h7¦***+g¼eÉ
°n] 9ÿ *'*i"
"27+=S)1"_`40+,1*g*!#"1"4Â,+,6+'-k6+ W "?-;*+g!a1+'0".0+,_+'0-*+h""`0(1023S+, J
"27e7i40?0+,e*
f# s
<
=>
?
*+a1+,(*c-*+7i+%6£*+,+ ` W ¡+g+,(102%'+*+Y°D0+,_+'0!a1+'0?"£(1043+,3Ã
[ 13 *-Ã [ | *Ã [ | *-Ã [ |=ÌEÌ=Ì ,!,¥h, A@
M©
²
·5³8¸2B¶º_À.ÀDCM¸S»}À.¼/·5³Å·á
mn*Ò"71_")7Ò***+û
ü
40h"*6
x
a1B+7Ò)10*i"(10+!+'(7¢"7-(6*7-*+7i+'%£*6+,+df+g4_+
a1+,_+'0(L+a!*+,_S+,7¢"#-*+ûü;40h"*6
"$#.%'¥(/(
W %,40?0+a!ûõü
W **¦+'}(a7¦*+*+,++*i"!+103 W
ùi)7i*!7g#"1"45a11+7,GÒ+ï40Si"*×+'}(a7*+£7i+'%f*6+,+!"*+£7Ë+!+,!7Ê#"1".4²
`¡+,_+, W
}a1+£_40)+7£+.((1*+aaS"?Ã+,+,B*h03
dú+,6+7##"1"4((1*+7103+£_``0)+ W
* ? s |=ÌEÌ=Ì!|E? [ ,!Ò+!_`40)+ ? È +,(+7i+,1*7g*6+Ë+,i"*-*+%')+,1*ga1+
%,%''aS"Ë**+e"°D0+,_+'0Ë}a1+'0 W Ã È ;ûõü
"$#.%'¥(/( (S77i+7_40)+7* ? aS"?Ã+,+,1*i03!-/_`40)+7S77i%6"*+a.¡"*
*+.(1023S+,Èa&_`40)+7!767i}%6"2*6+a&¡"*$*6+((L+,1*fm¢Çc(10243+,ñ\7a1+´U
* £a1+£¡+,+."*"27#*+
û
ü
(6(B*6+7
J
#_`40)+7 W
Ä
Technion - Computer Science Department - Technical Report CIS9609 - 1996
9:E<5>.?A@B5:E? û ü * Vcz.Y}|~`UG¥VAL-e7| *XÃ % É |E³´7´ í ³µ¶ ,¥,
^ `UG¥VAL-e \ Y
:E?Um.B5:N F| Ã % É * VSz&Y ,
?A =?
THUR HUWX{SG g 9@
K g û z&G=;G=J * Vcz.Y ,
^ <: +%6 YE{cB
AÅK
^ `G!VQL-e \ VSWXHUPG=I HWX{cG g à % É * YE{c ,
?A> =?
zVNV z.HI&`S|z~V}V HUWX{SG. g û ü * Y={c.|~`UG¥VAL-e
|/³´7´ í ³µ¨¶ ,
VcWXHUPG=I z.HI&`S|/VSWXHUPG=I UHWX{cG g û ü * z~V}V z&H I.`c|~`UG¥VAL-e f
[ | */Ã % É |/³´7´
^ VSWXHUPG=I HWX{cG THR UHWX{cG
THR UHWX{cG g VcWXHPUGEI UHWX{cG
THR z.HUI.` g Y={c
:E?Um.B5:N THR z.HI.`c|~THUR UHWX{cG
"
¿Á")6+X¦ØÙÚ
S, "
(10243+,ñ\7Ê*)f*c(1023
È
?
W
ü
³µ¶
D,!,
àläæ4ÞÜ¢åã¢Ùß
i* ? s E| ? s # N =| ÌEÌ=Ì , W ? È S+,*7g*+.41"40
ý* ? s c# " |E? s # L |EÌ=Ì=Ì , W ? È S+,*7
-kg_40)+7!77h}%6"2*+aj¡"*$*6+(10243+,
_``0)+b!c"*7!%61"?02a1+,5¿S_40)+7S77i%6"*+a&¡"**+£((L+,1*
? È _`40)+#-*+%1"?02ac**1_S+
*+g.4B"`0_`40)+#* ? È #c" ;¿Z+'!(10+ W *+g((S+,B*e+'0?"+,_+7
*U%,%'SaS"*g*+!}a1+'0X,¦***+(10243+,r+,_`40)*+7ra1+713 ? s # N *;g(10243+,ñÖ7¦}a1+ W *+S((+,1*
77h)!+7***+(10243+,¡"?0?0;7i+'0+%'*
*+#%1"?02ac¡"*¯.4B"`0 ?
s # N _`40)++,+'-/+ W *+g_`40)+#-*+
%')+,1*}a1+Ê-kS;*+g((L+,1*"27*+ ?
_``0)+-*+7h+'0+%'*6+a¯%61"02aø¡"*£*+e41".40 ?
s #c"
s # Nr_``0)+
*+
í
*.((S+,B*ñÖ7}a1+ W ¡¨+ga1#*+7Ë+
)**6+
0+7Ò-*+g((L+,1*
aË*+g(1023S+,Ò+7i¡"*%6+a
a V[2]=10
V[2]= 8
V[1]=-6
V[0]= 4
h
f2= 8
f1=-6
f0= 4
¿Á")+gÄØÙSÚ
²Bæ4éÜnÚì
à,ä?éÚ
V[2]= 8
V[1]=-6
b
d
V[2]= 7
V[1]=-8
V[0]=10
i
f2=-4
f1= 6
f0=-8
V[2]=10
c V[1]= 0
j
f2= 4
f1=-8
f0=10
Úâã¢ÞÜnç¦ÛÜ¢ÞÛBàlæ`à,ã¢Úîgêè
V[2]= 1
V[1]=-2
V[0]= 4
e
k
f2= 7
f1=-7
f0= 3
$" ü #.%'¥(/(
V[2]=10
V[1]= 0 g
V[0]= 6
f
m
f2= 1
f1=-2
f0= 4
l
f2=-6
f1= 7
f0=-4
ø
å?ÞéSç
G *)a1+,7i*a
G ¦*+g(10243+, W *+e((+,1*!a1+'0 W
àEãØ=Ò~ì7g5ã!çÒ$Õ~Ò-Ú$æEç_Ò8Ô¥×X×XÖ=å]ÒÙXÓ.Ô!Ùkè.ãÙXÓ3âáòÔ$Û~Ò$ÚX×8Ô!ÚXÒåÔ¯í~ÞßåÞ£¢-Ò$ÚX×
"$#.%'¥(/(
¿Á")6+^Ä#7i¡
7r+'!(10+e-/*+,+g7i(+a.13ûõü
Ú/Ô!ØÞßÙpÞßãà.Ô$áïáÂÛ×XÖ=Ý)Ó6ÔàEãØ=ÒÞß×kÝ!Ô$áïáßÒ$ØÔ]ê7698
o
f2= 2
f1= 0
f0= 6
ìLØÙå?çå?çãnÙSÚ;çUàlßÚ¦ãnÜ¢ÚÚà,çã¢ÙÚ;Þ4áÚ;ç¢ÙÞ ágå?ágãnÙÚrÛÜnÚ
Ì
+,*)70B .*}a1+
*+e¡3#**¨*6+^40Si"*Ó¡¨S }7l~Ò+,+
(10243+,7r+,_40)*+ga1+
45
n
f2=10
f1=-4
f0= 4
¬
a
*+£((S+,B*ñÖ7#!a1+'0^-g*+£(10243+,ó+¯S((+,1*ñ\7#Ë}a1+'0
-^*+c(1023S+,!+,_`40)*6+740?0g*+
Ã
Y$a57i+'0+%'*7ËBa
7i)%,%'+77h7Ë)7¢"$+,_40)*i"¹-k)%'*h"S
P
¡"*ób_`40)+b-!Yó+¯S((+,1*
!a1+'0
1`¡e7*6*"*Ë7f+'Ã+%'*S*+ïa1+%6"27¢"ó* +,Ç*a1+
G
Technion - Computer Science Department - Technical Report CIS9609 - 1996
*c!_+.mn*#*+,+'-/+S77¢"7ga1+
P
W
G
7¢"%'+£"*#"27#*6+£(1043+,4ñ\7*)
W
7i+'0+%'*+a&13b*+(10243+,#!a1+'0 W
)7U"j"*'7g¡
æ W -* æ.* P , \ {,'~Ò+g(1043+,e%'*)40?03£(+'-/+,7r}a1+ k ¡"*£*+g1"+,vÃ
[#_40)+
*+#_+%'*e((B*+aï-kS
a1+ G "27
i
¬ |
{ | Y ¼*6+**e*+#_``0)+7^"f*+
+,_``0)*i"£-/)%'*i"
*-Ã
[* k , \ }¬ ,g1)7
*+_40)+£-
W
_+%'*S7%'+7i(La£*6!*+g+7i)10*7Ò-~*6+g+%')7¢"_+#%,40?07"ø±)+g´
$" ü #&%'!()(
¿Á")6+#¬g0?"7h*7*6+û
40i"*¯
9;:=<5>.?A@CBD:E? ûõ"$ü #&%'!()( * Vcz.Y}|`G!VQL-e5| */à s | *tà s #c" | * Ì=ÌEÌ!| à [ , Ì=Ì=Ì ,$,!,
^ `UG¥VAL-e \ Y
:=?m.B5:} XÃ s * Vcz.Y , |=Ì=ÌEÌ!| Ã [ * Vcz.Y ,
? =?K g û z&G=;G=J * Vcz.Y ,
@ |=ÌEÌ=Ì!| 9@
? g !9
+
S
%
^ <:
YE{cB
AÅK
7h)%,% C
g û "$ü #&%'!(/( * YE{c.|`G!VQL-e 5 | * Ã s | */Ã s #c" |=ÌEÌ=Ì$| Ã [ , Ì=Ì=Ì ,$,
^ <_: +% y 77i%6"2*+aø¡"*¯%')+,1*(10243+,
^ 7i)%,% C# y 7 ? y ? y g 7i)%,% C# y ^ F
y D J
? y l g 7i)%,% C y ,
:E?Um.B5:N ? J |=ÌEÌ=Ì$/| ? `G!VAL$e
$" ü #&%'!()(3 §
¿Á")6+#¬B
Ò+e-k0?0¡"*6+,+,
Q$R
EJ
"
ÚÜnç¢å?Þá#Þ,Ý
ãnÙÚ
àläæ4ÞÜ¢åã¢ÙßÅã¢ÙBà,ãÛ1ÚÜ/ÝDÞÜnßçÞ4áä?ègÞáÚÛBàlçnçrÞ
ü
7i¡
7r**
û
?<_:E?ca Ô s @'@ºx1w#K¨yD:1EyL8 s $<@;EgJ T
J \ ûõ"$ü #.%'¥(/( * Vcz.Y}|~`UG¥VAL-e7| 84 sU
t =?Ky ?
9:E<< ^E^
¿Z
` \
YËa
77i)Ë+
**Òû
` \
$" ü #&%'!()(
acû
ü
=
+g+ S)1"_40+,B*4
K4yc
=K u K4=w#> t K4=<DJBH~Aq=?EKz,
4,ºË:BK4J
ÚÜã¢ÙÚçnÚàlÜnâiÙ!ãnÜ¢ÚÚ
\
?
J
i
S| \
û
ü
* Vcz.Y}|~`UG¥VAL-e7|
4 U
,M
s 4
8
g*+g(1-¦"27"!!+aS"2*6+
a£û
ü
$" ü #.%'¥(/(
6+,*)£*+7!+
_40)+g-/*+,+e-~a1+,(*
` \ `G!VQL-e
(_+g**¨*6+,3+,*6)¯*+76!+g_``0)+Ê-kS
`
`G!VAL$e $Sdf+e¡"?00
S¿¦+%6È7h)%,%'+77i*¦0+,_+'0 W ûõü~±7i*;a1+,*+,6#"+7¦*+B'a*0+,_+'0q[e7i+'0+%'*+a#13g"*7¦((L+,1*
p;3.*+e"a1)%'*i"b13B(L*+7¢"27 W
û
ü
* YE{c.|` ¹ |
·
r
k, \
]¸·
og;Ì
-" ü #.%'!(/( * YE{c.|` |
û
a.*+,+'-/+g*+7h+g¡"?00+*+g1a7¡"*c*+g41"40
?
·
;
k, \
]¸·
og;Ì
?
J l
J ,¢
[Zûõü£a1+,*+,6#"+7£*+c_`40)+f-+S%ó7i)%,%'+77i£135%,40?0?""*7i+'0?-!5*6+¯1aó7i+'0+%'*+aó13*+
((+,1*;pr3.*+e"a1)%'*i"bB31(S*+7¢"27
û
ü
*- |~` f[ | F ,Ø
a.*+,+'-/+g*+7h+û
ü
[ | F Øë I , \ ? J $" ü #&%'!()( *- |` f
_40)+7¡"?0?0++=S)40*6*6+ ?
J Á_`40)+7-~*6+7i+g}a1+7l
I
, \
û
{
û
"$#&%'!()( 767¢"7#*b+%657i)%,%'+77iS!*+ ? J _40)+G77h}%6"2*+a&¡Ò"*6*+£Ba&¡"*41"40
J 5lq_40)+¨ÒB)7 W *+g_`40)+!77U"S+a£*#+%6¯7i)%,%'+77iS
13ïûüÒ"7+=S)40*"*7 ? J _40)+
ü
?
Technion - Computer Science Department - Technical Report CIS9609 - 1996
767i}%6"2*6+7¡"*c+%6j7h)%,%'+77ie*+(10243+,ñÖ7_40)+-;*+LBac7i+'0+%'*+ab13£*+((L+,1*
ü
û
"$#&%'!()(
´Lûõü+,*)7*+g`B"40¦_`40)+-kS"*7
7h)%,%'+77i7 W ¡1"?0+Èûü
a
"*'77i)%,%'+77iS7,;+7h++e*+#7!+g_40)+7,
M©t
uY8¸A·5Á¾¶4z/¸ú8¼t¹¿
mn*0B 77û
$" ü #&%'!(/(
.
º³ãá
x
ü W
ü
+%,)7i+"*+'(a7¦0+77ra1+7,û
¡1"?0+¯ûõü#6+,°/+'}(a7B3fa1+7,b¡+,_+, W
"*g.3£(L+,i-/
Ã
[Ë-k0+4_+7 P | k |~W
J
-
(& x '*),+(-.-
"27r40¡Ò437¦(+'-/+,+a#_+,;û
a1+!"*+.*+,+!S%'+ W
K 6A}EJZ@º<D>JZ@¨*jû
·Há
?
+,*)7Ò*+`B"40
$" ü #.%'¥(/(
"$#&%'!()(
¡1"?0+¯ûõü
+'(a7+%6
(L+,i-kS7e0+77
!6+ïK u E=?x1Ey/<D>J@,gÌ
BS £-/
+'!(10+*^¿Á")+#´û
ü
+,_+,
T 7¢"%'+!*+,3£L+'0S¯*7h)*+,+7e)a1+,g1a7e**¡¨+,6+J>Sy~7h+'0+%'*6+a
13*+gS((+,1*
Ë}a1+'0nû
"$ü #&%'!()( )7h*
((103£+,_+,3.+,_`40)*h"S£-/)%'*i"£-/+,_+,63!0+4-
¿Z`031À'"$*6+G%'!(10+'1"*U3-ûõü.a5ûü
"$#&%'!()( W 0+,*.)7S77i)!+£**!*+G7h+%6*+,+¯S7!
)1"?-kSô%61"b-n%'*
!
aba1+,(*¸0"#"*.- ` GÒ+1)#+,!-e}a1+7#+'(a1+a&13$û "-ü #.%'!(/(
f
"27#*+.B)#L+,!-"1*+,40
a1+7"*+.*+,+ W "nµ+ W/10 #c" $û
"$ü #&%'!(/( 7i(7
0+4_+7l *#+%&0+4-e"*
S
#
"
/
±0?027!c_+%'*- `
+'0+,!+,1*7,b+,+'-/+.*+.B)#L+,-%,`0027#*6¯+,_``0)*i"&-k)%'*h"S&"7L)a1+a&B3
` f g`¡+,_+, W ¡+,b*+#!a1+'0"ø0+,_S+'0 W û F W "27g7i.40?0+,* ` W *+#B)#L+,g-¨%,40?027g*.+,_`40)*h"S
-/)%'*i"7e"27;U
* û F .,- f ¢* *+0S7i* ` ÍU* û F .
, +,1*i"+7g+eùi)7i*
*6+_40)+7g-
à Y.¡"*f40*+,6*i"
7¢"Â,
%,40?027
a
&)((+,#0?"#"*!&*+.B)#+,#-e}a1+.+'}(7¢"7!a&%,40?027#*b+,_`40)*h"S&-k)%'*i"&"ûõü"7
"_+,ø"c*+e-/0?0¡"Ë0+,!
?Sa
Oø:1K¯Jx1w[K4z£>i¤¯C'E=2=Ö@£y/>yD:BKbK.6A}EJZ@º<D>J£¤,x1JCy/<D>J
a K
l Ô
*- , f #c"
[K4zg>i¤CES=2=Ö@gy/>£K
P ø:BK#Jx1w
H[,û
u ES=x1Ey/<D>J¤ºx1JC4y/<2>SJ@
9:E<< ^E^
Ô2
z.UGEGEJ
+g1)#+,e-;%,40?07*û
"fû
ü
H[>SxBJ
<@
J[l¯ûõüb<@·[>x1J
FG> u KÏKJ
t K t
[,
*-
t K t
[,
., f
ë"*+,*i"7
"27+g(10)7e*+gB)#L+,
-;%,40?027"c*+
ü
/* -/*+S((+,1*
`UG¥VAL-e \ ` £
,
ac%'+#¡"* `G!VAL$e \ ` &[*ª-ke+,_``0)*i"¯*+#((L+,B*7h+'0+%'*h"S,^Ò+,+'-kS+ W *+
_+,*+
7h)%,%'+77i7l~¿Z+%67i)%,%'+77h;¡+%,40?0²ûür*U¡Ò"%'+ W %'+¡"*
7¢"#)102*i"
1)L+,
-+'}(7¢"7
%,¯++'}(6+77i+a.13£*+e-k00¡".+%')+,%'+
*n´A,
-Dr ¾ *- ~| ` , \
d¸+(_+#13£"a1)%'*i"f**
77i)!+g**"*02a7-/+,_S+,3
-
Y
"?
!
J
-5r ¾
D
-5r ¾ *$ |` .,
*- |` ,¨Å*$
` r ¾ *- |` ,
\
\
\
*-
-*
, f #c"
-5r ¾
*- |` f[,¢
"?-
` \
` \
*6+,¡"27i+
=
Ò+"+ S)40?"*U3¸02a7-k
ÿ* - r ¾ -* |~` 5., - r ¾ * ` $[,n
" *- ., f # N *- ., f # L $
#" *- [!, *- ., f # L%$
N
L
., *$ ., f #
., f #c"
Y
` \
Y£a
` \
g7i)%,%'+77i'7+,+,*6+a.-/*+#%')+,1*(B7U"*h"SG*6+40h"*6Ó"27e+%')7¢"_+'03
` ú#ac%'+!*6+,_40)*+!*+#+7h)10*h"
¿Z+%6¯-*+
Technion - Computer Science Department - Technical Report CIS9609 - 1996
%,40?0+af*i¡"2%'+Ê·%'+!*£7¢")102*6+!*+#((S+,B*g¡"*&a1+,(*
1ac¡"*Ga1+,(*6
` &[g+,+'-/+
*+,+g¡Ò"*6Ga1+,(*
%,¯L+g!+7i)6+a£13*+e-/0?0¡"+%')6+,%'+
`
d¸+!(_+!13b"a1)%'*i"**
77i)!+g**"*02a7-/+,_S+,3
-Drt¤
-5rX¤
D
J
"?"?-
-Drt¤ $* | ` 5.,
!
-5rX¤ *- |` , \
*UX,
*+#B)#L+,g-¨%,40?027g*!+,_40)*i"b-/)%'*i"7e13Gûüe-/
7i)%6f
W
`
*- |` ,
*-
` \
Y
-Drt¤ *$ | ` f[,n÷*+,¡Ò"7h+
., f
` \
=
c+!"+ S)40?"*U3&02a7-/
*$ |` ,
\
` \
Yca
-5rX¤ *- |~`
, -5rX¤ *- |` $[,n
f
#" *- ., S# " *- ., f # N%$
E*- [,!*$ ., f # N
*- , N *- ., f # N
*- , f
\
` \
\
a
"$#&%'!()(
Ò+
_+`037¢"27¦"!(10?"+7**;ûõüa!ûü
+%Ë7;!a1_1*+¦m/-
"*"27!+r"!(S*1*
*c+a1)%'+*6+!B)#L+,-a1++'(7¢"7*¸*+!1)L+,#-+,_`40)*i"¸-/)%'*i"Ç%,`0027*+,&+
"$#&%'!(/( c·*+,¡Ò"7h+
v¡Ò"*6¯û F *-;, \ W û ü
7i)102a¸)7i+ûü
(10243+,
ûü#7i)102a¸+.)7i+a
£¿g+'!(10+ W
W
+'}(a7
*i"!+7Ò!+ga1+7*6¯û
\X*i"!+7!6+^%,40?027e*#+,_``0)*i"c-k)%'*i"7l
M©p¨
uY8¸
+.F
9Ó
s9Ó
ajF
s
»}¸&z)ºS¹¼)·5³8ÀO¼/¶¸Q¹ý¸c¸S³ãá
Ñ
º³
x
` \
"$#.%'¥(/(
-kS
\
"-ü #.%'!(/(
Yca
)*
û
ü
Y£af
(+,h-k67
âèçBéæ
40Si"*ô 4027i(6(B*6+7_+%'*67-;_`40)+7e)(b*+#BË+*+,+e¡+,_+, W
Ñõ6+*6+,*+a£*gaS"?ò+,6+,B*g14027,eF
(10243+,7¡1"?0+.û
$" ü #&%'!()(
-" ü #.%'!(/(
J
Ñó"27gc+'}*6+,7¢"b-;#"1"4G*6*^%,¯aS0+
s
"27g¯+'*+,7¢"b-;*i¡°/(10243+,#"1"4f**
%,baS0+#°D0+,_S+'0;!}a1+'0?"gm¢*
"-#.%'!(/(
"27g(LB77¢"10+*6£+'}*+,a¸ûõü#abûü
_+%'*S¨-~!a1+'027 W
9Ó
û
*.°/(10243+,B!+7,mn7i)%f£%,7h+ W
+%6f(10243+,g¡"?0?0¨4_+!
+e-kS+%6£-~*6+g((+,1*7,
"-#.%'!(/(
m¢*e"27*6+,!(*i"£*G%6024"ò**ûü
+g(10243+,)%b!%6024"
"27¡Lmn¸F
"27c7i(+%6"240%,7i+!-
F
9Ó
s
Ñ
9Ó
s
Ñõ¡+,+Ë+%¸!}a1+'0¨7h*a7e-/
W "c+%6.0+,_S+'0~-*+7i+%£*6+,+ W 103£+(1043+,7
*+
S%'*i"_+0+e-²7h+'0+%'*h".g!_+¦+*+,r(1023S+,7(767¢"_+'03!(77*+'";_``0)+7Ò**;+e77i%6"2*+a
"$#&%'!()(
¡"*!*6+¨(10243+,ñÖ77i+'0+%'*i"mn.ûõü
W +,_+,3#7i+%'a#Ë}a1+'0"27;S77i%6"*+a¡"*!*+(10243+,;¡17i+*)
"27*(1043 W *6+,+'-kS+
`0-¦-*+e!a1+'07e7i+'0+%'*7!_+ W aË*+
(10243+,
_++%6£-*+,
(S77i+7"*7
_``0)+**"27e77i%6"2*+a.¡"*c*+7i+'0+%'*i"c-~*6+g!}a1+'0+'0¡ó"*
¿Z¦*1"27+7h
-¨._+%'*6"F
"Ó
s
W *+*i¡¨g40h"*67(S(B*+r_`40)+7)(!*+*+,+
aS"?Ã+,+,1*i03dó1"?0+e40?0*+_``0)+7
ÑÅ%'!+-k6
*6+È76!+g0+4- W
"$#&%'!(/(
*6+_40)+7g-¨Ë_+%'*S"$ûü
aS"?Ã+,+,B*Ò0+4_+7l
ÁË"0?0)7i*'*+!*+.aS"Ã+,+,%'+ W
*+40Si"*7*}a1+
G
0+,*g)7e0BS G*g¿Á")+Ä10+¯#7i¡
7e*+!7i*6+,(7e* +,cB3c+%b-
d¸+j%,ú)7h+jûü£*6Ç7¢"#)102*+F
*-æ | *-Ã
[ | *-æ | *$Ãq[ |ÌÌÌ ,!,!,!,
43¯%'!+-/
"Ó
s
# \
ÑÓ-/*6+j%,S7i+b-*i¡f(10243+,7.135)7¢"5*6+G(10243+,
Y
"ÿ
]
Technion - Computer Science Department - Technical Report CIS9609 - 1996
û
G
*a1+
-" ü #.%'!(/(
¼
W "*"27(10243+, ñ\7r*)£*!Ë`_S+
)7i+7r-/)%'*i"£-nY#*#+,_``0)*+LBa7
¦0243+,
7i+'0+%'*71a
ÅÃ
{,!,;aø((1*+7r*1"27_40)+* G
¦0243+,[
P
+,_`40)*6+7b}a1+
)7U"
+9((+,1*ñ\7
P
a.!_+7Ò*1"27_`40)+#*
G
P )7¢"ýæ
*-æ.* P , \ {,!,¨a.(6(B*6+7*+g_``0)+* G
:BKAq=?EK4z
G
J>Φ@yD:1Ey<Dy¨<@:1<@#y/xBz'Jfy/>¯w#> u K!<DJ
M;yD:1KziK/¤,>ziK#<2yC'EJ
k
»}¸&z)ºS¹¼)·5³8ÀO¼/¶¸Q¹ý¸c¸S³ãá
df+!4_+.40+a13$7i¡b**û
Ãq[* k , \
W a1+,°
**Ë"*£¡"?00
ac!_+7
¬*.a1+
4Y | { | ¬
9Ó
s
Ñóa£û
$" ü #.%'¥(/(
G
"¯a1+
"G¿Á")+Ä
º³ Á ¼/³8¼)Áº&ú
x
"27#£S+,+,40?"À*i"&-#"1".4²
ü
k
W
ýÃ
[
)7¢"
+g_+%'*6((B*+aË)(£"27
10+.+aS"?Ã+,+,%'+Ê"c*+g+,4_S"
L+,*i¡¨+,+,¸F
uY8¸
k
a
"7£¸+,*6*+,a1+$a
G
;Y | { | ´
P
+e(1023S+,¨+,_40)*+7L*
%6"2a1+7.**
[K
t KC4< t KΦ:1Eyw> u K!Î<D=2=
fy/E `K4J
Ãq[ W -* Ã
[* P , \ A´ ,!,
)7¢"
+g_+%'*6((B*+aË)(£"27
7i+'0+%'*7
*¢¡"*¯#_``0)+#-Y,
%'*)`003b7i+'0+%'*
M©
!a1+'0$-5*+9(1023S+,
+e((S+,B*Ë}a1+'0+,_``0)*+7a1+
G
¦0243+,e+,_40)*+7}a1+
P
1a
¦0243+,Ê B¡
7*6*r"*e7Ë+'ò+%'*e£*+a1+%6"°
7¢"ø"
W "*"27*+g(10243+,ñ\7Ò*)£*!(10243
*i".-nY#*#+,_``0)*+LBa7
*-æ.* P , \
W
G
+.((S+,B*ñÖ7g!a1+'0-*+(10243+,#)7i+7Ê-/)%'°
*n¡"*¯#_40)+#-¨Y,
P
*a1+
¦0243+,
=
b"1*+,+7i*h"b(S(+,*i3b-
û
ü
"27*6*"*40¡377i+'0+%'*7eË!_+g¡"*GÈ_`40)+!6+*+,e+ )40¦**6+g_`40)+!-*+!_+7i+'0+%'*+acB3
#"1"4¯**
7i+%6+7*+76!+g*+,+g¡Ò"*6£*+7!+7i**+,3
?Sa
a l
à % É ' } rtq Ë:1KJ
y J y THUR * Vcz.Y}|~`UG¥VAL-e7| Ã % É ' } Xr q ,5ûü* Vcz.Y}|~`UG¥VAL-e7| *-Ã % É '} rtq | 8;8
@'@ºx1w#K#yD:BEyû
s
*UÄ,
û
8;8
¤,>zEJL
F
ü
ESJ
O
FG<DJ<Dw#E xS@,K!yD:BK@ºESwK
t
\
F
Ð
7,!,
4
Ð
9:E<< ^E^
d¸+¨¡Ò"0?0(_+***61"7((L+,*i3+'1"27i*7~-k+,_S+,3a1+"!*6+*+,+
7h(+a#13*6+*i¡¨g40h"*67
"$#&%'!(/(
13!"a1)%'*i"b£*+#a1+,(*£-;7i+%6;¿Z%'B_S+,1"+,%'+#¡+g¡"?0?0~(6`_S+"*-k*+g_+,7¢"£-ûü
¿Z
*¬},
`UG¥VAL-e \
Y W -k13c·
;
]¸·
og;Ì
y J y THUR * Vcz.Y}|~`UG¥VAL-e , \ Ã % É '} rXq * Vcz.Y , \ û "$ü #&%'!()( * Vcz.Y}|`G!VQL-e5| -* Ã % É '} rXq |
Ì
+,*)7e77h)!+%'+%'*6+77r-/ ` `UG¥VAL-e a£(_+"*-/ ` \ `UG¥VAL-e m/- Vcz.Y "27
(10243+,ñ\7a1+ W
*U{,
û
û
y J y THR * Vcz.Y}|`G!VQL-e
, \
%,%'aS"*6!*+e"a1)%'*i"_+!77i)Ë(*i"
4
û
y J y THR * YE{c.|`G!VQL-e , Y={S
4
A
U
]¸·
og;Ì
* Vcz.Y ,¥¦
g
W
$" ü #&%'!()( * Y={S.|`G!VAL$e5| *$Ã % É '} rXq | ·
r ]¸·
ogrÌk,!, Y={S
\ ûõ"$ü #&%'!()( * Vcz.Y}|~`UG¥VAL-e | *-Ã % É '} rtq | ·
r ]f·
ogrÌk,!,
C,
·
;
Lû
A
U
* VSz&Y ,
Ê
W
Vcz.Y "27
£S((+,1*ñ\7a1+ W
Y +*+7i)%,%'+77iS
¡"*£*6+g41".40_``0)+#%,%'aS".*!*+gS((+,1*ñ\7Ë}a1+'0n
m/0+,*
Technion - Computer Science Department - Technical Report CIS9609 - 1996
*U,
y J y THR * Vcz.Y}|`G!VQL-e
û
., \
%,%'aS"*6!*+e"a1)%'*i"_+!77i)Ë(*i"
U
* VSz&Y ,!¦
Ê
W
$" ü #&%'!()( * Y}|~`UG¥VAL-e7| *-Ã % É ' } rtq | ·
r ]¸·
ogrÌk,!,
"$#&%'!()( * Vcz.Y}|~`UG¥VAL-e | *-Ã % É '} rtq | ·
r ]f·
ogrÌk,!,Ja
¹û
\
y J y THR * YE{c.|`G!VQL-e , Y={SãA
û
y J y THUR * Y}|~`UG¥VAL-e ,
¹û
È"
ûõü
Ò+."1*)1"*i"ó+,1"a¹*+¯L_+.0+,!c"27Ë**##"1"4
S77i)!+7!5a1_+,73f!a1+'0
-g*+
((L+,B*4bmn*!77i)Ë+7*6**+.((L+,B*# 1¡
7*+(10243+,ñÖ77i*'*+,3 W
aj"*7103f"1*+,+7i*"27#*
7i+'0+%'*e!_+7r**+g¡7h*ÒS%,%'aS"!*#*+(1043+,4ñ\77i**6+,3+e0+,!!7437 W **"?-3)£4_+g
H>4> t
+7h*6L+'0?"+,_S+**3S)¨S((+,1*ñ\7r!a1+'0
"7aS"Ã+,+,1**** W 3S)ï%')102aø103L+,+'±*B3
)7¢"¯û
"7h*+ac-#"1".4²g++a1+,g7i)102ac*+***+!L_+e0+,!£a11+7*e!+c**
ü
ûü7h+'0+%'*'7.!_+#**e"27Ê+,**+,S%,%'aS"¯*¯7h!+#ùi+%'*i"_+%'i"*+,h"S
3*/-k6¾*+(1043+,4ñ\7r("1*¨-²_S"+,¡
+,*6*+,!_+
)*'*+,£7i)ùi+%'*i"_+'03
W
c,
mª-*+e(1023S+,¨a11+7;S*;4_+
W %,%'aS"#*Ê"*'7¨7h**+,S3
#+'0?"210+#!a1+'0-¦"*'7((S+,B* W *+,¯(10243".#"1"4£"27
ÈBa£%,)*i")77i**6+,3
M©û
üE³A·5»}¶·5»}ºS¹}¼/³½`¸c¶C¹ý
+£û
·* À.¸cº» ¼)³5¹}· ¹ý¸±Á¾·;8¸z
*¢¡1"2%&"27#b7i(+%6"240
%,7i+.-^û
40i"*ô7Ê¡¨+'0?0
7#"1"4
ü
ü
,
W
"%'S('*+7#b"!(10?"2%6"*
77h)!(*i")*#*+£a1+,(*&*b¡1"2%&*+.((S+,B*7h+%6+7,Gûü#**7i+'%+7#£*6+,+-a1+,(*6
`
W 77i)Ë+7**"*7S((+,1*^7h+%6+7*#0+,_+'0
***+.(1043+,!7i+%+7*c0+,_+'0
` ¹[
W
`
W
a£*+,b77i)Ë+7**"*7((L+,B*g77h)!+7
+,*%b1"27"27#¯(L*+,1*i"240?03$¡6f77i)!(*h"S
W
)*"*#"27#
1}aba1+'-/+,7¢"_+.!+%1"7h¯+#(10243+,S77i)!+7e**e"*'7g((L+,B*7h+%6+77ga1+,+,($S7e*+!(10243+,
%,f7¢"#)102*+!*+Ë((+,1*ñ\7g7h+%6gmn*Ê"7eË+1"0+77Ê-ke*+!(10243+,g*¯S77i)!+#**"*7gS((+,1*
7i+'%+7Á*
ea1+,+,(+,¦0+,_+'07¢"%'+"*~%,*7U"#)102*+e7i+%6*6
7i)%6a1+,(*`¡+,_+, W 7²¡ÒS7aS"27%')77i+a
"£*+"1*a1)%'*i"
W "?-*+e(1043+,7Ò+7iË*#+'0?"+,_+**;"*7r((L+,1*Ò7h+%6+7*Ê0+77i+,ea1+,(*
*n*6È7h+,**(7
*+,ø"*e%,£)*i"?0?"À,+Ë*1"27+'0?"+'-
B4"7i**+((+,1*
` [Wþ
pr)*r¡B3#a1B+7r*+(1043+,r4_+*7i+%6!*e0+,_+'0
` "
*.0+,_+'0
` "
"?-
"*; 1`¡e7**¦"*7((S+,B*7i+%6+7*Ê0+,_+'0
0ómnfSa1+,g*£(+aS"2%'**6+!((S+,B*ñÖ7g7i+'0+%'*i"
` [
!¡+,_+, W
W
.,
W -/+'!(10+ W
"*Ê"27e"a1+,+a&+,)b*¯7¢"#)102*+#"*77i+%6
"fa1+,*fK u E=?xBESy/K*+#!+,h"*-¨*6+!((S+,B*ñÖ7g7i+'0+%'*i"f-/g*+Ë(1043+, W
"*
7i+'%+7S7a1+,+,(G7¨"*e%,
d¸+4_++'*+,a1+abûüe*aS0+#!a1+'027e-¨a1+,(*b-;7i+%6d¸+a1+'±+!7h**+,S3*.+!Ë(4"
*-Ã % É '} rtq ~| ` % É '} rtq ,
*$*-Ã % É '~} rXq | ` % É '} rXq , |
Aq=?EKz¦"27*6+,ïÊ(4"
7,¡+,+g]¸·
og;Ìø"27¨`07hÈÊ(1023S+,
]¸·
og;Ì
+£+'*+,a1+a5`0Si"*ò"27aS"?Ã+,+,1*#-k6ò"*'7!7¢"!(10+,_S+,7¢"&+£("1*&dó+,+7#*6+ï7¢"!(10+
40h"*6
40?0%,*+7e7.B3.+7iS)%'+7
S7(LB77¢"10+#**6+%,40?0**
7¢"#)102*+7*+S((+,1*ñ\7e7i+%6
*n¡1"2%6."27*+g%')+,1*Òa1+,(*6¯40?0%,*i"£È"1)7+.,
W *6+
+'*+,a1+a£40h"*6
40?0}%,*6+7*+g7¢")102*h"S
7Ë13+7i)'%'+77È"*£+'0?"+,_+7c**!*6+GS((+,1*£¡)102a4_+¯)7h+a
¾·-!%')7i+ W
*+#!a1+'0;-"*7e((L+,B*4ñ\7ga1+,(*b-Ò7h+%6
W
*+#(10243+,7i)102ab+,_S+,g+'Z%'+,+ab*6+*6*40~a1+,(*6b0?"È"*
40?0}%,*+a-/*+(%'+a1)+¿Á")+e{g7i¡
7#+'Z!(10+-²((103S"!û
*a1+,(*6
+,B'aS0+767!-
a¯£((+,1*!a1+'0**
7i+%6+7*.a1+,(*f
¿Á")6+0?"7h*7*6+g+'}*+,a1+acûõüg`0Si"*¯
[
ü
¡"*e(10243+,**;7i+%6+7
a M*(a,3,((f1,3),(f0,1)))= 0
Technion - Computer Science Department - Technical Report CIS9609 - 1996
b
f0=4 d
h
f0=6
i
¿Á")+g{¦ØÙÚçnÚã¦Þ,Ý
ÜnÚâ6éÜnç¢å
M*(b,1,(f0,1))= 6
c M*(c,1,(f0,1))=-1
e M*(e,1,(f1,1))=-7 f
j
k
f1=-8
f1=-7
m
l
7Ò"b*+Ê-kS!+,g%,7i+ W
n
f1=-4
G
Úâà,ä?ä?çæ4ÚáÚÜià,ã¢Úî#ê`ègâ'à,ä?ä?åáSæ
ü
o
f1= 0
>îï û ïEpü " ï û ïE1ü [ ï~õ ì ¶ àlâiÙâàlää1å?çMøÜ¢å?ãnãnÚ6ááÚëã
Úå?ççnÚä?Úâ6ãnÚî.êègÚà,âiÙ!âàlä?äkì
ø øÙBà,ãßÞ
ãnÞgã¢ÙÚáÞîÚå?ã¦åçâ'à,ä?ä?ÚîgÝ2Ü¢Þß.ìØÙÚîBàlçnÙÚ6î!ä?åáSÚçç¢ÙÞ
f0=-1 g M*(g,1,(f1,1))=0
f0=-2
*1"27
40Si"*%,G`07hL+¡i"**+,£"fË+,°/(77Ò-kS
*+,+g¡Ò"0?0+g+'(a1+a£S+*i"!+g103+40h"*6
7h***+1!+
((1*+7r_+%'*'7;-~_40)+7¡+,6+g+%£+,1*3
"27#77i%6"2*+ab¡"*¸+!-*+!Ë}a1+'027#7gL+'-kS+pr)*#¡1"?0+*+.7¢"!(10?"±+a
°/+%')7U"_S+ï40Si"*
P V L-e
%,h"+7+_40)+
(L+,;!a1+'0 W *6+¨+'*+,a1+a.40i"*Ó%,i"+7r0"27i*-
_`40)+7-/+%6!!a1+'0ª+
y
+'0+,!+,1*g-¨*+
i""40û
VßL-e
0"27i*Ê"27e*+#_`40)+767i}%6"2*6+a¯¡Ò"*6j.7i+'%b*£a1+,(*
*+,+Ë¡ÒS7e103GS+È7h+%6b-k6B*h"+,4+,6+'-k6+ W
ü W
103#+_40)+e-k+,_S+,3#!}a1+'0n;¡+,_+, W *++'*+,a1+a.û
*£13¯a1+,(*+,%'+ W
ü
P
13¯*+
y
VÿL-e
Ë}a1+'0nmnb*+
*+#°/+%')'7¢"_+_S+,7¢"f(S(B*+a
%,.%,40?0"*7i+'0?-+%')7¢"_+'03Ë-k6¾13a1+
+,_+,3£0+,_+'0"b*+!7i+'%c*+,+g"27e(L*+,1*i"240?03f£7i+%ø-k6B*h"+,e-;+#-;*6B7i+
+%')7U"_S+^%,40?027,*"7r*+e+7h!¡13!*+e+'}*6+,a1+a!°/6+%')7¢"_+40h"*6
%,i"+7e0?"27i*-_``0)+7
"-#.%'!(/(
-/
+S%b!a1+'0n^¿Á")6+ï4Y7i¡
7*+_`40)+!*i"2%'+7e((1*+a£)(b*+*+,+#¡+,f%,40?0?"Gûü
£*+76!+g*+,+S7r"£±)+#{
Ò+g+'}*+,a1+aø+,°/(77_+,7U"S."7e7i¡."£±S)+!
`0Si"*
¡"*Å*tÃ
ü
W %,40?0+a
s | */Ã s #c" | * Ì=Ì=Ì~| Ã [ , ÌEÌ=Ì ,!, W "27.+=)1"_40+,B*b*%,40?0?"¹*+
*¥*-Ã s |~` , | /* *-Ã s c# " |~` 5., ÌEÌ=Ì!| */Ã [ | Y, EÌ Ì=Ì ,¥,*n+%,40?0;**e"?- J W *+0+,_+'0¨-!a1+'0"ø"7
Ò+¯i""40û
+'*+,a1+afû
ü
¡Ò"*6
0+77*f*6+ïa1+,(*&-
7i+'%
W
¡¨+Ë+,(102%'+£*+!À,+,S°D0+,_S+'0Ò(10243+,B3$ø#"1"4$(10243+,*6¯ S+!*+
,
13£%,40?0".*+
+'*+,a1+a$`0Si"*
¡"*±*¥*-Ã |~` , | *$*Ã ., | *-*$Ã |~` $[, |EÌ=ÌEÌ , =Ì Ì=Ì ,!,!+,6+#"27g+aS"?Ã+,+,%'+#"&*+
|`
¡43¯**¡+!+'}*6+,af*+.À,+,°D0+,_+'0(10243+,g-/*+%,7i+.¡+,+
J D ` .mn&*+ï%,S7i+!-*+.+'}*6+,a1+a
40h"*6
¡
g
+
,
+
1
(
2
0
'
%
+
6
*
g
+
,
À
,
+
D
°
0
,
+
_
'
+
0
!
}
1
a
'
+
¦
0
B
3
!
*
/
*
Ã
,
$
*
*
Ã
, Ì=Ì=Ì¥| */Ã [ | Y, ÌEÌ=Ì ,!, W
[ |~` [ |
[ |~` [ 5
*(°D0+,_+'0¦(1043+,!a1°D0+,_+'0¦!}a1+'0+, 'mn£(6*i"2%')10 W ¡+%,.+,**+g#"1"4G40h"*6
ð
ñóò 8.5~
*3S5W<Zô5õö6T8ýqÒ_d3J÷Ò;ø
Ê'
98$8ýÊ
Ò
+#86
¼`¡*6*¡+4_+a1+,_+'0(+a£!40Si"*-/)7¢"#((S+,B*r!}a1+'027 W ¡+¡)102a0?" S+
*g+,_40)*+e*+
(S*+,B*h"`0+,+'±*;-q)7¢"*61"7`0Si"*¯¦df+%'a1)%'*+aËg7i+,*-q+'(+,h"Ë+,B*7¡"*#*+
û
ü
40Si"*
*!*6+7i**+g+'Ã+%'*-_`i")7(!+,*+,7r£*++,+'±*e-)7¢"ï£S((+,1*!a1+'0n
ª©
ùWú8¶¸S»}¼/Á¸c³7¹}ºS¹¼)·5³Á¾¸A¹·;8·7z)·½5¿
mnca1+,*6. +g*+g+'(+,i"!+,1**h"S¯!6+e-k+7U"10+ W
+'0?"5(10243+,7 W
¡+e0"#"*+ac*+g+'(+,h"Ë+,B*7*!S+,°D0+,_S+'0~!a1°
Í*U¹+,_`40)*h"S
7i)%6ó**.*+¯(10243+,£(LB77i+77h+7¸!a1+'0-g"*7!S((+,1*ñ\7£7i*6*+,3
Technion - Computer Science Department - Technical Report CIS9609 - 1996
9:E<5>.?A@B5:E? û ü * VSz&YN|`G!VAL$e5| *-*/Ã % É |~` % É , |/³´7´ í ³µ¶ 7,$,
^ `G!VAL$e \ Y
:E?m.BD:N F| Ã % É * Vcz.Y ,
?A> =?
THR HUWX{SG g 9@
K] g û z.G=;G=J * Vcz.Y ,
^ <_: +%6 Y={cB
A K
^ `G!VAL$e \ VcWXHPG=I UHWX{cG g à % É * Y={S ,
? =?
z~V}V z.HUI.`S|z~V}V HUWX{SG. g û ü * YE{c.| #"* `UG¥VAL-e |` ¡ %$% , |/³´7´ í ³µ¶ D,
VcWXHPUGEI z.HUI.`S|/VcWXHPG=I HWX{cG. g û ü * z~V}V z.HI.`c|~`UG¥VAL-e f[ | *$*tà % É |~` % É , |/³´7´
^ VcWXHPG=I UHWX{cG THUR UHWX{cG
THR UHWX{cG g VcWXHUPG=I UHWX{cG
THR z&HI.` g YE{c
:E?m.BD:N THUR z.HI&`S|THR HUWX{SG.
k§;á!Úëã¢ÚáîÚî
¿Á")+#
9
ÚÜ¢çnå?Þá#Þ,Ý
ãnÙÚ
,
-/)%'*i"úa¹fa1+,(*¹-^7i+'%
¿)6*+,!6+ W
W
í
³¨µ¶
7,$,
àlä?æÞÜ¢å?ãnÙßÅã¢ÙBà,ãâ'à,á#ÙBà,áSîä?Ú¨àeßÞîÚ6äBÞlݲîSÚÛãnÙ
ü
a¹77i)!+7È**"*7!S((+,1*#"27¸7i*a'a&#"1".45(1023S+,
*+#%'*)40¦((+,1*g)7i+aø-/)e+'}(L+,i"!+,1**i"f¡7¨"a1+,+a¸!#"1"4b(1023S+,m¢
a1+,*6g+,_`40)*+e*+40h"*67 W ¡+4027i#40?0¡+a!e+,S)10;#"1"4!(10243+,;*g(10243B4"7i*;*6+Ò76!+
((L+,B*4
Ò+,+'-kS+ W *++'(+,i"!+,1*7ea1+7%'i"+aø"¯*61"7e7i+%'*i"."1_0_+*+,+g(10243+,7
»
s
*!*-Ã
*!*-Ã
*!*-Ã
\
"Ó
F¯6Ñ
F
¡
s
0 1
V0 5
V1 1 7 2 0
0 1
V0 9 6
V1 -5 2 -7
0 1
V0 4
V1 7 6
d
V0 6
V1 2 -7
0 1
¿Á")+!YZØÙÚ
0 1
e
0 1
V0 10
V1 -8
0 1
f0= 6
f1=2
V0 -2
V1 9 7
f0= 3
j f1=-8
f0= 5
f1=1
V0 4 -1
V1 7 -4 0
f0=10
i f1= 6
V0 -8
V1 6
a
f0= 9
f1=-5
f0=-8
f0= 4
0 1
b
0 1
f0= 4
f1=7
h f1=-6
V0 4
V1 -6
% É '} rtq |~` % É '} rXq , | *tà ¢¡/f$r/É |` ¢¡/f$r/É ,!,
\
% É ' } rtq |~` % É '} rXq , |E·AF ,
8;89Ñ
rÑ
\
¡ %-% ¡ts&rXsEu |~` ¡ %$% ¡ts&rts=u , |/F ,
Òíâ.Ô!àEØ=×cÙXÓEÒ$×XÒ_âáòÔ$Û~Ò-ÚX×SÙXãMØ©ôáßÒ$ÕÒ)á.åãØEÒ)áïÞßà=ä8âáòÔ$Û~Ò-ÚX×AÖE×pÞßà=äÙXÓEÒ_ã~â=â.ã~×pÞßÙXÒóôÖ=àEÝ-ÙpÞßã~àÔ!àEØÔ7Ø=Ò$Ý$ÚXÒ!Ô!×pÞßàEä
F
ì íÒ$åÒ$åè&Ò$ÚAÙXÓ.Ô!Ù& ×XÒ$ÚpÞßÒ$×Mã¥ócØ=Ò$â=ÙXÓNì
N
k f1=-7
0 1
f
f0=-4
l f1= 7
0 1
V0 3
V1 -7
V0 -4
V1 7
f
à,ä?éÚrßà,ã¢Ünå?âÚçÛSÜnÞÛ1à,æ`àlãnÚî#êè
,´
c
0 1
f0=-2
f1=9
V0 -1
V1 -4 0
f0= 4
m f1=-2
0 1
V0 4
V1 -2
$" ü #&%'!()(
f0= 4
f1=7
g
f0=-1
f1=-4
f0= 6
f0= 4
o f1= 0
n f1=-4
0 1
V0 4
V1 -4
0 1
V0 6
V1 0
>îï û Eï 1ü " ï û ï/pü [ ï õ
ì
Technion - Computer Science Department - Technical Report CIS9609 - 1996
9:E<5>.?@CB5:=? û "-ü #.%'!(/( * Vcz.Y}|~`UG¥VAL-e7| *$*-Ã s |` s , | *$*-Ã s #S" |` s #S" , | * =Ì Ì=Ì~| */Ã [ |~` [ ,¥, Ì=Ì=Ì ,!,$,
<_<j ^ <: £
y \ J a1¡1*!Y
? y g !9@ |EÌ=Ì=Ì~| 9@ | Ã È * Vcz.Y ,
^ `G!VAL$e \ Y
:E?Um.B5:N ? J |EÌ=Ì=Ì¥/| ? \Y
?A> =?K] g û z.G=;G=J * Vcz.Y ,
^ <_: +%6 Y={cB
AÅK] ;
Y={S ? g ûõ"$ü #.%'¥(/( * Y={S.|`G!VAL$e | *-*/Ã s |~` s , | $* *-Ã s #c" |~` s #c" , | * Ì=ÌEÌ| * Ã [ | ` [ ,!, Ì=ÌEÌ ,$,!,
^ <: +S% y 77i%6"2*+a.¡"*b%')+,1*(1023S+,
` \ #"5* ` È |~`UG¥VAL-e ,
^ <_
: P \ Ê* `
,7 ? y n P ^ Y={S ? y n P
,
? y ¢ P g Y={S ? y n P 5
^ P \ ` a ;
J
y D
^ <_: kd\ g*Ë#"* ` ÐÈ ê " |~`UG¥VAL-e ,
,
? y ,n k g YE{c ? y ,n k
:E?m.BD:N ? J |EÌ=ÌEÌ-E| ? `G!VQL-e
C§;á#ÚëãnÚáSîÚî!áÞ4áO±/ÜnÚâéSÜnçnå ¿Á")+!
Ú
¼+£-e*+£(1023S+,7#)7i+a&()1"
W
,
ÚÜ¢çnå?Þá#Þlݲã¢ÙÚ
ü
à,ä?æ4ÞÜnå?ã¢ÙßóãnÙBàlãâà,á#ÙBàláîä?ÚàeßÞîÚä1Þ,Ý
îÚÛã¢Ù
¡1"%65%')102aS*!+'Ã+%'**6+£+7i)10*7!a1)+c*b*+.±}+aa1+,(*6
7i+'%£**¡7(+,h-k6!+a
df+Ë+7i)+g*+(+,i-/.%'+-;#(10243+,eB3¯%'a1)%'*h"£Ë*)Ë+,B*
8;8"Ñe;Ñ.qdf+g+,(S*+,+g*++7i)10*7¡Ò"*6¯+7i(L+%'*
*6S+a1+,(+,a1+,1*g_`i"210+
¡"*
ëÅ?l Ú
jk<_tcm= Ú k
j ?A: Ú Ícla´?
+#(10243+,g+,*7e[.("1*7-/Ë¡"
-k0B767,;+g_i"210+g"27*+7i)
W
#("1*e-/a1'¡¾a¯À,+,6£("1*7
-¦("1*7++aï-kS¨*6+g¡0+*S)!+,1*aS"_S"2a1+acB3.*+
B)#+,-~1!+7,
df+Ë+7i)+aË*+(L+,i-kS%'+)7U"#*6+,¨%'i"*+,i"2 W 0?" +e*+(1"+%'+a1_`B*+*;*++,a!-*+B!+ W
1)#-¡"7 W )*¡+aS"2a£S*±a¯7¢"1"?±q%,1*gaS"Ã+,+,%'+#L+,*U¡+,+,£*+7h+!+7h)+,!+,1*7,
Ò+g+'}(L+,i"!+,B*'7¡+,+g%'a1)%'*+aø)7¢"*+a1S4"£-~%6+% S+,7,;1"271!+g¡7%17i+,£+%,)7h+
"*r"7%'!(10?"2%,*+a.+,).*#%'a1)%'*+40?"27i*i"2%e+'(+,i"!+,1*7 W 3+,*¨7¢"!(10++,).*(L+,i-kS
7¢"1"?±q%,B*
1)L+,-BË+7,df+)7i+a+,_40)*i"#-/)%'*i"7¦7i+a**¦)7i+aÊ-kS;B#)+'0nñ\7%6+% S+,7¦(1023S+,Ä4n
0?"#"*!-YY.Ë`_S+7g(+,B!+Ë¡ÒS77i+,*#a1)i"G*6+!*)!+,B*4ïÒ+,+'-kS+#"*"7(LB77¢"10+£**
40*)S£+40i"*
"7e7i*6+,*¯S*+, W "*
%,£*¡Ò"."&YY!_+7 W a.*+,6+'-k6+e±1"7h
!B7h*B!+7¨"f!a1'¡;dó+,¯*6+a1_B*'+g-*+7h*+,e40i"*ö"%'+7i+7 W "*
.3.¡"¯Ë+
BË+7r"$YY!_+7 W *6B)7r"%'+S7¢"!"*7!+°/(L"B*7h°/(+,°/B!+
m¢£*+g+'(+,i"!+,1*7ea1+7%'i"+aï"c*1"277i+%'*i"
W ¡¨+7h*)aS"+a£*+e+'Ã+%'*-*+e-/0?0¡"!"a1+,(+,a1+,1*
_`i"210+7,
@7?j7m
R*Ú
@k¼Ì]?:E?SD>?
Ò+aS"?ò+,6+,%'+L+,*U¡+,+,f*6+a1+,(*&-Ò7i+%¸)7i+ab13G*6+!(1023S+,ab**g)7h+a
8;89Ñ
rÑ
B3
^ Bk5>.m=ô<7 Ú k
@ ¼Ì]?:E?SD>?
S)40?"*U3Ë+,*i¡+,+,*+e+,_``0)*i".-k)%'*h"S£)7i+a!13!*+e(10243+,
+gaS"Ã+,+,%'+e"
8;89Ñ
rÑ
a.**)7h+a.B3
X
@7?j7m
?A:.:=<:
R*Ú
Ò+¯aS"Ã+,+,%'+bL+,*U¡+,+,*+¯a1+,(*65-^7h+%6)7i+a135F
((+,1* W
Technion - Computer Science Department - Technical Report CIS9609 - 1996
^ Bk5>.m=ô<7 Ú A? :.:=<:
»
s
Å-/#*+£!a1+'0
-e"*7
K8r8"Ñe;Ñ.q
a.**
%'*6)40?03£)7h+a.B3
+¨aS"?Ã+,+,%'++,*i¡+,+,*6+¦-k)%'*i"#)7i+aÊB3F
.-/*+r!a1+'0B-1"*'7((L+,1*
s
»
W
8;89Ñ
rÑ
a.**
S%'*)40?03)7i+aøB3
S)40?"*i3óa*+ø-k)%'*h"Sú+,6
m¢5'a1+,*¸+f10+¯*&!+7h)+¯*+ø-k)%'*h"S
W
¡+Ga1+'±+a¹*+
-/0?0¡"Ë-k)%'*h"S7,
*Y,
Ñ,Ò ¹
'
* R , \ÜÛ
Ã7* R ,
"?-
*-Ã7* R , D H
Y
*+,6¡"27i+
à W ¡¨+¯%,ú%'+*6+¯f7i+=S )+,%'+G-
-/)%'*i"7ËB3_3S" H 7
ýS)40?"*i3f-Ò*6+!-/)%'*i"¸"%'6+7i+7,¯Ò+,+'-kS+ W ¡¨+Ë!+7i)+Ë*+ïaS"27i*'%'+!"±)`0"*i3
¹ Þ' Ý aÑÒ ¹ ¯' ß B3 H " H N H
¿Zb"_S+,5+,_``0)*i"¹-k)%'*i"
"%'+7i+7 W
·ÑÒ
+,*i¡+,+,
*+
&+'}(L+,i"!+,B*.¡7%'Sa1)%'*+a13f_`3"f*6+£_`40)+£-
S+£-
*+Ë"a1+,(+,a1+,1*£_`i"210+7.a
±1"Ë*+
_40)+7-*6+
+7h*¿+%6_40)+g-*+e*+7i*6+a!_i"210+ W ¡+
Ë+7i)+aË*+e(+,i-/%'+
-
8;89Ñ
rÑ
*+g(10243+,eB3Ë0+,**i"!"*(10243¯#7i+,*-YYB!+7B`"7h*
ª©
àý·5Á
¶8º»}¼/³C½á
x
¹}·ãâåäæKäsâèçBé
+£+'(+,i"!+,1*7.a1+7%'i"+a
"*1"27!7i)7h+%'*i"úa1+,!S7i**6+*+£(L*+,1*i"240^a1_1*+.-
)7¢"û
ü
¡"*¯#%'6+%'*¨S((+,1*!a1+'0_+,¨]ø"1"4;m¢£*+e±7i*+'(+,i"!+,1*e*+
(10243+,
7h+%6+7;*6Èa1+,(*6
9Ó
´¡1"0+e*+((L+,1*¨7i+'%+7*a1+,(*.[rF¯ÑeF
¡1"0+û
ü
s
]8;8"Ñe;Ñ.!7i+%6+7*a1+,(*6
77h)!+7**
B¡
7***+S((+,1*7i+%6+7*#a1+,(*£[++7i)10*7r-
*1"27r+'}(L+,i"!+,B*6+
7i¡
"¯*+e-/0?0¡"*10+
F¯6Ñ
F
F
s
»
d¹"7
oe4¡
7
Ì
B77h+7
¦"B*'7
´BY
´BXX
X
S X
´Ä
¬`
X
4[
_}7,98r8"Ñe;Ñ.
s"Ó
_7,]8r8"Ñe;Ñ.
m¢!*+
7h+%'a#+'(+,i"!+,1*40?0q(10243+,7;7h+%6+a#*a1+,(*.´rF¯ÑeF
\8r8"Ñe;Ñ.G)7h+a$S*+,!û
¡1"?0+
ü
"Ó
s
aËû
ü
)7i+a#S+-k)%'*h"S
1¡
7"*7gS((+,1*ñ\7^-k)%'*i"&¡1"?0+¯FcÑeF
9Ó
s
"!(10?"%6"*i03
77h)!+7**g"*7Ê((S+,B*#)7i+7*+.((17¢"*+£-"*7`¡b-k)%'*h"SG+Ë-/0?0¡"f*'10+ï7i¡
7Ê*+
+7i)10*7*`"+a
F¯6Ñ
F
F
»
s
98r8"Ñe;Ñ.
s"Ó
_7,]8r8"Ñe;Ñ.
_}7,
d¹"7
oe4¡
7
Ì
B77h+7
¦"B*'7
´
ÄÄ
Y
{Y´
[Ä
Ä{
XÄ
{1¬`Y
m¢¯*+g*61"'a£+'(+,i"!+,1*^`00(10243+,77i+%+aø*a1+,(*c´;+g*6+,+(10243+,7)7h+*+7!++,_40)°
*i".-/)%'*i"
à " $* z&HI.`c|)VcWXHPUGEI , \
`¡+,_+, W ¡1"?0+gû
ü
û
HNL-G=I y UH W *- z.HI.`c|)VcWXHUPG=I ,c$Y Ì YY´ ÄE- zEL-HW wy Oc{cI.G=Y *- z.HUI.` ,
B¡
7Á"*7((S+,B*ñÖ7¦-/)%'*i"
W
F¯ÑeF
9Ó
"!(10?"2%6"*h03¯S77i)!+7¦**"*7S((+,1*
s
)7i+7*6+g-/)%'*i"
à " -* z.HI.`c|~zVNVcz.J_GEJSL ,
\
\
\
à [ $* z&HI.`c|)VcWXHPUGEI ,
û H}L$G=I y HW *- z.HUI.`S|/VcWXHPG=I , û
H}L$G=I y HW *- z.HUI.`S|z~V}Vcz.JG=JSL ,
Ä
Y
Ì YY´ Ä- zEL-HW wy Oc{cI.G=Y *- z.HUI.` ,
Y Ì YY´
Ä.- zEL-HW wy Oc{cI.G=Y *- z.HUI.` ,
8;89Ñ
rÑf(+'-/+,7#+'%6+.(B7¢"*i"7
+,+'-/+ W
¡1"?0+
FcÑ
6F
W
9Ó
S77i)!+7**"*#(+'-/+,7#*
s
4_"2a.*+,¯mn¯*61"7Ò+'}(L+,i"!+,B*4¨+Ê-k00¡".*10+7i¡
7*+g6+7i)10*7S*4"+a
Technion - Computer Science Department - Technical Report CIS9609 - 1996
F
oe4¡
7
Ì
B77h+7
¦"B*'7
¬B
´BÄ
Ä{
{Y
[´B
X1¬
´
{XX
_}7,
s
»
d¹"7
98r8"Ñe;Ñ.
s"Ó
_7,]8r8"Ñe;Ñ.
F¯6Ñ
F
0?0*+,+.+'}(L+,i"!+,B*'7"*1"277h+%'*i"57i¡¾*+£a1_B*'+.-*+£!a1+'0"$((B%6$_+,*+
#"1"4f((B%6£¡+,b)7¢".*+%'6+%'*
Ë}a1+'0nm¢¯*++'}*g7i+%'*i"7e¡+*+7i**++'Ã+%'*
-!a1+'0
+,SÒa¯aS"?Ã+,+,%'+g"¯(10243S"£1"?0?"*U3c£*1"27ea1_`1*S+
·+43ï)+g**e7¢"%'+Èû
40?0¡
a11+7*()+ W
"?-;¡+
s
*£)7i+
s
7i¡Å¡ó()1"¯%,£+(+,i-/Ë+a."Gû
S7Ë_Ë|Ô
"Ó
"*43!0B7h+"*7a1_`1*S+_+,eF¯ÑeF
d254¹()1"Gaf`00%,*+!L*f(10243+,7e+=S)1"_`40+,1*!*i"!+#m¢+%'*i"$X.¡+
"Ó
F¯6Ñ
F
ü
ü ?´?Ì]?>më< ^ m R ? @kÕÌ?A:=?S5>.? t
jM lsÎkX_ͱl&ÖMppômEÎ
<7 m R
? Ök?c?×£mi< ^ a´<5@5?ApX_Í
Q$R
+e()(17i+g-*1"27+'(+,i"!+,1*"27r**+7h*¨*6+
+'Ã+%'*;-*+gaS"Ã+,+,%'+gL+,*i¡¨+,+,.*+e(1023S+,7,ñ1"?0?"*i3.
;*/"
*+#+,+'±*g-;!}a1+'0?"#+S7¢"2%131(*+7U"7Ê*+7i*+aï"27e**g7*6+È1"0?"*i3GaS"?Ã+,+,%'+#"%'+7h+7
,
-n_Sg-¨*6+!(1023S+,
W
*6+!(*6+,B*i"240!+,i"*-¨)7¢"b*+!#"1"477i)!(*h"S$a1+%'+7i+7l
(10243+, W ¡Ë B¡
7r**Ò"*'7((S+,B*"27¡+ W %,£+,+'±*-k6
7i*6+,
*1"27 1¡0+a1+
d¸+Ò%'Sa1)%'*+a*U¡e+'}(L+,i"!+,1*7,m¢*6+;±'7i*+¡+¨ S+,(**++,_40)*i"#-/)%'*i"#±+a!a_`i"+a
*+£a1+,(*aS"Ã+,+,%'+bm¢*+£7i+%'a¸¡¨+. +,(**+ïa1+,(*6ÇaS"?Ã+,+,%'+£±+a&af_i"+a&*+£aS"?ò+,6+,%'+
ýS)40?"*U3b-;*6+!+,_`40)*h"Sb-k)%'*h"S7,.F
+,*i¡+,+,b*+
)7i+acûõü¡1"?0+FcÑeF
9Ó
s
¿Z*+g+'(+,h"Ë+,B*g¡"*c*+a1+,(*6GaS"?Ã+,+,%'+ W
` % É '} rtq
[Ë*
û
»
5 1+,¡*+7h**+,S3£-f8;89Ñ
rÑfa
8;89Ñ
rÑ
s
)7i+a."*7¡¯7h**+,S37#!a1+'0-/
¡+_`i"+a
` % É ~' } rXq
-k6
` % É '} rXq
` ¡ %$% ¡Xs&rts=u W
`8r8"Ñe;Ñ.jaf*+,g*6)!+,1*e+,*i¡+,+,F¯ÑeF "
s ÓeW
g¿+S%f%'#1"*i"b-
a
a
` ¡ %$% ¡Xs&rts=u
-/
¡+#%'a1)%'*+ab+*)!+,B*e+,*i¡+,+,
a
ü
[Ë*{ W
¡Ò"*6
9Ñ
rÑ
8;8
` % É '} rXq
à % É '} rtq
a
a
W
Ò+.0+'-/*.(&"Å¿Á")+f[f7h`¡e7#*+¯6+7i)10*7#*`"+aÇ-kSa1+,(*úaS"?Ã+,+,%'+Å¿Z!+%6ú7i)%6
%'#1"*i"£-
"Ó
F¯6Ñ
F
` % É '} rtq
a
` ¡ %$% ¡ts&rts=u
¡¨+e!+7h)+
*6+
!+°/("1*7i°/(+,6°/B!+*4"+a.13û
df+!7i)*6%'**+#7i+%'aø-/
s
ü
a.B3
*+±7h*
ac*4"c*+L+,+'±*g-¨ûõüe_+,ÊF¯ÑeF
9Ó
s
` % É '~} Xr q ` ¡ %$% ¡Xs.rXs=u ;r%.("1* y 41"27+,(+7h+,B*740?0²*6+g*)Ë+,B*7r¡+,6+ ` % É '} rXq ` ¡ %$% ¡Xs.rXs=uk\ y +]ؾ_40)+e"7Ò*+4_+,+
+,+'±*-/**
y
1"27L+,+'±*"274_+,+aÈ`_S+,¨40?0*+e*)Ë+,B*7r¡"*.*+g7!+
*+
Ò+i"1*(£"f¿Á"S)+[Ë7i¡
7*++7i)10*7*4"+aø-kS-k)%'*h"SjaS"?Ã+,+,%'+ep;S*Gûüga
à % É ' } rtq Ù8;8"Ñe;Ñ.¯)7i+aj*+-k)%'*i"à ¡ %$% ¡Xs.rXsEu\
;Õ41"27
7i*'a7-/*6+ H -n%'**6*
¡7Ò)7i+ac*a1+,*+,i"'*+
*+-k)%'*h"S)40?"*i3dó+,."*;"%'+S7i+7 W 7i!a1B+7r*+aS"?Ã+,+,%'+gL+,*U¡+,+,Ã
% É ~' } rXq aýÃ ¡ %$% ¡Xs.rXsEu ~Ò+]Ø
F¯6Ñ
F
Ñ,Ò ¹
'
"Ó
)7i+ac*+!7Ë++,_40)*i"b-/)%'*i"
s
´Ã ¢£¡)f$r/É7\ à ¡ %$% ¡Xs&rts=u
ûüe)7i+a
41"27!+7i)6+7 W
+
7L+'-kS+ W *+gL+,+'±*g-)7¢"cF
s
»
_+,eF¯ÑeF 9
s Ó
pr*6f+'}(L+,i"!+,1*7+'}1"1"*+a7¢"#"?02!L+,_S"¯Ò+a1_`B*+!-gF
s ó
_+,!FcÑeF s9Ó "°
!8r8"Ñe;Ñ.²ñÖ7g7i**6+,3
»
%'+7h+ab¡"*&*+aS"?Ã+,+,%'+!"&(1023S"f1"?0"*i3fL+,*U¡+,+,&*+'"7i*6*+,3ba
)(f*££%'+,*`"¸("1*¡+,+#*+.a1_`1*S+È7h**+aø*¯a1+%60?"++#"%'+S7i+#"f*+#+,+'±*!%,bL+
¡ÒS7*1£%,+'-/)10"¸(+aS"2%'*i"£"*7e((L+,B*4ñ\7e!_+7 W
s9Ó
)*i"?0"À,+aÇ"*'7Ë}a1+'0
a¸+'}(10"*+a *+.¡¨+ B+77h+7-¨"*7((L+,1**6¯"*7#a1_1*+
+'(10`"+a¸13¯*+#7i+,6_`*i"b**#F¯ÑeF
¡1"?0+bF
»
s
+!a1+%60?"+!"f1"+,aS"?Ã+,+,%'+7%,GL+#+'}(1024"+a¸B3c*+#7i+,_*i"b**e¡+,b*+!aS"?ò+,6+,%'+#"
(10243"£1"?0"*i3¯L+%'!+7Ò02+, W
F¯ÑeF
9ÓW
s
7¡+'0?07^F
%,7i+g*6+,+g"270?"**i0+#(102%'+g-/r"!(_+,!+,1*13Ë}a1+'0?"
¬
»
s
W ¡"£"G40!17i*
40?0B!+7,m¢G7i)%6¯
benefit (points/game diff.) x 10-3
Checkers
Checkers
benefit(points/game diff.) x 10-3
210.00
200.00
500.00
Technion - Computer Science Department - Technical Report CIS9609 - 1996
190.00
180.00
450.00
170.00
160.00
400.00
150.00
140.00
350.00
130.00
120.00
300.00
110.00
100.00
250.00
90.00
80.00
200.00
70.00
60.00
150.00
50.00
40.00
100.00
30.00
20.00
50.00
10.00
0.00
0.00
-10.00
0.00
1.00
2.00
3.00
4.00
5.00
depth diff.
6.00
0.00
200.00
400.00
600.00
functions diff. x 10-3
ÉáÞøå?áæ#ãnÙÚgÞ4ÛÛBÞ4áÚáãeçnãnÜià,ã¢Úæè¯àlç
àÝ2éáSâãnå?Þá.Þ,Ý;ã¢ÙÚÛSäà'èå?áæ#à,êå?ä?å?ã/è!îåÿþ}Ú6ÜnÚáâ6Úì
ø
²
Ê ü Þ Ú6Üßå?áå?ßà,ë
~²
¿Á")+b[ØÙÚê1ÚáÚ Bã
ÞlÝ
ÿþ
í#ÚàlçnéÜ¢Úîêèeîå }Ú6ÜnÚáâ6Ú¨å?áeßÚàlá
Û1Þå?áãnç
Û1Úܦæ`à,ßÚìØÙÚä?ÚiÝDãæÜià,ÛÙgç¢ÙÞ çãnÙSÚêBÚáSÚ Bã¦Þ,Ýéç¢åáSæ
ÿþ
Xø
ø
à,çàÒÝ2éáâã¢å?Þá#Þ,Ýã¢ÙÚîå }ÚÜ¢ÚáâÚåáîÚÛã¢Ù#çnÚàlÜnâiÙ!ê1Úã Ú6Úá!ãnÙSÚ¨ÛSäà'è4ÚÜàláî#ãnÙÚÞÛSÛBÞ4áÚáãì
ØÙÚÜ¢åæ4ÙãæÜià,ÛÙ#çnÙSÞ ç
~²
ßþ
¨®éBàlä?åã/ègÞlÝqã¢ÙÚÚ ãnÙSÚ¨ê1ÚáÚ Bãà,ç;àÒÝ2éáâã¢å?Þá#Þ,Ý
ãnÙÚîSå ÚÜ¢ÚáâÚå?á#ãnÙSÚ
àläé1à,ãnå?Þ4áeÝ2éáâã¢å?Þáì
Ò+¨+'(+,h"Ë+,B*7¡"*#*+r-k)%'*h"SaS"?ò+,6+,%'+
aS"2a#**+7i*¦*+7¢"*)*h"S!¡+,+*+*i¡
((L+,1*7
4_+g7¢"#"?0(1023S"1"?0?"*U3£)*)7i+aS"?Ã+,+,1*+,_`40)*i"ø-k)%'*h"S7,mnb+%'*h"S¯Ä
¡+^7i¡¹**û
ü
7Sa1_`1*+`07h#"G7i)%6¯!7¢"*)*h"S
S7Ë_Ë
Q$R
?´?Ì]?>më< ^ m R
? a´<5@5?ApX_ͱ?:.:E<_: <5±m R ´
? jM lÎ_?A:ÏÐ=
jk?: ^ <:Na l5>.?
+c(+,_")7Ë+'(+,i"!+,1*7.a1+,!7i*'*+a&*+£L+,+'±*£-^)7U"&((L+,B*.!a1+'0n5m¢5*+.-/0?0`¡Ò"
+'(+,i"!+,1*7Ë¡¨+.*+7i*6+a&*+£i"27i &-^)7U"&b¡b!a1+'0ndf+¯%'a1)%'*6+a*i¡¯+'(+,i"!+,1*7È"¹*+
à % É '~} rXq
\ à ¢¡/f$r)É\ à ¡ %-% ¡ts&rXsEu ac_`i"+a¸*+
_``0)+7- `
% É '} rtq a ` ¡ %-% ¡ts&rXsEu "!*++
\[ | Ba ` ¢¡/f$r/É "!*++g |` % É '} rtq ,¢¿S*+7i+%'Sa
+'(+,i"!+,1* W ¡+e±}+ac*+#a1+,(*¯(!+,*+,'7-*+#7i**6+,"+7 `
% É ' } rXq\ ` ¡ %$% ¡ts&rts=u\ Ä |` ¢¡/f$r/ÉD\ X.a
_`i"+a.*+-/)%'*i"7Ã
Ã
a
Ã
1
!
3
`
_
3
"
!
6
*
+
H (!+,*+,-¦*+"ÑÒ ¹ ' *U7i+,+e+=LY,
¡ %$% ¡ts&rts=u
% É '} rtq W ¢¡/f$r/É
-/)%'*i"7"c*+g+!ÖY | ,¢
a1`"¸-Ò%6+% S+,7,#¿g*+±7h*g+'}(L+,i"!+,B* W
¡+±+a
*)!+,B*.-YSY&B!+7Ë+,*i¡+,+,ó*+f(10243+,¯a¹*+b((S+,B*£¡7.%'a1)%'*+a¹-/£+%6
%'#1"*i"-e_`40)+7,+.(+,i-/.%'+£-
*+£(10243+,"27#!+7h)+a&B3¸!+°/("1*7i°/(L+,°/BË+¯7
+'-/+r¿Á"S)+
7i¡
7*+
6+7i)10*7*`"+a
r+0+'-/*(*;-*+±)+g7i¡
7*+e+7i)10*7r*'4"+a
-/!*+fa1+,(*¹+,
+'!(10+ W
` ¡ %$% ¡Xs&rts=u
},
S(02L+'0+a
P
` % É '} rtq ` ¡ %-% ¡ts&rXsEud\
7i*a7#-/40?0%,7i+7Ë¡+,+
*+#'(¯ +a£13GY.6+,(+7i+,1*7*6+ÈS%,%'))102*6+ab+7i)10*7-/e*+!%,7i+7e¡+,+
P
Å¿
` % É '} rXq;\
` % É ' } rXq ` ¡ -% % ¡ts&rXsEu3\ [ *n´°n[ W X° W İ/´ W ¬°
41"277i*'a7#-/`00%,7i+7Ë¡+,+
` ¡ %$% ¡ts&rtu ` ¢ ¡/f$r)É
\ R ¿
af*+Ë(& S+abB3&[."27-/*+.%,7i+7¡+,+
R ¹*+
R \ -/*+S(#02L+'0+a£[Ê+,(+7i+,1*7¦*+*)Ë+,B*7¦¡Ò"*6!*+
%'#1"*i"7
* ` % É ' } Xr q |` ¡ %$% ¡Xs&rts=u ~| ` ¢ ¡/f$r)É , A *UÄ | ´ | ., | *¬ | X | [, | *ª{ | Ä | , | *n | ¬ | ´Q,!¦+Ë+7i)10*7Ê-kSg*+#-/)%'*i"$+,6 W
X W {°nÄ W °i¬ '5_+,63("1*
+'!(10+ W *+("1*
7i¡¸"$*+.i"1*#(*-
*+˱)+ W
H % É ~' } Xr q H ¡ $% % ¡Xs&rts=u \
H ¡ %$% ¡Xs&rtu H ¢¡/f$r)É5\ R 40?0¨%,7i+7e¡+,+
P
¡+,+S%,%'))102*6+a7¢"#"?0i03
ac+,_+,63¯("1*
R
b*+
'(f02L+'0+a
P
7i*a7g-/
41"27g7i*a7-/g40?0Ò%,7h+7e¡+,+
pr*6¯+'(+,i"!+,1*7g7i¡7¢"#"?0+,4_S"g+L+,+'±*-;)7U"¯GS((+,1*
Ë}a1+'0"27e41"40
{
-/£*+f%,7i+b-#+,9dó+,ó*+f+,S!"%'+7h+7 W
·_+,+7h*i"*i"
Performance vs. error in depth model
performance(points/game.)
Technion - Computer Science Department - Technical Report CIS9609 - 1996
*+b+,+'±*b-)7U"¹*+f!a1+'0a1+%'+7h+7,
*n+,B*i"_+g+,6,¦"27r0+77#-/)10*£)a1+,6+7i*i"*i"
Performance vs. error in function model
performance(points/game.)
0
1
2
3
4
5
1.55
1.50
1.45
1.40
0.8
0.7
0.4
0.2
1.85
1.80
1.75
1.70
1.65
1.60
1.35
1.55
1.30
1.50
1.25
1.45
1.20
1.40
1.35
1.15
1.30
1.10
1.25
1.05
1.20
1.00
1.15
1.10
0.95
0.00
5.00
s
¿Á")+e
ØÙÚÛ1ÚܪÝ2ÞÜ¢ßà,áSâÚ
Þ,Ý
Þ,Ý
ü
¥
ãnÙSÚ¨Û1ÚܪÝ2ÞÜ¢ßà,áSâÚÞ,Ý
ª©pª
ü
error(f_op - f_model) x 10-3
0.00
ø
à,ç
àÁÝ2éáâ6ãnå?ÞáÞ,Ýã¢ÙÚßÞîÚ6äå?áæ¦ÚÜ¢ÜnÞÜ'ìØÙSÚä?ÚiÝ2ãæ4ÜUàlÛÙçnÙÞ çLãnÙÚÛ1ÚܪÝ2ÞÜ¢ßà,áSâÚ
àlçqà¦ÝDéáâã¢åÞ4á
ÞlÝãnÙÚ¦ÚÜnÜ¢ÞÜå?áîÚÛSãnÙgßÞîÚ6ä'Ý2ÞÜ
ü
1.05
error(d_op - d_model)
à,Ünå?Þ4éç
ù % É '~} rXqOÆ ù ¡ %$% ¡Xs&rts=u
àläéSÚçÞlÝ
à,çàÝ2éáâã¢å?ÞáÞlÝqã¢ÙÚÚÜ¢ÜnÞÜrå?ágÝ2éáSâãnå?ÞáßÞîÚäSÝ2Þ4Ü
à,Ünå?Þ4éç
ø
Dü % É '} rXq4Æ ü ¡ %$% ¡Xs.rXsEu
ìLØÙÚÜ¢åæ4ÙãæÜià,ÛÙeçnÙSÞ ç
àlä?éÚç¦Þ,Ý
;y8Á¾Á¾º».¿
ì
+#+'(+,i"!+,1*7a1+76%'i"+ac"f*1"27g7i+%'*h"S$%'1±!+ac*+#13B(L*+7¢"27e**"*e"27gL+,+'±q%6"`0¨*.)7i+!
((L+,B*!a1+'0 W
aj***6+!+,+'±*"27g6+*+,B`"7h*¡+ +,g((L+,1*7,!m¢*4027iba1+,!7i*6*+7
*+!#-/)10+767g-)7¢"¯¡£!a1+'0nm¢f*+!+'*7h+%'*i"f¡+a1+7%'i"L+ï&40i"*
-kg)7U"b
¡SË}a1+'0¡"*¯#L)a1+ac+,
Ç
È
'
#_3'
+#86
mnË*+(+,_S")77i+%'*i"7 W ¡¨+e77i)!+aÈ***+(10243+,"7r%'+,*4")*"*'7((L+,B*r!a1+'0ªr¡+,_+, W
S)1"*6+g(B77U"10+#**(10243+,"27)%'+,*`"¯)*"*7Ò((+,1*!a1+'0 W
"*"27
"270++a.B3Ë*+
(10243+,mn.*1"27¨7h+%'*i"
+7i(L+%6"`003c¡+,£*+g!a1+'0
W ¡++,+,`0"À,+#ûõü*6+,¡Å40i"*
)%'+,*`"c!a1+'07l
0
W ûü
**%,.aS0+
Ò+,+!+!7i+,_+,`0;(B77U"1"?0?"*i"+7#-/e+,(+7i+,1*i"¯)%'+,6*4"B*i3fL)*g*+!!a1+'0n#df+!77i)Ë+!**
*+g(10243+,e(B77h+77i+7£)((L+,g)aø£*+aS"27i*%'++,*i¡+,+,¯*6+g!}a1+'0nñ\7+,_``0)*i"c-k)%'*i"ba
)1"*+.%'!!faø"27e+40?"27i*i"2%#¡+,
*¢df+7i¡7i)%6¯0+61"
*+!%'*)`0~((L+,1*ñ\7-/)%'*i"Ò1"7eË}a1+'0;-)%'+,*4"1*i3£"27
*+#(10243+,
*6i"+7e*!0+6G*6+S((+,1*ñ\7Ò-k)%'*h"Sf7i+aøc"*'7e(7i*e!_+7
,
40h"*6
"f+%'*h"SjÄ ;m¢*"27Ò-k+S7¢"10+#*Ë+,%'+#*1"27
40Si"*
¡Ò"*6GË(}%'+a1)6+g-/+7i*i"2#*i"£
S)a£."*7+,4
Ò+û
-/)%'*i"
0ü
40i"*
77i)Ë+7***6+(10243+,
(LB77i+767i+7 W "faaS"*i"b*.!!a1+'0-"*7((L+,B*4ñ\7
W #)((+,)a
-/)%'*i"
+,6+'-k6+ W
W
0
W !*+aS"27i*%'+
L+,*U¡+,+,!*6+!a1+'0-k)%'*i"£a*+
S%'*)40((L+,B*4ñ\7
*!*$Ã | 0 , |
!(10243+,"27¡a1+'±+aGS7
û
b
F ,
W ¡+,+]¸·
og;Ìj"27g4027i£#(1023S+,
]*¢*+(1043+,4ñ\7
-/)%'*i"
1"27¦!+7¦*6*+S%#+,_``0)*i"#-/)%'*i"!**((L+7Á"(1043+,
-/)%'*i"
Â,
W "*7!a1+'0nñ\7!a1+'0
-k)%'*h"S¯+,*%
W "*'7¦!}a1+'0nñ\7
W S7¨¯77i%6"2*+aË)aø!"*7Ò+,r)%6¯#(10243+,¡"?00
+g%,40?0+a£bxBJLCK4zy/E<DJ(1043+, W ¡1"0+Ê(1043+,¡"*Ë+,6;L)a
W 7i)%6ï7r*++
)7h+a!13ûü W ¡"?00
Technion - Computer Science Department - Technical Report CIS9609 - 1996
+g%,40?0+a£.CK4zy/E<DJ(1043+,4;"%'+g*+e1"+7i*r0+,_+'0(1023S+,"27%'+,*4"£)*r"*7r¡#-/)%'*i"
W *+e+,
S)a¯77h}%6"2*+aø¡"*£*+g*S(.0+,_S+'0-/)%'*i"¯¡Ò"0?0+gÀ,+,L
*!*-Ã " | Y , | -* Ã [ | 0![ ,!,;767i)!+7***+g((L+,B*e)7i+7-k)%'*i"ÁÃ À [
(10243+,
 ·
R Ã
à [ * R ,5 0¥[
à À [ * R ,
à [ * R ,
**
7*i"27¢±+7
0![dÄ
+!+,6gS)a7S77i%6"*+ab¡"*f*6+!!a1+'0ªñÖ7e-/)%'*i"7g+,(6+7i+,B**+!(10243+,ñ\7e)%'+,6*4"B*i3fL)*
"*7Ë((+,1*óm¢*!"27Ë(B77U"10+b**Ë*+¯((L+,1*!"27.4027iÇ5)%'+,*`"ó(10243+,¹mnó7i)%Åf%,S7i+ W
"*7
-/)%'*i"7¦¡"?0?0q4_+¨S77i%6"*+a+,¦)a7ldf+¨a1+'±+*++,¦)a7r77i%6"2*+a¡"*#*+r!}a1+'0nñ\7
-/)%'*i"7*!((103.*#*+g+ga1+,*+,È"+a.13.*+
S((+,1*ñ\7+,6;L)a7,¿+'!(10+ W
*!*-à N | Y , | *!*-à " | 0 " , | -* à [ | 0![ ,!,!,;77i)Ë+7**"*7Ò((+,1*"7®*!*à À " | Y , | ** à À [ | 0¥À [ ,!,;¡+,6+
*,
R Ã Ã " * R ,5 0 " TÃ À " * R ,
Ã
R à à À [ * R ,5 !0 À [ | à À [ * R , !0 À [ Ä]Å ßà [ * R ,5 ¥0 [ |
Â
*[,
Â
0
Ò+"()*Ê-kg*6+ûõü
(10243+,
"*R , 0"Ä
à [ * R , 0¥[ 40i"*÷"27Ê*+!7!+!7-/g*+!ûü`0Si"*
W
)*¡"*f.(1043+,**
±*7S)¨+'*+,a1+aca1+'±1"*h"S+g)*()*"27aS"?Ã+,+,1*m¢7i*+a.-+,*)61".#Baa¯_`40)+ W
+,¡40i"*÷+,*)7e.7i+,*-;LBa7ea¯ËS+g-;_`40)+7,e+Ë+1"£-~*6+S+g"27 W
¡+g¡)102a£4_+6)¯û
ü
*+
**"?-
¡"*b13ï7i+,*e-¦-k)%'*h"S7**
7*i"27¢-/3£*+g+,6
%'7i*64"B*'7 W ¡+g¡)102a£4_+
+%'+'"_+a.g_40)+
**-n40?07¡"*1"£*++,*6)+a#S+~¿Z;+,_S+,3LBa".*+g7i+,*-+,*)+aÈB'a7 W
*+,+Ê"27
!7h+,*--/)%'*i"7
76*i"27¢-/3".*++,6
%'7i*64"B*'7 W -k¡1"2%fû
ü
¡)102a£4_++,*)+aø**
1a£7*6+7i+'0+%'*+ac!_+
Ò+40i"*
S+,+,*+740?0*+7h)%,%'+77i7a£%,40?027"*7i+'0?-;+%')'7¢"_+'03£¡"*¯*+gS((+,1*!a1+'0
*nLBa7~,¦*+g((L+,B*%,¯%61B7i+-kS¾-/r+%£-*+g7i)%,%'+77i'7,
*!a1+,*+,È"+g¡1"%6b@,KyL-!_+7
¿S+%6f1ac"7i)%¯7h+,*-1a7 W
*+(10243+,#%,40?07Ê"*7i+'0?-
6+%')7¢"_+'03$*6Ga1+,*+,È"+.*+!'+
-_`40)+7**g*+ËB'a£"27Ê¡¨S*c-kSg*+!(10243+,."%'+*6+(10243+,#a1B+7Ê*g B¡¡1"2%6$LBac*+7i+,*¡"?0?0~%'*6)40?03.+7h+'0+%'*6+a£13!*+g((L+,B* W "**' +7r*+e)1"S¯-*6+7i+g+77r*+eS+
_``0)+7e*6**+7i)%,%'+767i"7¡*.-/*6+(10243+,rmn¯*61"7e7i*+ W *+(1043+,7
¯77i%6"2*+a£'+
-_``0)+7¨-k+S%.-
"*7;40*+,6*i"_+e!_+7,+0¡+,;S)a.-*+S++,*)+aË13*6+^40Si"*
"27*6+41"40#"1").7e-`00*+7i++7l++S7i£"27**+,_S+,G¡Ò"*6¯*+¡¨S7i*(B767¢"10+.7i+,*
-¦-k)%'*h"S7
7*i"27¢-k3S"*6+g+,
%'S7i*4"1*7 W *+g(10243+,"7)'B*6+,+a!*Ë_S+*0+7i**6+41"40
#"1"#)7,$Ò+)((L+,L)a¸"27*6+£41".40¨`B"#)7-^40?0S+77U"%'+£S+£-
*+.Ba7
%,£4_+Ë_`40)+¡1"2%c+'Z%'+,+a7e*1"27.4B"#)¯+7i+,*-~LBa7+,*)+aË"27*+7h+,*-;40?0Ba7
**%,.4_+g.4B"`0²_40)+;mª-*+g1"+7i*_`40)+g-~+^S77i%6"*+a.¡"*¯Ba#"27r0+77Ò*
+!-Ò*6+!#"1").7 W
Í*¢*+!*+,gLBaG"7
*+,+È"7c(B77U"1"?0?"*i3$***1"27gLBaj¡"?00L+ï7i+'0+%'*+a
.,
)'B*6+,+a.*#_S+#1"+,
_40)+ Ò+¯40i"*ô6+,*)7!b7i+,*!-^LBa7#¡Ò"*65b+.-g_`40)+7,
(10243+,
%,£+,!(103._`i")7e7i+'0+%'*i"b7i**6+,"+7,
mn5Sa1+,#*$7i+'0+%'*£¯Ë`_S+ W
O
*6)40%,aS"2a*+Ê"7w#E `<Dw#<2Jc[ W (+[1¬X° [Ä W
!7i*6*+,3Ë**7i+'0+%'*7*+gBa£¡"*£.4B"`0#"1")c
̲B ¯*
*+Ê0+'-k*g*6+,+"f¿Á")+£,´e+S((+,1*7i+'0+%'*7g.!_+*ea1+
***+g_`40)+#***+((L+,1*
77¢"7*!a1+
?¬
|
8Ã#"27~"Ë*+S+^Ö |
BaÈ*++-/
¡"?0?07h+'0+%'*ea1+
W
`
ô
G
e+#(1023S+,g B¡
7
G "27Ò"
, W *+S+e-k}a1+
lnqÒ+,+
+#%6"'%')7i*%'+7)a1+,;¡1"2%6!*+S((+,1*
7¢"%'+Ê"*'741")ö"277i`00+,e*c*+g#"1")÷-
"27g(LB77¢"10+*6*g}a1+
|
"27"b*+g+ g
±*/"?-Ã [ * G , \
¡)102ab+7h+'0+%'*6+a
[Y
´Ã [ *$Ã5, \
£a
G
a.*+g#"1"#)
Cæm¢*
-
#+,+"274027ib.(B77U"1"?0?"*i3
M*(f1,f0)=[2,5]
Technion - Computer Science Department - Technical Report CIS9609 - 1996
M*(f0)=[8,11]
d
M*(f0)=[7,9] c
b [0,5]
e
M*(f1,f0)= [0,0]
f1= 10
f0= 0±1
M*(f1,f0)=[5,5]
a
f1= 0
f0= 8±1
f
M*(f1,f0)=[5,5]
f1= 5
f0= 10±1
[2,2]
M*(f0)=[7.5,8.5] c
M*(f0)=[9.5,10.5] b [5,5]
e
M*(f1,f0)= [0,0]
g
h
M*(f1,f0)=[2,2]
d
f1= 1
f0= 4±1
f1= 2
f0= 8±1
f1= 10
f0= 0±0.5
°
f1= 5
f0= 10±0.5
O±
[2,2]
h
M*(f1,f0)=[2,2]
g
f
M*(f1,f0)=[5,5]
f1= 0
f0= 8±0.5
3§rá!Úëà,ßÛä?ÚÝ2ÞÜrãnÙÚeçnÚ¯®`éSÚáâÚÞ,Ýâàlä?äçÛÜ¢ÞîéâÚî.êèZ
ø Ùå?ä?ÚãnÙÚÜ¢åæ4ÙãÞáÚå?ç¦Ý2ÞܦãnÙSÚâà,ç¢ÚÞ,Ý ° ú ö ì
° ú õ8
¿Á")+£,´
ãnÙSÚ¨â'à,çnÚÞ,Ý
a
f1= 1
f0= 4±0.5
f1= 2
f0= 8±0.5
4²Bæ4éÜnÚeçnÙÞøçã¢ÙÚ
â'à,ä?ä?çÝ2Þ4Ü
ü ìØÙÚ
ã¢ÞÛ /ä?ÚiÝ2ã
Ãb¡Ò"0?0+#7i+'0+%'*+a´*/"?-Ã [ *-Ã7,5,+,+'-/+*+g(10243+,
%'%60)a1+7e***+g((S+,B*e43
7i+'0+%'*+'"*+,}a1+
à !+##"1".40¨(LB77¢"10+_40)+*6*^%,f+S77¢"7e*£a1+ýB3c*+
G ea1+ý¦
((L+,B*Ò"7+g41"40(B767¢"10+_`40)+Ê"7Ê+,+'-/+e}a1+
¡6*#-/*+g((S+,B*
_40)+
"$\ | ,¢¼}a1+ G ¡*¯Y-k*+g(1023S+,
a£a1+ëc
à ¡*6¯X W *+,+'-/+ga1+ë
¡*£*6*6+(10243+,
_`40)+Ê"fÖY | X¢¦"È"02e+7h1"È7h`¡e7;*6*}a1+ ¡*¯¬ | ¢ 13'*i"40a1+%6"7U"SG40h"*6
W
"%60)aS"c41"#" W ¡)102a£7i+'0+%'*e}a1+ **}a1+
0ü
¿Á")6+!X!7i¡
7r*+û
40i"*c
9:E<5>.?@CB5:=? û 0 ü * Vcz.YN|`G!VAL$e5| *$*-Ã % É | 0 % É , |E³´7´ í ³µ¶ 7,$,
^ `G!VAL$e \ Y
:E?Um.B5:N ·AF| Ã % É * Vcz.Y ,D 0 % É | Ã % É * Vcz.Y , 0 % É
?A> =?
K g û z.G=GEJ * Vcz.Y ,
^ <: +% YE{cBA K ;
^ `G!VAL$e \ Y={S I.HJAOcG g ÿÃ % É * Y={c ,5 0 % É | Ã % É * Y={S , 0 % É ? =?
z~V}V z.HUI.`Y}|z~V}V I.HJAOcG. g ûõ0 ü * YE{c.|`G!VQL-e |E³¨´
´
^ <: +% z.HI.K
` Az~V z&HI.`UY
VSW z.HUI.`Y}|)VcW I.HJQOQG g
ûõü */ z.HUI.`S|`G!VAL$e f
[ | *$*-Ã % É | 0 % É , |E³´7´ í ³µ¶ ,¥,
0
Y={c I.HUJQOcG=Y g YE{c I&HJQOQGEY]VcW I.HUJQOcG ¦
Y={S I.HJAOcG g #"* y , | 4A* P n, >¸ È ¹ rº¼» ( o=nXn q ' Ms ½ r (
I&z.zEL I.HJAOcG=Y g I.z&zEL I.HJQOQGEY]&YE{c I.HJQOQG ¦
I.z.zEL ¢ È s |I.z.z=L ¢ ~' ¾ g µ.4c* y , | `* P ¢, >¸ È ¹ r~º¿» q-¡)¡Xu q ' Ms ½!r (
I.z.z=L z&HI.`UY g A´K ¢ ' ¾ I&z.zEL ¢ È s ¦
:E?Um.B5:N I&z.zEL z.HUI.`Y}| I&z.zEL ¢ È s |~I&z.zEL ¢ ' ¾
,
¿Á")+!XZØÙÚ
Ò+,+6+g*+,
a1_S+,73£7i+'%b40i"*7 1W
`¡+,_+, W
°
ü
í
³µ¶
,
à,ä?æ4ÞÜnå?ã¢Ùß
,´S**6+,*)GÈ+-~_40)+7 W
*+7i+£40h"*67¡¨+,6+a1+7¢"+a¸-/!aS"Ã+,+,1*#()(17i+7,b+ïp®£40i"*
7a11+7eû
0ü
+,*)7
S+-;_40)+7a1)+#*.)%'+,*4"1*U3b77i%6"2*+aø¡"*b*+#(1043+,4ñ\7+,_40)*i"b-/)%'*i"¼+,_+,*6+'0+767 W
)10?" +!*+!û
0ü
40h"*6
W
p®a(*7*+#7¢"2%#À,+,S°n7i)
[
S77i)!(*i"fac(S(B*+7_``0)+7^"f*+
7Ë+!+,#7a11+7g#"1"4¯+.%'7i(1"%'3f1)#+,740i"*
,´¨`07h¯1"()102*+7+7
-_`40)+7,e¡+,_+, W *+7i+'+7(_S"2a1+ÈÈ+,)i"27i*i"2%#!+7h)+g-;*+#%,%')%'3£-*+#"1".4G1*
_``0)+7-¦"%'!(10+,*+!7i)*+,+7l
0L
Technion - Computer Science Department - Technical Report CIS9609 - 1996
m¢*r"7r+7h3!*7h+,+g**¨ûü
=
+,Sg)a7Ê+ )40;*6£À,+,
0
"27+,+,40?"À*i"£-ûõü4;Á+,*ûü¡+
103!+,+a.*6%,40?0~ûü
!¿Z)*+,!S+ W
40h"*67,;+e-/0?0¡"*+,S+,
7i**+7r**r"?-;¡¨+%,`00ûõü¡"*G#%'+,*'4"¯(10243+,¡B7h+e-k)%'*i"7
-n40?0;¡"*1"$*+#+,6g)a7Ê-Òf)%'+,*4"f(10243+, W
0ü
S)a7+,*)+aøB3.û
*+,b"*g¡"?0?0¨+,*6)$._``0)+.**"27g¡"*1"&*+
-kS¨*6+
)%'+,*'4"¯(10243+,
aÈ"*¡"?0?0~+,*6)ï!_+e**r"27#Ë+,L+,r"c*+
7i+,*-~!_+7r+,*6)+a£13ïû
0ü
-k*+g)%'+,*4"c(1043+,4
K4y5[6KEJË)%'+,*4"cAq=?EKz,Kyk+
.[
VSz&YN|`G!VAL$e5| n ,º#Á:BKJ HUWX{SG]A y |P EJ t A +
?<_:E?ca
Q$R
¡"*£40?0
¡+È%,f(_+c7i*S+,
6+'0*i"7i1"($L+,*i¡¨+,+,¸*+#*U¡
| y |P \
û
0 ü * VSz&YN|`G!VAL$e5| ;,º4Ky5 n [6KE#%'+,*4"
ý!Ky/ |~UHWX{cG \
Aq=?EKz£C'>JZ@º<@ºy/<DJBHc>i¤¯Ez ,<Dy/ziEz'#¤ºx1JC4y/<2>SJ@£yD:BEyg@,Ey/<@2¤,byD:BKbKz'zi>z£C>SJ@,y/zUES<2JLyD@È>h¤
û
ü
*
9:E<< ^E^
`
pr3&"a1)%'*i"
W
*6+ja1+,(*6ú-*+b7i+%6ó*+,+¿
"!!+aS"2*+'03
` \
Ya
` \
b*6+G(6B-g-/0?0¡
7
` DJk df+¡"?00¨7h`¡**Ò"*e"27e*6)+g-/ ` \lk %,%''aS"
**6+±7i*Á"a1)%'*i"_+77i)Ë(*i" W -kS13 E
Y {c]A´K] ; * Vcz.Y , W û ü * YE{c.| k |/ û b F *- n ,¥,+,°
*)7ÁBa¦**
+'017*9+ W *+)(g-}LBa7
6+,*)+ag13gûõü * Y={S.|
|E û ;b F *-;,!,
k g
0
,
%
'
%
S
a
"
¯
*
.
*
!
+
i
7
+
'
%
ø
a
"
1
a
)
'
%
i
*
"
_
£
+
7
i
7
)
Ë(
i
*
"
k
S
B
3
+
û
*
&
[
n
, 6+,*)7 W ._40)+
ü
A
|
|
W
W
k
"5 |P W *+!++,*6)+ab13Gûü *- |
[ | ;, g¿Z13 Y={cU
A±K] * Vcz.Y , W ûõ0 ü +,*6)7g!'+
y
k
0
H " |H N ;7i)%6b** H " "27Ê*+##"1".40;_`40)+!-Ò40?0;*+#S+7-Ba7"\+ W a H N "27g*+#41"40
_``0)+-~40?0*6+7i+g+7l;+,+'-/+ W HUWX{SG W *+ûü_40)+-£
**r"27e77i%6"2*+a.¡"* YE{c W +'0B7
**+
'+e**¨û
0 ü 767i}%6"2*6+7;¡Ò"*6 Y={c ¿Á"40?03 W û ü +,*)7r*+e41"40_``0)+!*6+ YE{c
_``0)+7 W aï"¯*+7!+g+,
S7r-kS `UG¥VAL-e
\ W *1"27_40)++'017*Ë*+g+g6+,*)+a£13£ûõ0 ü W
a£40?0Ba7¡"*£*1"27eû
ü _``0)+ W ¡"?0?0~L+'0*6!*+gS)(£-~LBa7Ò+,*)+a.13¯û
0 ü Ja
m¢*¦"27"1*+,+7i*h"*g*+**¦"?-¡¨+%,40?0û
0 ü ¡"*eË}a1+'0
**7.1"*63((L+,B*-k)%'*h"S
77i)Ë+**e*+*+,+,ö"27e*)+g-/
a¸"1±1"*+b+,Ë)a7 W
O
a&¡"*5*+fw#E <2w#<DJ$7i+'0+%'*i"Å7i*6*+,3 W
¡¨+£%'*)`003+,*.b#"1"4
(10243+,1)7 W ¡1"?0+e"£7i+%'*i"£[g¡+"1*+,(+,*+aË#"1"4ïS7;(1023S+,;¡#77i)!+7**"*'7;S((+,1*
*n¡Ò"*6!((17¢"*+
7¢",
)7i+7~"*'7¡-k)%'*h"S
W +,+¡¨+r"1*+,(+,*r#"1".4ï7re(1043+,r¡S7¦!a1+'0
-¦"*7((L+,B*gaË"27*+,+'-/+g**'40?03.)%'+,*`"bS)*"*76+%'*i"7,dó+,+,_+,gû
((L+,B* W
40?0;7i)%,%'+77iSñ\7B'a7¡"?0?0;+#+,*6)+a¯7¢"%'+!S+-;*+,
0ü
7¢"#)102*+7*+
%,c+#+'Z%60)a1+a
Ò+(10243+,
¡"?0?0*+,ï%'S!()*+"*'7;_40)+7r-/;*+7h+
LBa7aË¡"?0?0Á7h+'0+%'**+e+e¡"*.41".40q#"1"#)
_``0)+ W
¡1"2%6£"27+'%'*i03.¡*È"1"`Ga1B+7,
0
¿Á")6+,´7h`¡e7
c+'Z!(10+-;*i¡%,40?027-kS
ûõü
%'+g¡"*
0]\
ÌX
Y
ÿ
Ì XZm¢¯*+%,S7i+
- 0 \
0 \
0
7i+'0+%'*7*+7!+g!_+7ûüea1B+7,
0
W ûü
Y
뺮8k#®º®
_
W û
ü
S¯*+#7!+#7i+%6¯*6+,+ W
%'+¡"*
06\
#a
7i+'0+%'*7*6+7!+g!_+g7r#"1"4Ga11+7,m¢¯*+%,7h+
-
v274G(6)1"#*+%61"1)+e¡1"2%6
W -;*6+g*+,+7h+%6+a£13*+#40i"*÷*)*" #
?¬n
·+e-q*+!B7i*7¢"1"?±q%,B*r+'}*6+,7¢"7r-*+#"1"4ï`0Si"*"27*+
%,c+a1)%'+*+#4_+,+g'%1"Ë-ªS%'* W
1àó1Ô!Ý$Ù$
æ B¡ãèEÙ/Ô$ÞßàEÒ$ØèÛÔ8×Xâ.Ò$Ý)ÞòÔ$áï£Þ ¢!Ô¥ÙpÞòã~à
ã!7
ó B¡ Ó.Ô!×Ô¥Ø=ÕÔ¥àÙ/Ô!ä~Ò8ã¥ÕÒ$ÚkÙXÓ=Ò8ã~ÚpÞßä¥Þßà&Ô á¥ B¡¥ì*pàvÙXÓEÒ8ã~ÚpÞßä!Þßà.Ô$á7 B¡!æ.çÒØÞߨ
à=ã~ÙÓ&Ô¥à=ØáßÒÙXÓ=ÒÝ!Ô!×XÒ]ç5ÓEÒ-ÚXÒ]Ô]à=ãØEÒ]Ó&Ô¥××XÖ=Ý$Ý$Ò-×X×Xã~ÚX×8çD¤ ÞßÙXÓÒ§¦Ö.Ô$áÕÔ áßÖ=Ò$×$ì¨ ¤ ¡ Ý!Ô áïáßÒ-ØéçÞßÙXÓ]¢$Ò-ÚXã3Ò$ÚXÚXãÚkè.ãÖEà=Ø
çÞïáïáÝ-ã~ÚXÚXÒ$Ý$ÙpáßÛ
Ô¥×X×XÖ=å]ÒÙXÓ=ÒkçãÚX×XÙÝ!Ô!×XÒ7óôãÚã~â=â.ã~à=Ò$àÙ5à=ãØEÒ$×5çÞßÙXÓ3åã~ÚXÒkÙXÓ&Ô¥àvã~à=Òkâ.ã×X×pÞßèáßÒã~Ö=ÙXÝ$ã~å]ÒæEç5ÓÞïáßÒ ¡ ç_ã~Öáߨ3Ó&Ô ÕÒÖEà.©£ªXÖ=×XÙpÞ¬«}Ô!èáßÛ
ÚXÒ$ÙXÖ=ÚXàEÒ-ØÙXÓE
Ò «&ÚX×XÙ$ì
[[
1"2740Si"*
4_"2a77i+%1"7h)*+,+7**Ò%,*+'ò+%'**+g#"1"4£_`40)+-*+g(+,1*a1+
mi7"*!(177¢"10+¯*$Saa7i)%5+'}*+,7U"S*fû
´Ã % É '~} rXq
à ¡ %$% ¢£¡)f-r)É W
"a1+,(+,a1+,%'+!L+,*i¡¨+,+,
a
1-k6*)*+'03
|0
7#¡+'0?0
ü
"-¡+£77i)Ë+¯b**`0
W
"*"7Ò+7i3.*!7i¡Å**7i)%6G(%'+a1)+%,*+'1"7h*
à " *¢
ûü¡"*c*+
(10243+,8*$Ã
" | *~Ã " |/F ,!,!,%,!_S"2a7i+'%1"e}a1+ O W ¡+,6+7û ü *6*)7i+7Ã [ "7i*+a-5Ã " W %,*
(+,h-k6
*1"27e()1"Æe`¡Ò".**e*+((+,1*g¡"?0?0;4_+*0+7i*
XË-kS
a1+ a11+7*e4_+
13!"!(10?"2%,*i"7eS£*+_40)+-}a1+ -/*6+g(1043+,4
Technion - Computer Science Department - Technical Report CIS9609 - 1996
¿Á")+.Ä"?0?0)7h**+7e.7¢"*)*i"c¡+,+)7U".#"1"4¯¡"*c+,_`40)*h"S¯-/)%'*i"
V1=9 a
V1=8 b
V0=-6
V1= 9
V0= 6
d
f1=8
f0=-6
c
?
g
e
f
f1=9
f0=-9
f1=4
f0=-5
f1=9
f0=6
§;á!ÚëàlßÛä?ÚÝDÞÜ;àeçnÚ'à,ÜnâiÙ.ãnÜ¢ÚÚvøÙSÚÜnÚeãnÙÚçnãUàláîBàlÜnîZ#%ø¦Þéä?îÙ1à
Ú6Ü
÷ çnéSâUÙËÛÜnéáSåáSæ
ø Þ4éä?îâiÙBà,áSæÚãnÙÚ, ü àlä?éÚ;Þlݲã¢ÙÚãnÜ¢ÚÚì
¿Á")+.Ä
Bì
;Þø¦Ú áÞîÚ
ÚÛÜ¢éáÚîËãnÙÚêSÜUà,áSâUÙËäÚ'à,îå?áægã¢Þ
7¢"#"?027¢"*)*h"Sïh"7h+7¦"#)10*i"°/(10243+,B!+*+,+7l~̲)% 1a1*a#m¢1" qa1+7%'h"L+^7i+%6
40h"*6
HURSJ
û
W
W -k#)10*i"°/(1023S+,
B!+7a£%'%60)a1+#**()1"!"27Ò"!(B767¢"10+#¡"*S)*-k)6*+,
+7i*6i"2%'*i"7L)**6+!(1023S+,7,ñ
+,_``0)*i"f-/)%'*i"7,£Æeh-ÊY4;7i¡¨+aj**£7i40?0¡Ó()1"¯-/
HURSJ
û
"27(B77U"10+Ë"?-;¡¨+#77i)Ë+ȯ)((+,gL)abc*+7h)
-~*6+(10243+,7lñ-/)%'*i"7 W
ac#0¡+,
S)a££+,_+,63(10243+,ñ\7¨-k)%'*i"
D27G
4 40h"*6"27r**à % É ' } rtq à ¡ %$% ¡Xs.rXsEuM\ Y
n* *+À,+,°
,1"2777i)Ë(*i"!"27)7h+a!*Ê"1-k+,
Ê)a.SïÊ_`40)+g-}a1+-/¨(10243+,7i+a
aS"+%'*i03!Ë*+((+,1*ñ\7_`40)+
*6)40+'024*i"!*6g*1"27;77h)!(*i"#"27
à % É ' } rtq à ¡ %$% ¡ts&rts=u
1"27.77i)!(*h"S5!+7Ë**Ë¡1"?0+Ã
a
Ã
43$+,_40)*+GjB'a5aS"?Ã+,+,B*h03 W *1"27
¡ %$% ¡Xs&rts=u
% É ' } rtq
Ò+7¢"2%e77i)!(*h"S!)7i+aÈ-k*6+
h""`0
7i)
77h)!(*i"
aS"?Ã+,+,%'+e"27)a1+a
¿S;+'!(10+ W *+e(1023S+,¨.3Ë(+'-k+,1 #_+, 11"1*¨¡1"0+g*+gS((+,1*
(+'-/+,7*+£((LB7¢"*+fm¢Ç7h)%5b%,7i+ W
((L+,B*4ñ\7_``0)+ W
40*)*+.(10243+,ñ\7_``0)+."27#S*ÈbaS"+%'*!S((B7¢"*+£-
*+
¡+^%,."1-/+,g#)aø£*+g(10243+,ñ\7r_40)+7i+aø£*+g((L+,1*ñ\7_40)+a
$" ü #.%'¥(/(
Ò+¯_+¯767i)!(*i"ó%,+c)7i+a&"5*6+¯%'B*6+'}*!-^*+bû
S)a£
?
È
?
È #c"
40i"*ð*$a1+,*+,È"+b
*¨*6+e0+4_+7¨0+,_S+'0ªpr)*"¯'a1+,*#+#10+g*!()+g)7¢".*1"2777h)!(*i"
W ¡+
±7i*+,+a£*!a1+,*+,È"+#¡ó*+7i+)a7e+g(S(B*+aË)(¯*6+7i+%6£*+,+
?Sa
a lS
5<@
E!JL>
@'@ºx1w#KgyD:BEy
s
t Ke<DJ£yD:BKg@,KESzUCl:y/ziK'Ke@nAEJJLK t
[6>x1J
EziK.<2yD@@ºx1CC'K,@'@,>z@ʤ!yD:BKziK!K `<@ºyJ>J T JKiHESy/< u K
EJ
¤,>zK'ECl:ïw#> t K=
t
, ? oEq
y
M
y
* ,
?m y 9:E<< ^E^
77i)Ë+
?m
y \
? oEq
y
a
*+7!+7i)%,%'+77iS W *+,+'-/+ W
*,´Q,
y ,d
? oEq
?m
y b, \
?m y ?m
È
`+
?m
$" ü #&%'!()(
y nmª-
y 5,
`+
[
`+
È
P
\ k
È
`+
+
Ä
[
È
[
@'@ºx1w#K
yD:BESy
s
K " E| Ì=ÌEÌ!|K*n
Kr M
f
y
J;
È #c" Ì
y a
?m
W
M@,xBCl:GyD:BEyÁ¤,>z#KECl:£@ºx1C'CKl@@,>z
Ë:BK4JM¤,>zgK'EC,:£w#> t K4=
y ,
? o.p
[,
û
[ |=ÌEÌ=Ì!| + s
]+
t @
?m
Ä + È S# " Ì
y b,¡+,+e((1*+a-/
m/-
PB\l
j k
W
"27
7Ë+7i)%,%'+77i
y
K*n
*X,
Technion - Computer Science Department - Technical Report CIS9609 - 1996
y ¹,a
?&m
ñµ*6¯(10243+,
a1+!ac*+,+'-/+
¦mn*-/0?0¡
7Ò**
È #c"
+
?_o.p
y ,
mn*Ò"7+7i3.*7h`¡ó**-kSÒB3£7i)%,%'+77iS
*Ä,
?
K
=
È
+
Ä + È S# "
[
y f[
? o
Ksr
)!#".)(¯*+g*i¡"+ )40?"*i"+7e-/*+
*¬N,
y ,
?_o.p
?_oEq
y f[ Ì
?m
y , Ì
W
y A
o
y f[
?*o.p
y &[¦¡¨+,6+((B*+aË-/÷*+
?m
È
+
È #S" Ì
7i)%,%'+77i W ¡+g+,*
y
? oEq
`+
? o.p
y 5l \
?m
? o p
y
y , ,+ È ,+ È &[ Ä
y
=
¿S*+7i+%'ac7¢"2a1+-¦*+g"+ S)40?"*i3 W
*{,
?m
tM©
uY8¸vw4xi¶8»y³8¼)³½¾º_zX½M·5»N¼X¹8Á
y
y 5l \
?m
p7i+abS¯0+,!.
W
? o q
y
? o p
y ¹,
¡+#4_+!a1+,_+'0(+a&$40Si"*
? o p
W
274
ü
W
+
È #c" Ì
a
**%,G(L+,i-/¯7h40?0¡¾a¸a1+,+,(
()1"¯77h)#"L)a7e¯*6+7i0)*6+7i)÷--/)%'*i"7-;*+(10243+,ga."*7((S+,B*e!a1+'0ª
+¯40Si"*×* +7#7Ê"()*c(B7U"*h"S
)((+,)a
È
à È
Ã È #c" W
W
ba1+,(*&0?"#"* W
af.%')*Ã$_40)+
2 È
y
aj-kS+S%!a1+'0
W
à È
b7i*6*+,3
W
Êmn*g+,*)67e*+!ûõüÊ_`40)+!-;*6+6B*g13£103
7i+'%1".}a1+7*6*#"1*
4Ã+%'**1"27_40)+
6274 `0Si"* W )*Ë"27##)%Ë+£+7i*6i"2%'*+a¸"
i274ü#40i"*ò103G(6)+7g%6+7g*6*!E=D=!a1+'027S+,+!*
Ò+¯40i"*ô¡6 }7#7¢"#"?02i035*6G*6+£i""40
¡*7h)*+,+7%,GL+(6)+a
+
i254
()+&mn6+,)102
W
*+£(10243+,!%,)7i+£*+£S((+,1*ñ\7_``0)+c-^ca1+£*fa1+,*+,#"+¯¡+,*+,
"*7¯%6%'+.*£+,*ø_`40)+.**Ê"7Ê+,**6+,*¸*+ï%')6+,B*#%')*Ã_`40)+¯Ò1"7Ê"27gS7i+afSf*+
*n+'%'+,(*g-/e*+!7¢",mn274
((L+,B*4ñ\7e_`40)+!L+'"G+'%'*i03c*+7Ë+7e*+#(10243+,ñ\7e_40)+
(10243+,ñ\7Ê-/)%'*i"5a&*+.((S+,B*ñÖ7g-/)%'*i"56+*"2a1+,1*i"2%,40 W
+e)aË
?
È
?
È #c"
ü
W
*+
)*!*+'"aS"?Ã+,+,%'+."27#)a1+a
a1+,(L+,a7.*+
aS"27i*%'+-/9*+0+4_+70+,_S+'0ª
**+0+4_+7¦0+,_+'0 W "*%,!L+
È *aS"27i*%'+ ` W *6+g)a£%,£L+%'Ë()*+aË-k6Ó*+g)a7
` 77h**+a.13!0+,!. O{ %')*Ã5_`40)+2 È -/ø}a1+
"7*+Ë1"S+7i*!%')+,1*_40)+-`00*+£%'+7i*'7e- -/ò*+
("1*-_S"+,¡
-(10243+,
2 È "27Ë}aS"?±+ab*}a1+7Ò¡+,+e"*"27(10243+,
y
y ñÖ7;*6)£*#(1023 W aË"27)7i+a.-/
()1".¡+,+g"*"27(1023S+,
ñ\7*)6£*!(10243 *+%6¯a1+ W -/+%6 y S77i%6"*+a£¡"*c*+g(10243+,
y
¡B7h+!*)¸"*"27*c(1043 W 2 È "27.4B"#"À,+a13b*i"!+
È"7
? È "27!aS"±+a
b¿S+%6 y 7i)%6$*6* y ó
77h}%6"2*+a.¡Ò"*6£*+%')+,1*(1043+, W *+40h"*6
%6+%6 }7¡+,*+,*+
(10243+,¡1*7"*7!a1+'0n
* *+
y
Ê(1043+,;
, *%'1*i"1)+e"*'7
7i+%
y
aS"+%'*i03¯%'!()*6+a.)7¢"*+e"()*
-/0+,_+'0
¿Á")6+ï¬7i¡
7g£7i+%6f*6+,+È7U"È"02#*£*6+S+#"b±)+¯4Ä.¡"*f+!aS"?Ã+,+,%'++,_S+,3c0+`7*h"7U±+7*+L)a¯%'S7i*4"1*
-%6
O
à " * W ,
à [*W,
W 1`¡Ò".***+_40)+-~a1+
5[r1"27L)a¯40?0¡
7*+(1023S+,*!(L+,i-kS
!%')*Ã
X+,+'-/+
-k*+((L+,1*
¡"?0?0+!*0+7i*
W "*7
_``0)+¡"?0?0~L+*ËB7i*e¬g-/*+g(10243+,
¿Á")6+!{!7i¡
7(6)1".(+,h-k6!+a¯*e!a1+,+,(+,0+,_+'0~-~*+g*6+,+¨Ò+g((+,1*ñ\7!a1+'0-*+
*
à " ,"27+a13b*£()+.}a1+ P ùU)7i*Ê0?" S+!"¿Á"S)+¯¬B.¡+,_+, W *¯7i*S(f*+
7i+'%b*+#(10243+,e"*7i+'0?-*n¡"*Ã L ,#)7i*g4027i¯6+,+*6£()+Ò+(10243+,g+S7i7e**"*7gS((+,1*
S+7g*+7h+'0+%'*h"Sf"&}a1+5
à aj*+!((L+,B*#!a1+'0v*¢¡"*à N ,e7`06+a13f£_``0)+£-Ò*Ê0+S7i*
(10243+, n*+Ë(1043+,¡"*
[´
W
Technion - Computer Science Department - Technical Report CIS9609 - 1996
V1=8 a
α1=8
V0≥-5 c V1≤2-(-5)≤α1
V1= 8 b
V0=-6
d
f1=8
f0=-6
g
e
f
f1= 9
f0=-9
f1=4
f0=-5
| f1 + f0 | ≤ 2
§rá#Úëà,ßÛSäÚÞlÝÛÜnéáSåáSæ
Û1ÚÜ/ÝDÞÜnßÚî#êèH#%
¿Á"S)+!¬B
α1=8
d
α3=7
α1=8
V1= 8 e
V0=-6
V2≥-2 f
V0≥-5
h
i
f1= 9
f0=-9
f2=-2
f1=4
f0=-5
g
f1=8
f0=-6
ü ì
V3≤3-(-2)≤α3
V1≤2-(-5)≤α1
j
| f3 + f2 | ≤ 3
| f1 + f0 | ≤ 2
k§;á#Úëà,ßÛä?Ú;ÞlÝqîÚÚ6Û!ÛÜnéSáå?áægÛBÚ6Ü/Ý2Þ4ÜnßÚ6îêèH#%
¿Á")6+!{
[ep;+%,)7i+-;*6+%'7i*64"B* MÃ L
P
ÃON
W
ü ì
*+(10243+,"1-/+,7e**"*%,¯+,*e*
ËB7i*
XÈ-k6
¡+,_+, W "*
40+a13£S7#_``0)+-¬g-k6Ó(+,_")7e7i+'%
a1+
W *6+,+'-kS+e"*
+,+7Ò*!()+
6274ü40Si"*9"7¨0"27i*6+aË".±S)+4+-/0?0¡"*6+,+,Ó(_+7*6*254ü40¡Ò437+,*6)7
Ò+
*+û
_`40)+-~*6+gBS*
ü
K4yc ?]| + \ 254¦ü* VSz&YN|`S| 84 sU | *9@
4 U
s 4ZX<@£8 s ÷Î<2yD:B>SxBy#yD:BK\[6>x1J t @
A}>@,<2y/<D>J Vcz.Y
y/>
` M Ã È * W , Ã È #S" * W
t ¤ºzi>w
t KiA
yD:
Φ:BK4zUKc8
@¢A}EJLJK
9:E<< ^E^
s
"-#.%'!(/( * Vcz.YN|`S| 8 s YX ,
W >h¤GyD:BK£HESwK¸y/zUK'K
\
?WV
ûü
@'@ºx1w#KbyD:BEyr¤º>Sz£EJ&=?KEi¤
, Í
È
Ë:BK4JM
?
\
?]V
"$#&%'!()( * Vcz.Y , \ 274üE* Vcz.Y , W "*g"27#7i)1ýï%6"+,1**b7i¡**#B3c}a1+.()+a¸B3
"$ü #.%'¥(/( * VSz&Y , 77i)Ë+g** { "27.((L+,B*4ñ\7a1+ W a£4-/*+,7i+%61"
¿g(_S"¯**ûü
274
ü
%,£_S+
!+'Ã+%'*£û
+!-¨"*77i)%,%'+77iS7 W
-/g+,_+,3b!a1+'0
y
?
?
y
y c,_`+ È
`00*+!!a1+'027#77i%6"2*+ab¡"*¸*+!(10243+,S+,+!*£()+£1"27!+7Ê**
{
2 È + È ? y úln!¿S÷0+,Ë W "*g-/0?0¡
7Ê**
y n"%'+2 È "27r*+
%')6+,B*r+7i*r_`40)+-k ? y -q*6+(+,1*
y q¡Ò"0?0;4Ã+%'*r"*'7-n*+,ñÖ7;_40)+¨Ò+7!+6)!+,1*¨0a7
S77i%6"*+ab¡"*f*6+!(1023S+, W
v2 È
aÈ*+,+'-/+
a1+ W *+,+Ê"7Ò!¡3Ë***+%')6+,B*
"?-
|=ÌEÌ=Ì!| 9@±,!,º4Ky
?<_:E?caTS
Q$R
9a
"27e(10243+,ñÖ7}a1+
?
?
bdc=ã~Ú5×/Ô¥ñÒã¥óAÝ/áòÔ¥ÚpÞßÙpÛæ}ÙXÓEÒ8Ô$áßä~ã~ÚpÞßÙXÓ=å
Ý$ã~å]â=Ö=ÙXÒ$×kÙXÓ=Òè.ã~Ö=à=Øfe ó ã~ÚÒ!Ô!Ý)Óà=ãØ=Ò~ì*g7ã!çÒ$ÕÒ$Ú$æóôã~Ú5Ò)ÜÝ/ÞßÒ$àÙÞßå]âáßÒ-å]Ò$àÙ/Ô¥ÙpÞßãàæÔ
Ù/Ô¥èáßÒã¥óÙXÓEÒe´ÕÔ áßÖ=Ò$×MÝ!Ô¥àè.ÒÝ$ãå]â=ÖEÙXÒ$Øvã~à=Ý$Ò8Ô¥Ù_ÙXÓEÒè.Ò-ä¥Þßà=àÞßà=ä3ã!óAÙXÓ=Ò×XÒ!Ô¥ÚXÝ/ÓNæ=×pÞßà=Ý$Ò8ÙXÓEÒ$ÛØ=Ò$â.Ò$à=Ø3ãàáßÛvã~àÙXÓ=ÒhicÔ¥à=Ø3ÙXÓ=Ò
Ø=Ò$âEÙXÓ}ì
[X
Technion - Computer Science Department - Technical Report CIS9609 - 1996
2 4 ü * Vcz.YN|`G!VAL$e5|
9;:=<5>.?A@CBD:E? 7
*!*-Ã s | s ,¥*!*-Ã s #S" | s #c" , | * Ì=ÌÌ!| *-Ã [ | [ ,!, Ì=Ì=Ì ,¥,!*-2 s |=ÌEÌ=Ì!| 2 [ ,!,
^ `UG¥VAL-e \ Y
:=?m.B5:} /Ã s * Vcz.Y , |=Ì=ÌEÌ$| Ã [ * Vcz.Y ,
?A> =?
+ g r!()*+pr)a7~* `UG¥VAL-e7| / s |=ÌEÌ=Ì!| [ ,
@ |EÌ=ÌEÌ| 9
@
? g -9
K] ; g û z&G=;G=J * Vcz.Y ,
^ <_: +%6 Y={SB
A K
7i)%,% C
4 üE* Y={S.|`G!VAL$e
|
g 25¦
*!*-Ã s | s ,!*¥*-Ã s #c" | s #c" , | * Ì=ÌEÌ$| *-Ã [ | [ ,¥, Ì=ÌEÌ ,!,!*-2 s |=ÌEÌ=Ì$| 2 [ !, ,
<_<j ^ <_: +S% y 77h}%6"2*+a.¡Ò"*6¯%')+,1*
(10243+,
^ 7i)%,% C# y 7 ? y ? y g 7i)%,% C! y ^ ;
y D J
? y , g 7i)%,% C# y ,
2 È g .4_*$2 È E| ? y X,
^^ <_: +,_+,3 y S*ÒS77i%6"*+a£¡"*¯%')6+,B*(1043+,#ï2 È +Ë y A ? y ,
:E?m.BD:N ? J |=Ì=ÌEÌ$/| ? `UG¥VAL-e
:E?m.BD:N ? J |EÌ=ÌEÌ-E| ? `G!VQL-e
9;:=<5>.?A@CBD:E? ;S!()*+pr)a7~* `S| X s |=ÌEÌ=Ì!| [ ,
^ ` \ Y :E?m.BD:N X s |EÌ=ÌEÌ!| [
?A> =? 7h)%,% p g ;!()*+pr)a7* ` ¹ | p s |=Ì=ÌEÌ!| [ ,
<<_j ^ <_: +%6 y S77i%6"*+a.¡"*G%')6+,B*(1043+,
+! y g Y={c +! y [ Ä Y={S +Ë y ,
^ F
y D J +! y , g 7h)%,% pe y ,
:E?m.BD:N p
G#%
¿Á"S)+ZØÙSÚ
ü
à,ä?æÞ4Ünå?ãnÙSß
k*n)*;¡Ò"*6!((LB7¢"*+
7U"S, W
W "c¡1"2%¯%,7h+254 ü
7¦*+(10243+,-k)%'*i".+%'S!+7r!+7¢"#"?02*g"*7((S+,B*r!}a1+'0
*+e0+,_+'0-()1"#"%'+7i+7)(£**+g("1*¡+,+g*6+,3!)7i+g*+7!+-/)%'*i"
274;
()+7e7#)%6¯7
¿Á"aS"#)((+,L)a#*+7i)-*++,_`40)*h"S-/)%'*i"7Á"27#+7i3g*'7i e-/ËB7i*(S%'*i"2%,40
£1-/*)*6+'03
+,_``0)*i".-k)%'*h"S7
Ö¢
W *+e)aË!*+e7i)Ó-_`40)+7"%'+7i+7r¡"*.*+
aS"27i*%'+
-/÷*+#0+4_+7gac*+,+'-/+#+a1)%'+7Ê*+!!)1*
-¨()1"!+,6+'-k6+ W
)7¢"£0B17i+#)a7Ê¡"?00
(103£(61"1"*e()1"
±274
m¢$Sa1+,#*b+,*7iS!+!"2a1+b-
¡¾#)%
£)1"?-/÷*+,+7-;a1+,(*
`
(6)+7 W
ü
W aï±}+aø%61"Ë-n%'*
±254
¡+4_+.)b7¢")102*h"S5((103"
W ¡17i+e0+_S+7¡+,+77¢"+aø*i¡¨#a1
ü
H+Å*¢¡¨+ 7i)%6Èe*6+,+B3 "
- I
i*$ |!`c| + ,!,
++'}(L+,i"!+,B*r¡7¦)
¡"*c+,°D0+,_+'0;!a1+'0"c(1043+, W *-Ã
Ã
Ò
,
h
7
)
%
c
*
*
Ã
Ã
" [. J+Ë
++'}(L+,i"!+,1*7e¡+,+%'a1)%'*6+a
"| [
-/
+ ¡+%'+*6+a5YSYY.*+,+7af'274 ü f*6+,¯cÉ
([Y
-"I
é*U[ | 4Y | +
,!¿Z+S%fL)aK÷
7i¡
7*+£(L*i"&-0+4_+7#()+a¹7#ø-k)%'*i"-
*+£S)aK!
+ 7È+'}(L+%'*+a W *6+¯!)1*_``0)+7¡"*e7i)S)a1+a!13
()1"!a1+%'+7i+77¦*6+
aS"?ò+,6+,%'+g+,*i¡+,+,!*+-/)%'*i"7"%'+7i+7,r+41"#)0+,_+'0q-²(6)1"
[Ä
0.7
Portion Pruned
0.5
0.4
0.3
0.2
0.1
100
90
80
70
60
50
40
30
20
10
0
0
Technion - Computer Science Department - Technical Report CIS9609 - 1996
0.6
Bound
¿Á")+#[Y¦ØÙÚÛBÞ4Ünã¢åÞ4á#Þ,Ýqä?Úà
,+
\
"27
S%1"+,_+aï-k
$#&%
ÚçÛÜ¢éáÚî!êè
ü
(' ü
à,ç;àÒÝ2éáâã¢å?Þá#Þ,Ý
ãnÙÚê1ÞéSáî#Þá
ë274üÒ"7+a1)%'+aø*7i*aa274;
d¸+
+,(L+*+a.*+g*61"'a.+'}(L+,i"!+,B*e-;)7h+%'*i"
'Öì
Y W ¡+,+
Ö[g+,(102%6".*+û
+gL)ac)7i+aË-kS()1"!"27,
à "
"*) ü [
à [ \
$" ü #.%'¥(/(
274
`0Si"*
¡"*
ü û NH L-GEI y HW *- z.HUI.`S|/VcWXHPUGEI ,c&Y Ì YY´ Ä.- Ez L-HW wy Oc{cI.GEY *- .z HI&` , û
HNL-GEI y HUW *$ z&HI.`c|~z~V}Vcz.JG=JSL ,$Y Ì YSY´ Ä/- =z L$HUW wy OQ{SI.G=Y $* z&HI.` ,
Y Ì YY{
.Ä - z=L$HUW wy OQ{SI.G=Y *$ z&HI.` ,
dó1"?0+*+4_+,'+eB)#+,e-¦0+_S+7i°/(+,°/Ë`_S+e-ke!7i+%6£*!a1+,(*c´!B3£û
274)7¢" Ã "
+a&*f%6+'"_S+G54_+,+-ÈSY£0+4_+7i°/(L+,°/!_+
0+4_+7i°/(+,6°/!_+
S)+7i*i"
Ò+e0S7i*
6+7i)10*7'4"27i+."1*+,+7i*h"
-" ü #.%'!(/(
¡7e¬`[ W
254
ü
%1"+,_+a¹54_+,S+-ÄÄ
77i)Ë+g**¡+`00%,*+##!a1+'00?"¯(10243+,ga
!)Ë}a1+'0?0?"!((S+,B*r*+
7Ë+
7i+%6#+7i)'%'+7,mU7*+L+,+'±*ÒS%1"+,_+a#13#!a1+'0"#+,).*
0
_+,%'Ë+*++'}*6a1+,(****+S°/!}a1+'0?0?"(10243+,%,7i+%a1)+r**+L+,**+,¦()1" ïÒ1"7
)+7i*h"S
"27gc"7i*%'+!-;*+# B¡b*Sa1+,Ã$+,*i¡+,+,b B¡0+a1+!ab7i+%+!7i¡+,g*.*1"27
!B7h*i03Èa1+,(L+,a7r!*+(*i"2%')102*)+-²*6+B!+*6+,+
a#+,_40)*i"Ë-k)%'*i"7ldf+4_+*6+7i*+a
)+7h*i"&"5*+£%'1*+'}*#*+£L`_S++'(+,i"!+,1*¸df+£¡*6+&"*+,*h"_S+ïa1+,+,(L+,1"+,_17¢"727f
4 a274 ü aÈ0+,**+e*U¡g(10243ïB4"7i*;+S%.*+,)7¢"#*+7Ë+
7i+%6.+7i)%'+7l£274 ü *
¬`{Ye(L"B*'7;S)*;-;ÄYSYdf+g%,7h+,+
L*!(10243+,7r¡+,+
40!17i*;+=S
)1"_`40+,1* W *1)7 W *6+ B¡0+a1+gL)*
*+g((L+,1*^7i*6*+,3Ë¡ÒS7"a1+,+ab#¡*6B¡1"?0+g*Sa1ÃG*Ë*+aaS"*i"40;7i+%6£(+,h-k6!+a.B327¨
4 *+
S*
1
3¦'53x
2 4
;
76 98$8ýÊ;:nS73]q,<
mn.*+±7i*r(*;-²*61"7¡ #¡+4_+(+7i+,1*+aË!+,*a7-k)7U"Ë((+,1*;Ë}a1+'027,m¢£*1"27;7i+%'*h"S
¡+a1+7%'i"L+ïø!+,*a103c-kS% )1"i"f7i)%6$.!a1+'0;-/
((L+,B*4ñ\7#a1+%6"7U"S7#"27#"_+,5S7e"()* W
+'!(10+7,
7i+,*g-1a7g¡Ò"*6f*+
a&*+Ë0+1"f(%'+a1)+¯(a1)%'+7!c!a1+'0
7)*()*
)*6+
1"27Ò-k!+,¡ #"277¢"#"?02g*#*+76%'+,i")7h+a£13ïÆg7h(_7a1+76%'i"+aø"¯*+(+,1"
Áb +.*+Ë0+1"$*'7i f!S+!-/+7¢"10+ W
¡¨+£77h)!+£***+£((S+,B*"27!b#"1"4(10243+,
aË"*7Ë}a1+'0*6+,+'-kS+%'7¢"27i*7Ò-~*i¡!%'!(L+,B*'7,!a1+,(*¯-~7i+%6¯a££+,_40)*i"ø-k)%'*i"
[1¬
"%'+£*+7i(S%'+-(B767¢"10+ba1+,(*f_40)+7"27#)%6$7i`00+,#*¸*+7i(%'+.-(177¢"10+.-k)%'*i"
7i**13$a1+7%'i"1"¯Ë+,*}aj-/0+1"f*+£((L+,B*4ñ\7#a1+,(*°/-/°n7i+%6
W
W
¡+
a¸(%'+,+a&¡"**+
Technion - Computer Science Department - Technical Report CIS9609 - 1996
!6+%'!(10?"2%,*+a£*'7i .-0+61"Ë"*7r-/)%'*i"
¨©
¬3¸cº»}³¼/³C½
¹8¸8¸S¶¹ · À.¸Sº_»
É"_+,#
7h+,*-+'!(10+7 W +S%#%'7¢"27i*i"g-qLBaÊ*+,*6+,¡"**+;Ë`_S+¨7i+'0+%'*+a13g*+;((L+,1* W
* ` ¡ %$% ¡Xs.rXsEu ,)a1+,*+S77i)!(*i".***+gS((+,1*
"*"27+'02*i"_+'03.+7i3.*g0+6."*'7a1+,(*£-~7h+%6
7i+'%+7#*6jø±+a5a1+,(*5a&**"*7#+,_``0)*i"-/)%'*i""27# 1¡&*f*+Ë0++,ó"%'+c*+,+
"27#103f7i40?0
7i+,*#-g(102)7¢"10+G_40)+7#-/
*+b((L+,B*4ñ\7a1+%6"27¢"7l÷·+G(LB77¢"1"?0?"*U3
` ¡ %$% ¡Xs&rts=u
W
¡+ï%,*+7i*#¡1"2%¹-
*+,
"7.*6$*+7h*.-k£7i*h"%'*£6+,+,!+,B* W
"nµ+ W
+,+7+7h*!¡"*
-k£13+'Z!(10+
*- z.HI&`S|Tz.G ,¡+%,c+,¡a.B3£a1+,(* ` -/¡1"2% T y J y THR *$ z&HI.`c|~` ,6+,*)7 Tz.G *- z.HI&` ,
dó+,
Ã
¢¡/f$r)É5\ à ¡ %$% ¡ts&rts=u W 7i)%!!+,*ag+,+a7¦103#-/+,¡$+'!(10+7¦*"1-k+, ` ¡ %$% ¡Xs.rXsEu ¦¡+,_+, W "
*+
%,S7i+¡+,+3Ã
¢¡/f$r)É aS"?Ã+,7-/ à ¡ %$% ¡Xs.rXs=u W 7h)%£40Si"*"27(6+*+,Ámn.a1+,r*e"!(_+
*+Ë0+1"Ç1"0?"*i3f"*+£(6+7i+,%'+£-^c¡S£-/)%'*i"!a1+'0 W
40h"*6
a1+,(*
`
a1+,(*
`
¡+4_+£a1+,_+'0(L+a5¸"!(_+a
**%'7¢"2a1+,7Ê*+#+'02*i"_+!Sa1+,i"£+,*i¡+,+,f(177¢"10+!_+7lgdf+#¡B*e*£+,¡'a£+,_+,3
"?-~*+gË`_S+
"_+,."c*+g+'Z!(10+7"a1+,+ac*+g1"+7i*#"1"4¯_40)+¡+,¯7h+%61".*
¦mª-"*;"27r*+
7h+%'a!L+7i*!_+ W ¡+
7h*i"?00¡1**6+,¡Òa"* W )*¨Ê0?"**i0+
1"*r0+77,;Á"!(10+,!+,B*
7i)%6!(L0?"%'3#**~%'+aS"*7ga1+,(*6ÈS%,%'aS"g*e*+;+'02*h"_S+¨(102%'+-*++'Z!(10+¨Ë`_S+ W *+40Si"*
+,¡a7+,_+,3ba1+,(*b¡"*b*+#B)#L+,g-;!_+7**e_S+0¡+,
È"1"`j_40)+!*c*+#+'Z!(10+
!_+ W
a£(+,`0"À,+7Ê"*
¡Ò"*6G*6+1)L+,
-¨40*+,*i"_+!_+7¡"*b1"+,g#"1"4b_`40)+#*c*+
+'!(10+!_+1)7 W
Ëa1+,(*£-/¡1"2%6¯*+g+'!(10+!_+g7*+g1"+7h*
#"1"4¯_40)+¡"?0?0+,*
#1"c+,¡a
+`0Si"*Ó"277i¡."ø±)+[Z
9;:E<D>?@CBD:E?F8GEHUI.JbýG¥VAL-e *-;
ûÅ F K ,
^ <_: +%6/ z.HI.`c|~Tz&G. 0 ; eû F K
z.HUI.`Y g YE{cG=YEY=z.I.Y *- z.HI.` ,
^ <:
` ^ :=<7a g*.û HRcbdG!VAL$e
û
g T y J y THR * Tz.G *- z.HI.` , |!` .,
z&{SJSL ` g z.{cJSL ` 0 z.HUI.`Y T y J y THR *- |~` ,5û ¦
0 z.HUI.`Y T y J y THR *- |~` ,5û ¦
Ò
¡
:E?m.BD:N` "*6£41".40 z.{cJSL ` M§rá!à,ä?æÞ4Ünå?ãnÙß¹Ý2Þ4Üä?Úà,Ü¢áå?áæ^àßÞîÚ6äÞ,Ýqã¢ÙÚÞ4ÛÛBÞ4áÚáã çîÚ6ÛãnÙ1ù ¢¡/f$r/É
¿Á"S)+[S
m¢Sa1+,*!7i*)a13!*+g+'Ã+%'*-*+aS"27i*%'+g+,*i¡+,+,*+gË}a1+'0
-/)%'*i"¯aË*+%'*)40L-k)%'*h"S
£*+e0+61"¯1"0?"*i3£-*+40i"*
W ¡+g_S+g(+,i-/!+a.*6+e-k0?0¡"+'(+,h"Ë+,B*
à ¢ ¡/f$r)É W )7i+aøB3.*+e0+1"¯40h"*6 W ¡7±}+a
[ZÃ
¡ %$% ¡ts&rts=u ¡7¦a1+'±+aË*
6+,*)!_40)+
%,%''aS"*ge(!+,*6+, V
¡"*£(1"?0?"*i3 V
a£Èa1
_40)+¡"*c(1"?0?"*i3f
V S
7i+,*-¨B!+7e+,*i¡+,+,f*i¡#"1"4f(10243+,7 W
YYBa7**e·
r·
¼e;¼e¹-ªS%'+a
Åa!8r8"Ñe;Ñ.
s 8
à ¢¡/f$r/É
¦+-/)%'*i"#+,*)7
W
¡7e%'a1)%'*+a
W *+,*6+,¨¡Ò"*6."*'7
%6B7i+,£!_+7 W ¡+,+e"_+,b7r+'ZË(10+7
*#*+e0+1"¯40i"*ö¡1"2%c)(qa*+a
` ¢¡/f$r)É [{
´L
-/*+,+%6!_+ W *+%')6+,B*
` ¢£¡)f$r/É
¡7%'!(+aB4"7i*
!+,B* W *6+%,%')#)10*i"_++,6*6+
¡7r"%'+,Ë+,B*+a
V
Technion - Computer Science Department - Technical Report CIS9609 - 1996
XZ+g+'}(L+,i"!+,1*
¡76+,(+*+aË-/_i")7
` ¡ %$% ¡ts&rts=u
¦m/-*+,3.¡¨+,6+"¯aS"276+,+,°
¿Á")6+¯[[b7i¡
7*+¯%,%')#)102*i"_+£+,S'*+£-
*6+¯40i"*7#7Ë£-/)%'*i"¹-
*+¯aS"27i*%'+
à ¢£¡)f$r/É
à ¡ %$% ¡Xs.rXsEu
=
+,*i¡+,+,
a
m¢*
4027i7h`¡e7*+
`
Cà ¢¡/f$r/É
%')B*6+,74-k*6+,YY+'!(10+7-/*+%,7h+g-
¡"*c(1"?0?"*U3£-~+,6+ )4027*YÖ[X
Depth Learning: Accumulative error rate
error
No. of samples
1.00
260.00
0.90
240.00
Learning the op. depth
220.00
0.80
200.00
0.70
180.00
0.60
160.00
140.00
0.50
120.00
0.40
100.00
0.30
80.00
60.00
0.20
40.00
0.10
20.00
0.00
0.00
0.50
1.00
0.00
function diff.
1.00
2.00
3.00
4.00
5.00
depth
3 ÚàlÜnáå?áæãnÙÚgîSÚÛãnÙ.Þ,ݦçnÚàlÜnâiÙéSçnå?áæõöEöeÚëà,ßÛSäÚ6ç÷øÙÚÜ¢Úù ¡ %$% ¡ts&rts=uMúÍû àláîýü ¢¡/f$r/É îåÿþÚÜnçÝDÜnÞß
ü ¡ %$% ¡Xs.rXsEu øå?ãnÙ#ÛÜ¢ÞêBàlêå?ä?åã/èeê1ÚãXøÚ6Úá ögàláîõìÁØÙÚäÚhÝ2ãæÜià,ÛÙ#çnÙSÞøç;ãnÙSÚ
à,ââ6éßgéäà,ã¢å ÚÚÜnÜ¢ÞÜrÜUà,ã¢Ú
à,ç;àÝDéáâã¢åÞ4á
Þ,Ýã¢ÙÚÚÜ¢ÜnÞܦÞ,Ýã¢ÙÚÝDéáâã¢åÞ4á
ßÞîÚä/ìØÙÚrÜnå?æÙãæ4ÜUà,ÛSÙgçnÙÞøçã¢ÙÚ;â6ÞéáãnÚ6Ünç¦Þ,ÝãnÙSÚ;ä?Úà,Ü¢áå?áæà,ä?æÞ4Ünå?ãnÙSß&Ý2ÞÜ
ÝDéáâã¢åÞ4á
ßÞîÚ6äBÚÜ¢ÜnÞ4Ü;Þ,Ý5ö ì
¿Á")+![[
Ò+
%,%')#)102*i"_+e+,*+"27*+(*h"S-*+Ò0+1"È7h+77¢".¡+,+*+Ò0++,7;Ê¡
!a1+'0
-e"*7#((L+,B*4ñ\7!a1+,(*ó+£+'(+,i"!+,1*ï7h`¡e7#**"a1+,+a¹¡+,5*+.-/)%'*i"¹!}a1+'0"7
%60B7i++,)£*!*6+g((+,1*ñ\7¨-k)%'*i"
`¡+,_+, W
*+40Si"*
W
7h)%,%'+,+a7"£0+61".*+((+,1*ñ\7a1+,(*
¡+,f*6+!((S+,B*ñÖ7e-/)%'*i"f"27#7¢"1"?±q%,1*i03ÇaS"?Ã+,+,1*g-/ò*+!!a1+'0 W
*6+40i"*cñ\7
+,S¨'*+e"%'+7i+7l
̲+1"¯*+Ë((+,1*ñ\7ga1+,(*6$-7i+%6b43¯(_S"2a1+!_+,3b)7i+'-/)10;"1-/*h"S
W
+7i(L+%6"`003¸"$
BË+ï14"7i*c¡+ +,#((S+,B*&dó+,5c(1023S+,#"27!4¡Ò+-
*+.0?"#"*-e"*7#((L+,1*ñ\7#7i+%6
i"À,
¨©
W "*g%,.0+aø*+g((S+,B*e*#*(£S
7i¡"aS0+#(B7¢"*i"7\Ä4n
¬3¸cº»}³¼/³½Í¹8¸ ·5¶8¶·7³8¸c³7¹
1ÀÀE¹}»}ºS¹}¸A½5¿
+e(+,i-/.%'+-*+e(+,_S"S)7¨`0Si"*
-/)%'*i"
+¸*)40+'*¯7i*+,(
a1+,(+,a7S!*+e B¡0+a1+e-*+((L+,1*ñ\7r+,_`40)*h"S
"7ø*5a1+,_+'0(Å40i"*
-kS.0+1"ú+,_40)*i"
-k)%'*h"S7,
`¡+,_+, W 0+1".*+g((S+,B*ñÖ7r-k)%'*h"S£¡"?00(103£a1+,(+,aø£ 1`¡Ò"Ë*+g((S+,B*ñÖ7¨a1+,(*6
-¨7i+%Êmnba1+,e*£+ £*61"7g%6"%60+ W
¡+4_+#a1+,_+'0(+a&f40i"*ö-/0+61"c*+-k)%'*h"S
a£a1+,(*c-;7i+%6¯7¢"#)10*'+,)7¢03
"%'+.0+1"f1"*3b+40-/)%'*i"&"27a
7i)!(*h"S7,
[
W
¡¨+Ë_S+!a1+!*6+!-/0?0`¡Ò"f7¢"!(10?"?-/3"7i°
Ã5*$E, \ ZÅÄ e *$E, \ÇÆ´È}Z È e È *-E,
"27*++,_`40)*+aøBa£a e È *$E,~6+,*)7*+ y *.-/+*)+g-¦**LBa
S+S((+,1*ñ\7^-k)%'*i"&"27#ø0?"+%'#1"*i"&--/+*)+7,
¡+,+
Technion - Computer Science Department - Technical Report CIS9609 - 1996
[Z
7i)(L+,7i+,*e-*+Ê-k+*)+7r)7i+a£13.*+((L+,1*"27 1¡£*#*+g0+6+,
+g((S+,B*ga1B+7Ò*
%6+e"*7Ò-k)%'*i"c¡1"?0+(10243"
a1+,
*+7h+77i)!(*h"S7
®* Z¢¡/f$r)É |` ¢£¡)f$r/É ,
W *+e0+1"£*S7i #"276+a1)%'+a£*#±aS".*+(`"
Ò+0+61"c(%'+a1)+!%'!()*+7-/e+%b(LB77¢"10+ïa1+,(*6
`
!¡+'"1*g_+%'*
Z3f
W
7i)%6b**e*+
* Zf8Ä e5|` ,ËB7i*
S+,+7r¡"*¯*+gS((+,1*ñ\7a1+%6"27¢"7l¨+#a(*+a.Ë}a1+'0"27*6+g+7i*e(4"
7i*'*+,3
-/)aË-kS
40?0~a1+,(*7,r¿+%6¯a1+,(*
W *+`0Si"*
(L+,i-kS7#1"?0?0°n%60?"#1"2b7i+%6
¡+'"B*_+%'*)B*h"0£-/)*+,!7¢"1"?±q%,1*"Ë(_+,!+,1*%,fL+ïS%1"+,_+a
77i)Ë+**
*++7i*_S+%'*-/)a¯7h#-ª-kS*+%')+,1*ga1+,(*¨¿Z+%6£-*++'!(10+7 W
b7i+,*!-g%'7i*`"1*7**#+'(+77#*6+ï7i)(L+,i"i"*i3-
*+£7i+'0+%'*+a
40h"*6
d* Z npoq)q-rts=u Ä e7|` G,
(L+,i-/7rg#"1"4ï7h+%6)7¢"
-*++'!(10+eBa
W "!(_".*+
Z npoq)q-rXsEu
*+#40i"*
"7
)1"?02a7
!_+_+,"*7!`0*6+,*i"_+7,&+
W 7h**i"Ê-kS¾+%6!-*+g7i)%,%'+77iS7
**++,aË-*1"27;7i*'++%6!-*+
40*+,6*i"_+!_+7r%,!L+
77i%6"2*+aˡ"*
Ê,a1#"1*Ëc1a¸**a1+,*6+,#"+7#"*7#"1"4_``0)+ 767i)!+.**i nX ¡ ( rts "27#*+£a1#"1*
1a£-;*+%6B7i+,b!_+ W a
" |=Ì=ÌEÌ$| s 6+*+#a1#"1*
LBa7Ò-kS
*+#40*+,*h"_S+Ë`_S+7,+
40h"*6
aa7*+ J
%'7i*`"1*7
¦ *"*7%,%')#)102*+ac7i+,*
ZÍÄ * e *- nX ~¡ ( rXs ,M e *- È ,!, Y Nyk\ |EÌ=ÌEÌ-|~J *+
-;%'7i*'4"B*7l
=
Ò+;+'*7i*S+¨%'7¢"27i*'7-7i0_S"g*+¦"+ )40?"*i"+77i37i*+,
Z
W "nµ+ W ±aS"
*6*7*h"7U±+7*+7i37i*+,¯
+!+,*a¡+¨)7h+ag"27r_i"2*i"!-*+r0"+r('!#"e!+,*a)7h+a#B3#oe)aga#
*¨ ´`1-/
(**6+,#+%'1"*i";pr+'-/+*+40h"*6
7i**7²"*'7Á"*6+,*i"7 W "*¨7i+,*7S7¢"2a1+
e(6*i"!-L"*'7¦+'ZË(10+7
-/(6+77¦Ë1"*i"1"277i+,*¦"27¦*;4_``"0210+e*e*+(}%'+a1)6+**¦)1"?0a7r*+%'7h*4"1*7,
7i0_S"£*+%'7h*4"1*7
7h3}7i*6+,
W
*6+40h"*6
-/*+,
*+7h*7*6+7i0)*i"b_+%'*613£!+7i)i"."*7e%,%')%'3
"$(6+aS"%'*h"b*+Ë((+,1*ñ\7Ê!_+7Ò-k*+#*+7i*+'ZË(10+7l+!(L+,i-/%'+Ë-¨*+Ë+,¡¾_+%'*Se"7
%'!(+a£¡"*b**
-*+!%')+,1*e_+%'*mª-¨*+,6+g"27eï7U"S1"±q%,B*Ê"Ë(_+,!+,1* W ¡+767i)!+#**
*+#%')+,1*g_+%'*"27e*+#L+7i*
*6*^%,GL+-k)aï-kS
*+#%')+,1*a1+,(*
W
aø*+!40i"*÷+,(+*7
*+g(%'+77Ò-kS¨*6++'*
a1+,(*¯)7U".*+%')6+,B*_S+%'*r-kS"*7r"1"*i"240¨7i**+,3
Ò+"+,0B(.-*6+
40i"*
W *6*¨7i+%+7-/;*6++7i*r-/)%'*i".-/¨Ê"_+,ïa1+,(*6
W "27¨7U"È"02
vÊÉ^e¾\X4acB3
*.*+Ë+,*}aba1+,_+'0(L+ab13f;i"27i*6+,7i+,fabÆei-
\{4ac)7i+acB3bog;;óÊ·
¨mU¼Ê·g·
Æ{4-/g*)1"¯*+'"#+,_``0)*i"f-/)%'*i"b-/
LBS GË`_S+7,!¡+,_+, W
=
*6+7i+!(S7
77h)!+
±+a!7i40?0a1+,(*#-/*+'"7i+%6]¯+,)10+,b\[Y4)7i+a!7i+,*-
"+ )`0"*i"+7r-/LB 0+1" W
)*1"27(
S77i)!+7S103£S+e0+,_S+'0a1+,(*c-;7i+%6
Ò+40i"*Ó"27Ò0?"7h*+a."£±S)+[ Ò+^7i*6*+,30+1"£40i"*Ó¡7r*+7i*6+a!13*6+-/0?0`¡Ò"Ë+'}(L+,i"!+,B*4¨6+,+±+a£7i**+,"+7
*-æ | { , W -* Ã
[ | { ,aÎ*-æ | Ä ,
væeaiÃ
[6+*i¡¨g_i"2*i"7-B#)+'0nñ\7
W ¡+,+)7i+a.7((+,1*7 W ¡+,+
-/)%'*i";%6Ç7h**+,S3f¡7g)7i+a&*6G(10243&B!+7)1*i"?0ÄYYø+'Z!(10+7!¡+,+.+,+,*6+aaf"_S+,
*c*+!0+61"&40i"*¯b+£40i"*ס74027ib"_+,Çb7i+,*-
*+,¸-/+*)+7 W
"%60)aS"*+7¢"?
-/+*)+7S%'*)40?03£)7i+a£13*6+7i**6+,"+7,
Ò+40h"*6
¡7)c¡"*f!a1+,(*c0?"È"*g-ÒS¨++'Z!(10+7e¡¨+,6+aS"_S"2a1+ab13£*+40Si"*
"B*6ïË*4"1"¯7i+,*ga¯Ë*+7i*h"£7i+,*e-;7¢"À,+È{SYY¿S
+%6¯-;*++'0+,_+,fa1+,(*b_``0)+7 W
(+,h-k6!+a£[°
*+(S
"*+,'*i"7L+'-k6+g!_"#*#*+e+'}*a1+,(*;%6Ë"*6+,*i"Ë"%60)a1+ac)7¢"!*6+e0"+
!Ï
(S!#"!Ë+,*}aË-/
#7i+,*-7h+,_+,40*6)7a£%'7h*4"1*7 ,
Ð$ÑÒ3Ó&Ô$Õ~ÒÖ=×XÒ$ØdÙXÓ=ÒÕ~Ò$ÚXÛ
Ò/ÜÝ)ÞßÒ$àÙáßâ ×Xã¥áßÕÒ3â=ÚXã~äÚ/Ô¥å3æ_ç7ÚpÞßÙXÙXÒ-àéèÛëê;ì í£ì îkìïê;ìQðSÒ$ÚXñ~Ò/áòÔ~Ô¥Ú$æcóôã~Ú8×Xã¥áßÕ~Þßà=äiÙXÓEÒÝ$ãàE×XÙXÚ/Ô$ÞßàÙX×
×XÛ×XÙXÒ$å
Y
Technion - Computer Science Department - Technical Report CIS9609 - 1996
9;:=<5>.?A@CBD:E?F8G=HI.J7KML-I.HNL-GOQP * G=RSHUT
VSWXG=Y ,
Z3[]\ ^ <_:
` ^ :E<7a g*6Èû HRcbdG!VAL$e
Zf3g Z3f #S"
hi?Ajk?lAm
Z3npoq/q-rts=ukg Zvf
Z3fvgxwy J`AKz.WX{AL y z.J * G=RcHT
VcWXGEY}| Z]npoq)q-rXsEu |~` ,
VcI.zOQI.GEY=Y g Y=z.I.G * Z f |~` , YEz&I.G * Z npoq/q$rXsEu |~` , N
0
cm=p !(6+77
, ¡Ò"*6£*+`B"407%'S+
:E?m.BD:N * Z3f |` ~
9;:=<5>.?A@CBD:E? wy J`AKz.WX{AL y z.J *$eû F KM| Z8npoq)q-rXsEu |~` ,
z&JYL-I.H y JSL-Y g
^ <_: +%6/ z.HI&`S|eSz.Y=G=J Tz.UG 0 ;eû F K
K g û z.UGEGEJ *$ z&HI.` ,
^ <: +S% YE{c 0 K ;
`z.T y JHJcL ( o=nXng û y J y THR * YE{c.| Znpoq)q-rts=u |` .,
z.J_YL-I.H y JSL-Y g z.J_YL-I.H y JSL-Y Z * e * `Uz.T y JHJSL nX ~¡ ( rts ¢£¡t¤r ,M e * `z.T y JHUJSL ( o=nXn ¥, , Y ¦
:E?m.BD:N Z **Ò76*i"27¢-/3 z&JYL-I.H y JSL-Y
k§;á!à,ä?æ4ÞÜnå?ã¢Ùß¹Ý2ÞÜä?ÚàlÜnáå?áægàeßÞîÚä1Þ,Ý
ãnÙÚÞÛÛ1ÞáÚ6á`ã~ ççnãnÜià,ã¢Úæè
¿Á")+#[
Ò+£+7i)10*7#-g*+£+'(+,i"!+,1*Ë-k#*+£*6+,+¯7i**+,"+7"27!7i¡&"±S)+¯[´+¯40Si"*
7i)%,%'+,+a1+a¸-/g*+.*+,+%,S7i+7 W
S%1"+,_S"&$%,%')S%'3G-Ê4YY Ó-/g*i¡¨¯7h**+,"+7af Ó-k*+
*1"a
;¿Z)*+,Ë+ W *+e1"S+7i*
%,%')S%'3¡7%61"+,_+aË-kS¨*6+^%'*6)40a1+,(*£)7i+a.13!*+7i*'*+,"+7,
¨©pª
«¬®M¯±°²´³µ«¶8¶·5³8¸S³5¹´¬v¸Sº_»N³8¼/³C½¾k¿8À=¹}¸SÁ
mnca1+,*!*6+7i*`¡ó¡+'0?0*+û
)1"?0*
gB!+
(10243S"#(S
ü
`0Si"*
W ·
̦½
a.*6+e0+1"¯40i"*
%,£+e"B*6+,*+a
W ¡¨+e4_+
W *6*"2710+g*S% )1"+gaË4"B*'4"ï!a1+'0q-Á"*'7;S((+,1*
aË)7i+"**Ê"*7a1_`1*S++g7i37i*+,
%'S7¢"27i*7r-*i¡g4"£%'!(S+,B*7 W ÊB!+,°/(10243S"#(S
-" ü #.%'!(/(
abÈ0+1"¯(S¯e+#B!+,°/(10243"¯%'Ë(+,1*g)7i+7e*+!û
40i"*c^Ò+0+61"
(S%,%')#)10*+7¦*+!_+7Á(+,i-/!+a13*+((L+,1*;a!}aS"?±+7*+((+,1*!a1+'0}¡1"0+
(10243"
;*'YË"$)+'}(L+,i"!+,1*7~,
-k*+,g+S%f b!_+7
)7¢".*+40i"*
Ò+·
Ì~½
W
*+7i37i*+,ò)(qa*+7^"*7g!a1+'0¨-ÒS((+,1*
"b¿Á")+[ =
7i37i*+,÷¡ÒS7e*+7i*+aø"$.+40?"27i*i"2%.7¢"*)*i"¸B3ø0+,*6*i"£"*(10243f£7i+ )+,%'+Ë-Ò%6+% S+,7
]*U`00(6.7¦)7i+a#()1"
BË+7B4"7i*r+,)102;È"1"`(1023S+,7
W )*"*¨aS"2a!S*+'ò+%'*r*++7h)10*'7 W
Â,
7¢"%'+$40?040i"*.77i+%+a¹*¸*+f7!+ba1+,(*Åa&*i"!+c0?"È"*c¡7#"_+, ×·
̦½×a
=
"*7
*i¡!((S+,B*7e+S%b)7i+ab£aS"?Ã+,+,B*7h**+,S3)*g¡"*f.S)103¯+ S)1"_40+,B*(1023S"G1"?0?"*i3+,3
40?0
7i+%6+a&*fa1+,(*5Äba&*+,340?0)7i+a&_`i"2*i"7#-B#)+'0ªñÖ7!7i**h"%Ë+,_``0)*i"-/)%'*i"&df+
7i*S((+a£*6+BË+7
7i+,_S+,40*i"!+7*!*6+7i*
¡Å¡+'0?0·
̦½¾(L+,i-kS714"7i*"*7((L+,B*'7,+#*+7i*
¡7%'a1)%'*6+a¯13.*)1"£Ãf*+g0+61".!+%61"27iaø(+,i-/#"£#*)6!+,1*-
YYÈB!+7
+,*i¡+,+,¯*6+g*U¡
=
¿Á")6+
[Xg7i¡
7*++7i)10*7-*1"27+'(+,i"!+,1*;Ò+
(10243+,7r7i**¦¡"*£40!B7i*+ S)1"_`40+,1*
1"?0?"*U3
`¡+,_+, W
`-k*+,#103&-/+,¡Ó+'ZË(10+7 W
*+!0+61"&(S×+%'Ë+7!7¢"1"?±q%,1*i0357i*S+,*6&"*7
Strategy learning
success/guesses
(f1,8)
(f2,8)
(f1,6)
1.00
Technion - Computer Science Department - Technical Report CIS9609 - 1996
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
5.00
depth
10.00
ÚàlÜnáå?áæeÞÛÛ1ÞáSÚáã ççnã¢ÜUà,ã¢Úæè
¿Á")+[´
°D0+1"c((+,1*7,Ò+(+,h-k6%'+-·
̦½¾ +,(*"%'+7¢"£)1*i"?0)*gXY#+'!(10+7e¡+,+
(%'+77i+a
-/*+,*6* W *+e0+1"£%')_+¡70+,_+'0+a
df+e"7i(L+%'*+ac*+%,%')'%'3-~*+gË}a1+'027e7
7i)%,%'+,+a1+ac"f%S
)1"h"c!a1+'07
-/*+,#YYË+'Z!(10+7 W ·
̦½
S)40?"*i3c-ÒXÓ-ke*+±7i*af{9-kSg*+!7i+%'a
#Ì
+1"¯Ë+
a1+,*+,È"+ajB3£*+0+1"c(}%'+a1)6+
-/e"*7e*i¡¨.((L+,B*'7e¡"*$
+'!(10+7
aS"2aø*
%,)7i+#13#-k)6*+,!aS"?±q%,*i"b-*+g!a1+'07l
ÊÊÊ!Ê
+#"1".4j40Si"*
S77i)!+7***6+S((+,1*
)7h+7*+#7!+#7i**+,3£7a11+7e*+(10243+,m¢
*1"27g((+,g¡+#(+7i+,1*+af.+,+,`0"À,+a¸_+,7¢"b-*+##"1"4$40i"*ò**g%,b)*i"?0?"À,+¯aS"?Ã+,+,1*
((L+,B*c!a1+'07lÓdf+b(+7i+,1*+aó*+fû
40h"*6
ü
a1+,*+,È"+È"*'7e+'}(L+%'*+afa1+%6"27¢"b-/e*+#+'}*g!_+ W
W
¡1"2%õ7U"#)102*+7ø*+b((S+,B*ñÖ7¯7i+'%ó*
aø+,_`40)*+7Ê*+#+7i)10*+ac1acB3b7i+%61"
"*7767i}%6"2*6+a£7i)*+,+e)7¢"!"*7Ò`¡ï7i*6*+,3Ádf+g*+,£(+7h+,B*+a.*6+û
_+,7U"SG-¨û
ü
**g+'(a7e*+#*+,+#103¯S%'+ W
$" ü #&%'!()(
40Si"*
W ¡1"2%6."27
)*g((B*+7
`00;*6++%'+7673c_`40)+7,#+!û
+*U7¡+'0?0~7]£"1"4,¨767i)!+7***+g((L+,B*e7i+%6+77a1+,+,(b7*+g(10243+,ñÖ77i+%6
ü
40h"*6
i"À,Ëdf+!+'*+,a1+ab*+.û
ü
abû
-" ü #.%'!(/(
((L+,B*4ñ\7a1+,(*c-7h+%6
40Si"*7g*6£+,10+*+#)*i"?0?"À*i"-!a1+'07-*+
~}(L+,i"!+,1*7.(+,i-/Ë+a"Å*6+ja1S4"ó-%+%6 +,7£a1+,Ë7i**+a*+fa1_1*+b-û
ü
_+,
#"1"4£)*
4027i!7i¡+a!*6*¨.+,S;"£*+e!a1+'0~%,£a1+,*+,i"*+
(L+,i-/%'+7U"S1"±q%,B*i03rdf+
a1+,_+'0(L+aú540i"*
40h"*6
W
û
0ü
W
-kS*6+¯%,7i+7#¡+,6+£*+¯(10243+,! 1¡
7#b)a
%'1_+,+7¦*6ûü;¡+,!*+e+,L)a((B%6+7À,+,
*+£+,6+
W a!%'SB_+,6+7¦*È"1"`È¡+,
*+gL)a£1+7*È"1±1"*i3
1
3254¸()1";d¸+g+'}(10+aø*+
61-/*)*6+'03 W (6)1"ø"27"Ë(B77U"10+Ë"
·+g-~*+gËB7i*r"!(6*1*¨*6+%1" S)+7"G7h+%61"ËB!+g*6+,+7"27
1
(177¢"1"?0"*i3f-¨((103S"G7¢"#"?02()1"¯*+%61" )+7Ê*ïû
ü *+#+,+,`0¨%,7i+!7¢"%'+!*+#À,+,°n7h)77i)!(*h"Sja11+7e*g02a
#¡+,_+, W
40h"*6
W
274
¡+!a1+,_+'0(+af.(6)1"
ü W **)*h"0?"À,+7+'024+a_+,7¢"-*+À,+,°n7i)77h)!(*i"*6
40?0`¡&()1"6)1"
"27
`00¡+a."_+,GÈ*i"B*L)ac£*+7i)
-~*+g(10243+,ñÖ7a.!a1+'0nñ\7-/)%'*i"7,dú+,c*+gS)a
8254r
((1%+7ÒÀ,+, W *+!)1*¨-~()1"£((BS%+7*6*-
[
performance(points/game)
Learning system
1.36
Opponent 1
Opponent 2
Technion - Computer Science Department - Technical Report CIS9609 - 1996
1.34
1.32
1.30
1.28
1.26
1.24
1.22
1.20
1.18
1.16
1.14
1.12
1.10
0.00
50.00
100.00
150.00
examples
200.00
¿Á")+[XZØÙSÚÛBÚ6Ü/Ý2Þ4ÜnßàláâÚrÞ,Ýqã¢ÙÚä?Úà,Ü¢áå?áægçnèçnã¢Úßà,ç;àÒÝ2éáâã¢å?Þá#Þ,Ý
ãnÙÚáéßgê1ÚÜÞlÝqÚëàlßÛä?Úç'ìqí#Ú'à,çnéSÜnÚî!êè
ßÚ'à,áÛ1Þå?áãnç¦Û1ÚÜ;æà,ßÚ4ì
m¢$*6+ï7i+%'Sa&(*-
*+.((+,#¡+*'% S0+af*6+(S10+,ð-0+1"$&((+,1*ñ\7!a1+'0B3
)7¢"c"*7!_+77+'!(10+7,cdf+a1+,_+'0(L+a5&40i"*ò-/g0+1"$$((L+,1*Ë}a1+'0 W
L*
a1+,(*óa&+,_`40)*h"S5-/)%'*i"5Ò+G40h"*6ô¡ 713¸"*6+,*i"_+'03&"%'+7¢"*+£Ë}a1+'0^a1+,(*6
ac0+61"&.-/)%'*i"$*6*g+7i*#(+aS"2%'*7*+Ë((+,1*ñ\7Ê!_+7-/**a1+,(*£¿S*+7i*h"c*+
40h"*6
W
¡+!*h"+aj*.0+f#"1"4&(1043+,'7g**7h+%6G*6G˱+a$a1+,(*&ab)7i+f+,_`40)*h"S
-/)%'*i"$S7i+abfË0?"+!%'#1"*i"$-;-/+*)+7 W
1¡f*£*6+#0++,.+!+7i)10*77i¡**
-/+,¡õ+'!(10+7g++,+a1+ac-/0+1"GË!a1+'0~*6*
6+,+7
40!17i*
(L+,i-/+%'*i03¯¡"*f7i)%6GË(1023S+,m¢
*+-k)*6)+ W ¡¨+e!+.*Ê"1_+7i*h"1*+e*+40i"*cñ\7;1"?0?"*i3*#!a1+'0!+e7i(1"27i*i"2%,*+a.(10243+,7ldf+
*+7i*6+a¯*6+È`0Si"*ö"&.B!+,°/(10243".0+1"b7i3}7h*+,
Ë+40?"27i*i"2%#7¢"*)*i"
W
!+a¸·
̦½#+!7i37i*+,÷¡7*6+7i*+aø"
W 0+1"."*7S((+,1*g¡1"?0+(10243S"¯B4"7i*"*4+7i37i*+,ö"a1+,+ab7i¡¨+ac
"!(_+,!+,1*
7Ò!+e+'Z!(10+7-((L+,B*4ñ\7!_+7L+%,!+4_``"0210+
Ëùi
"277i)+r-/-/)*+,¦+7i+%Ê"27¦*+*a1+,Ã!L+,*U¡+,+,#*+r(1043S"1"?0?"*U3B4"+aB3!a1+'00?"
_+,7h)7*+1"?0?"*U3.0B7h*B3Ë*+g+a1)%'+a£(6)1"L¨]c+g¡6 +,+aø*#+#a1+e"¯a1+,*!7i¡ú)a1+,
¡*e%'7¢"2aS"*i"7e*1"27*Sa1+,ï"27+,+'±q%6"240n
d¸+L+'0?"+,_S+£**g*1"27g¡6 £(+7i+,1*77¢"1"?±q%,1*(6+77"f*6++.-gx@,<DJBH#((L+,1*g!}a1+'027,
+ûüÁ-ªÈ"03-q40i"*.7(+7h+,B*+aÊ"#*1"27¦((+,;+¨40?0+,+,40?"À*i"7¦-#"1".4!**40?0¡)7
*g)7i+°D0+,_S+'0q((L+,B*r!a1+'07*+,*6+,¡"*#)a7r#*+'";+,67,¦1"27¡ #4027ig(+7i+,1*7~"1"*i"240
7i*+,(7^"*++ø-=?KEz'J<DJBHÈ((+,1*#!a1+'07lG+Ë*7i b-0+1"f°D0+,_+'0!a1+'027g"27+'*+,!+'03
aS"?ýï%')10*a£a1+7i+,_S+7r-k)6*+,+7i+%
þ
ÿ
#
q
df+r¡¨S)10a^0?" S+¨*6* og4_S"aÌ
+,Àag½;!+'0?02-/
0+,**i"g)7Á)7i+;*+'"+'ýï%6"+,1*~%6+%6 +,(10243S"
%'a1+7g.7¢"27^-kgS)7i3}7h*+,¯df+!¡)102af4027i.0?" +.**6 *+e+i03£7i*+7r-*1"27¡6 q¿Á"40?03 W
Á+%0S3 W
i"+pr+,°n(4"
-/g+'0(1"f)7"
1"_S+,7¢"*i3-
¡+e* ]&\¶g
\]&pr+, +'02-/Ó¦"a1_+,
+#¼+,*6+,i02a7-/e S"!1"27e+'}*6+,!+'03£+'ýï%6"+,B*0(
()10?"2%
7h0_+,
(6÷_4"?0210+*6!*+
;4r
l^pe
7i5~}(L+%'*+aS)*%'!+
Technion - Computer Science Department - Technical Report CIS9609 - 1996
8Ey/y/Kz'J
JLE=?@,<@
EJ
s
S+,+,40
!a1+'0
-7h**i"2%.+,_``0)*i";rziEJZ@l>SJ
t
FGESC,:B<DJKJLy/K=D=<?HK4JC'KO P MnO PT O W
4Y
\[^pr+,i0?"+,b+'%$af 1¡0+a1+!m¢58zU>4CK'K t <2J1H£>i¤yD:1K.Jy/K4zJLEy/<2>SJE=¦¥><DJygv;>SJ,¤,KziKJLCK!>SJ
z'y/< §¨C'E=
JLy/K=D=<?HK4JC'K©ª¥
v
s
s
VSV¢«W (+71¬X4|}1¬` W 1¬S¬B
^qpr+,i0?"+,g+
®*6+,+7i+'%¯40Si"*¯
O P M
P T N³ W
+7h*°D±7i*e(1-~(6}%'+a1)+
µ´S^¶g·oe)a£a¸e*4.8¨Ey/y/K4zJ¹v;=?E@'@º< §¨C'Ey/<D>JbEJ tc»
7 W
s
z'y/< §CE=²Jy/K4=KiHK4JC'K
1¬`Z
C'KJLK
s
JLE=?@,<@º¼+,¡¾½S qed¹"?0+,3$a
1¬ \X^¿µ°n
7h)
W
g
B*'
abÂÁB%6+'Ã+, W +aS"*7 W
W ]&L¨!(L+'00 W
a
¼¡*À,31
oe+,+,(£*)SB*Ámnbg]G7¢02a
v;>SwA
xBy/K4z@'Mgv:BK,@'@EJ
v;>HSJ<Dy/<2>J
t
W (+7eXX4|¬`{Z
(i"+,g¼+,¡½6 W
SY
\Ä^¦Â7i+,610+,*i"2%r(B7U"*h"S7a7i(L+%')102*i"_+(102431mn!g]b7U0aa#ÂSB%+'ò+, W +aS"*'7 W
v;>SwA
xBy/K4z@'M
v:1K,@'@^EJ
t
vr>HJL<2y/<D>J
?¬4^o#Æe1)*Ça$¶g]c1+
P T P1RW
W (+7gÄ`|q{[Z(i"+,g¼+,¡Å½S Ç403}7¢"27-^`0(°/+,*c()1"
Y
z'y/< §¨C'E=¦Jy/K4=2=?<?HK4JC'K
s
R MgJ>1\N1M
1¬`XZ
\{^¶gÆeSi-heoe+,(*°D±7i*"*+,*h"_S+Èa1+,+,(L+,1"
HSKJC'K
W
PV M¨ º
V T O³B W
b(*i"`0~a1#"277¢"10+.*+,+7h+%6
s
z'y/< §CES=
Jy/K4=?K T
S{X
\^¶g¦
Æei-!Ée+,+,40?"À,+af1!+*6+,+7,mn8zU>4CK'K t <2J1H.>i¤!yD:BK.Jy/K4zJLEy/<D>JE=²¥}><DJyÊv;>J,¤,KziK4JC'K>SJ
z'y/< §¨C'E=
JLy/K=D=<?HK4JC'K©ª¥
v
s
S «W (+7
s
[S{4| ZW oe+,*"* W ]cm W
Y^¶g
Æeh-h;]c)10*i3B°/(10243+,
`0(°/+,*Ë()1"
)L{Z
zy/< §¨C'E=LJLy/K=?KUHSKJLCKN`M¨ T OSOO W
s
l^o#̲+,_B3¯a£]&¼+,¡LÍg>ÎÅvr>wAqxBy/K4z@8¨=?E£v:BK,@'@ºÁd\L¿+,+,
W
Z
4
[^¶g
o#Ì
)%'+a¯¶4"?ÃeÏEw#K,@#EJ
tËÐ
f40i"*È"%Ë7i0)*i"f-;°/(L+,7i¸B!+7,em¢8¨zi>C'K'K t <DJBH
Ì
)% 1a1*#abÆ#pem¢1"n
>i¤yD:1K#Ñ^<DJyD:ËÑ^Ey/<D>JLE=¦v;>J,¤,KziK4JC'Kg>J
s
K'C<@,<2>JZ@º¼+,¡Å½
¦d¹"0+,3bac7 W
zy/< §¨C'E=LJLy/K=D=<?HK4JC'K© ses
s
4X1¬B
T R6«W (+7g4X{4|qÄS[ W
)S)7i*
S{Ä
,´S^o#
]b%
0?0+7i*+,Ò;7i(1"%'3.1)L+,7r-/#"°/4G7i+%
s
z'y/< §¨C'E=JLy/K=?KiHKJLCKe Q M
P V'T BO³ W
S{{
X^o#¦Â
Ée#3B*'7¢"+,¡"2%'Àa£o#Æ#df+,+
a1+%6"27¢"¯*6+,+,*i"2%g((1%£*6È%'1aS"*h"
#)10*i"2+,1*£"1*+,%'*i"7lÓmn8zU>4CK'K t <2J1H@£>i¤fyD:BK¸Jy/K4zJLEy/<2>SJE=¥}>S<2JLy£v;>Jl¤ºK4zUK4JC'K¯>J
JLy/K=D=?<HSKJC'K!©ª¥qv
Ä
O «W (S+7Ä[4|}ÄS{ W
z'y/< §CES=
S
B#)+'0nc!+£7i*)aS"+7Ê"$S%1"+Ë0+1"G)7U"b*+ËB!+!-Ò%6+% +,'7,cÔFÕ¥>x1zJLE=ÖM^M
P OSO TUPP W
¬4
s
s
XZ
B#)+'0n£S!+!7i*)aS"+7Ê"&%61"+#0+1"b)7¢"¯*+ËB!+#-
%6+% +,'7""?|B6+%'+,B*#(S+77,
ÔF×¥>xBz'JES=\M¨OSO4M
R ³1O ThR O V,W
{^Â1B%6+'Ã+, W Â1r)10L+,7i
%,40?"+,%6+%6 +,7(S¯
4Ä1¬B
W ¼Á+'0B W peÆe1"SB* W ¦`Ì
)
s
z'y/< §¨C'E=LJy/K4=KiHK4JC'K
´
Q M
W a#o#À`-kS
PV TUP S W
[Z
¡i02a%6!(1"7i1"(
!#"$&%'!()*+,.-/!(10243"5%6+77,98;:1<2=?>@,>'A:B<DC'E=FGE4HEI,<2JLK,MNO4M
PBQSR4TUPV4QW
XYZ
Technion - Computer Science Department - Technical Report CIS9609 - 1996
\[Y^]&L_`ba1+,^]c+,)10+,edf+'"B*g77h+77i!+,1*"b+,_`40)*i"j-k)%'*i"7lmnbo#qpr+40 W
<DJ&v;>wAqx1y/Kz!v:BK,@'@
QW (+7{Z'|}{~07h+,_"+,B%6"+,%'+!)10"27i+,'7 W
X
.7i*+,a
+aS"* Wsetu EJC'K,@
W
4{
© Copyright 2026 Paperzz