Real Time System Iit Notes

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Version 2 CSE, IIT Kharagpur

Timing Constraints Performance Constraints

Delay

Deadline

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Behaviorial Constraints

Duration Delay

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Deadline

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Duration

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FH>0 G€[“ @‰232$)-RG1>0#(H!%@)(W1*9".#@Q1G#(U(*>0_]7@ O@€O+?E %* Ž0EKŠ4(U(W@:"%(G1*>0V>)".@?30(W!1I(G$#O-B1*3S0"1*#!U#(U(*1U1*=!^2/0O ".-B1*!†, * (W57".@)31>0J(WC2(W1* 1"%@)(WO1(1 *> Š4]U"%<1BVI".$)F%@0$K (D19"_1%Eqr-M1>0432I".$01*%@)J"./0/M"%(KA?(W1S%1*>0@1*>04(*C2(D1! 1*9".@)(*<19( €#+%&0

*

Version 2 CSE, IIT Kharagpur

Await Second Digit first digit

dial tone / set timer (30 s)

Await First Digit

Idle

*-*£(,@L232!$Z.-[".@

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timer alarm / idle tone

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Await First Digit dial tone

handset lift / set timer (2 s)

* 2(

timer alarm / beeping

Await Dial Tone

€O+&0* %Ž ,@L232!$Z.-[".@L“ l

Await Receiver On−hook

FGC/M\%- V"32$#O@0@)(W1*9".#@1

 $ Š5 $­  Š5 "8Q < qX@ “5.E – E E Ž0S=]‡>?"%3w@)(*#32*53 1*>0y-R%$#$O_]7#@0+v!^2"%/0$Oy.-:".@ l4l 1DC/?‡%-:@2N (W1*9".#@Q1(  ‘=œ;›’ ›–=‘ ’2™;‘ ¬¡ =R¥2}‘ ’2™ ’- ¯œ;—£¦1¦ §’ ¯œ;™-––2 ¡R• ™}‘2¤7$‘ –=$‘¢Ÿ —2œ ¢’2™;‘ '” ˜7¡v’¢Ÿ  7R¥2;‘ ’2™¢’ ¯œ;—£¦1¦--–›­R’=$‘ ’ ™¬¡; œ;™}‘2¤8¡<§ ™;’ -–=¡; ’2¬œ}—£¦1¦¯¡ ’2-–}”<‘ —-’2¤

FH>0 G€[“ @ 232$Z-RJ1*>)#(,@)(W1*9".#@1V#(\(W>)_]7@LO@n€#+)E*  E4qX@c€O+?E*  Sk"%(4(*%@i"(J1>0=O@)+ 1*@0=I(4/)*232&)!3'SM1>0 (*C2(D1!‰1"%@)(WO1(V1f"h]H".O1,*#@0+.Nr6)"98 1%@0 (W1".1*".@)3d" 1*##((*1,1!^2/0OO@ f * “(!Eqr-G1>0*#@0+.Nr6)"98 1%@0 "%/0/?5".9( A)9(W1S1*>)`(WC2(W1*! 1*9".@)(*<19(1* *> Š4]U"%<1J€O9(W1 V#+%O1w K (D19"_1%S2!$I(*\O1J!@Q1!9(`> ŠJ]H".O1 l !! P !‚V± @2Nr>0%8£K (D19"_1%S0".@?3;1>0!".$#$'I( 1!#@)".1*3E

1& `5 ! / 2# =! ( £š’2-–“— ¤7 7R’W¡›¤8}—£¦¤ §Á’ ›‘  1’“¤7 =R’¬¡R ™}˜8¦-¤Ÿ  ¤7}—£¦-¤‘ ™¬¡;™1™;‘ ;–¢’ —;‘ R°4”<;¦1¦2›¡; œ;™;‘£¤8¡ s*;, *-, > £K 5  / £!$#& 5 <  `5 )! / 2*# =! qX@"%(*G.-M"432&09"_1O@)".$@)(W1*9".#@15S_".@` P !@Q1K#(aQ&0#*53\1*\2!&)Z-RK"V(W/M!0 !^2"%/0$O%-K" 30&0".1*#%@)"%$@)(D1"%O@Q17]U>)"%3f@)(*#32*53:#@n“5.E – E 2E ŽI(H1*>)-R%$#$O_]7#@0+8(  š“©2™}˜“•1–2 ¡1¡'’2¢Ÿ1˜1’1’2™;‘ª™š ’  —;‘£¤8¡;-’ š£™;–¬¦- ¡1¡4’ —; ‘ ¬¡; œ}™;‘2¤8¡ R’žœ;™}‘1‘  œv’8¡'’2™›’ ª¦™ œ;—£¦›™}• -–2—-’£™-–  š“©2™}˜“•1–2 ¡1¡'’2¢Ÿ1˜1’1’2™;‘ š£™-–ª—;‘£©ª¤-˜£–£—-’=}™;‘“¦— ¡v’=‘ ›Ÿ -’21; ‘ ¢’2™“®1°¬¡; œ;™;‘£¤8¡<§nR’Wœ;™;‘‘  œv’7¡ ’2™ ’- ¬$‘£’2-–-‘ —-’=}™}‘ —£¦'™;• -–2—;’2™-–  š ©2™;˜ 21}•ª’- ¢Ÿ˜£’1’2™;‘¢•1–2 ¡1¡;-¤›š£™-–”=™-–2›’2—;‘¯®1°¯¡; œ;™;‘£¤8¡<§’ }‘¬™}‘ª–££¦1— ¡1$‘ ¬R’  ™;˜ ¦-¤•£–2™¤;˜=œ;’  ¤7}—1¦ 5 )    Š 32!$I"hCn@)(W1*9".#@1`6M1D]U!@d1D]L P !@Q19(,#(=%@0 ]7>0*;".-B1*=".@e P !@Q1`2!!&0(S"f#@0#=&0 1*# &)(D1\!$I"./)(*`6M-R*1>0=%1*>04 P !@Q1V".@i2&05E7uc>)"%3n%@?(WI32!3 1>0`-R$O$#_]7O@)+;^0"./0$#`%-G32!$I"hC!%@)(W1*9".#@Q1JO@d“%E    

– E 2E Ž  FH>0#  G€[“ @ 232!$M-R%HO1UI(7(*>0_]7@;#@€#+)E 0EqX@€O+?E 2SO-Z1*>)V@0!^1730O+<17"./0/M"%(6M-R%V1*>0".$I".fSQ1*>0@;1*>)\6M!/0#@0+ (*%&0@?3:I(7/0232&)53 "%@)3 1*>0(*C2(D1! 1"%@)(*<19(U1* *"h]H".O1H".$#$O7%@2Nr>0%8 (D19"_1%E

*;,

Version 2 CSE, IIT Kharagpur

Await First Digit ring−back tone

ring tone / set timer (2 s)

*  (

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timer alarm / terminate call

Await Ring−back Tone

Await Receiver On−hook

@i30!$%-".@

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first digit / set timer (10 ms)

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next digit / beeping

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@i30!$%-K"%@f“0“FC/?,.-,V!$I"hC%@)(W1*9".#@Q1



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*

Version 2 CSE, IIT Kharagpur

Local Operator button release button press set alarm (15sec)

Await Event1 button release

Inter− national Operator

timer alarm

set alarm (15sec)

Await Event2

timer alarm

*  (Š

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Dial Tone

button release dial tone Await Button Release

@i30!$%-K" V&0".1*#%@)"%$%@)(W1*9".#@Q1

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

absolute deadline of Ti(1) = Φ+d relative deadline of Ti(1) =d Φ

Φ+ d

Ti(1)

Ti(2) Φ+ pi

0 arrival of Ti(1)

deadline of Ti(1)

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Version 2 CSE, IIT Kharagpur

T2 T1

T4

T3

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Version 2 CSE, IIT Kharagpur

ei

Φ =2000 0



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 Φ





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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

Φ+ x= pi

M

M

Ti(1) Ti(2)

Ti(k−1)

Φ

Ti(k) Ti(k+1)

Ti(2k−1)

Φ

x

x

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Frame Number

T3

F1

T1

F2

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F3

T4

F2

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Version 2 CSE, IIT Kharagpur

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Major Cycle

Minor cycle

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f2

f3

f4

f4n

f4n+1

f4n+2

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Version 2 CSE, IIT Kharagpur

Task arrival Deadline ∆t

t

0

kF

d

(k+1)F

(k+2)F

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∆t

Task arrival

t

Deadline

d

(k+1)F

0

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur



&=(*),on?)>6=,?i?)m&=CMDE),onj,5C-+-, V +-)ZTaI/6m)q:)>17S:&C-I?$xI5TK&(*) V ,?13GOi?6=I?S*$L9l&=,/.YG:.fC-$x)>n?)>6clS*$ CA&pI?TK&=CMDE)dCX. Ì  » F Qr 6I/D &(*CX.>[ ‹FqO4 6>6Q J:Q  TaI/+M+-Ij8.)>,?.C-+Mc/Q}x)^$*Ij8tC-+M+-S .Y&63,j&)&=(*)P,54*4*+-CX17, V CM+-CA&0c'I5Tz‹Fq:4*6>Q6J:Q  &=(*6=I?S*i/(p&( )^TaI?+-+MIj8PC-$*i&(*6=)7)P.CMDE4*+-)^)q*,5DE4*+-)>.>Q ]­ 2À”´ OB ]µ KI?$ .CX9:)76f,6=)>,5+M@B&C-De)Z.Yc:.Y&)7D C-$_8P(*CX13(l&=,/.YG:.f,56=)'.=13(*)>9:S +M)<9lS .CM$*ikTaI?6=)7i/6I/S*$ 9O@ V ,?13GOi/6I/S*$ 9k.=13(*)<9:S*+-CM$*iLQ Œ^(*)>6)^CX.2I?$*+-c'I/$*)^4L)>6C-I:9:CX1UTaI/6)>i?6=I?S*$L9m&=,/.YG'º ˜K3¼  . Ë [  . (5ËfDd.Y)<159[ & . Ì Ë/ËbDd.)>15([  . Ì Ë/ËbDd.)>1>¾2,?$ 9'&(*) V ,/13GOi?6=I?S*$ 9 &3,?.G V )mº . ¼ & . Ì Ë?Ë/2Ë 1 &657¾Q KI/DE4*S:&)H&=(*)p1I/DE4*+M)7&C-I?$&C-DE)HTaI?6 V ,/13GNi/6I/S*$ 9&=,?.GQ QJÊLÀhÇJ²<Á”Êz±Uµ cS .CM$*iE&=(*)m)q:4*6=)>.=.CMI/${K¼ J*Q  ¾U&I1I/DE4*S:&)f&=(*)H&=,/.YGk17I?DE4*+-)&=CMI/$&=CMDE)/[*8K)H( ,on/) 5  . Ì Ë/Ë?(5Ë Ë 1 &65 . J5Ë/Ë?2Ë 1 &65 Ì  Ì Ë?Ë  –OIL[O&(*) V ,?13GOi?6=I?S*$L9d&=,/.YGkº v8I?S*+X9k&=,5G/) J5Ë?Ë/Ë'DEC-+M+-C–O)<1I/$ 9*.&Id17I?DE4*+-)&)/Q ]­ 2À”´ OB Jµ #%$d,H.CMDE4*+-)P4*6=CMI/6CM&0cN@™9:6=C-n?)7$'4*6)>)7DE4:&=CMn/)^.=13(*)<9:S*+-)76<[j&08KIf4z)76=CMI:9:CX1&=,/.YG:.2Gº FK,5$ 9ZOº HP,5$ 9e, V ,?13GOi?6=I?S $ 9m&3,?.G ,?6)'.=13(*)<9:S*+-)>9\QfŒ^(*)'4z)76=CMI:9:CX1m&3,?.GGº Fm( ,?.&(*)e(*C-i?(*)<.0&f4*6C-I?6=CM&0c,5$L9l)q:)>17S:&)<.bI/$ 1)')7n?)>6c J5ËDeC-+-+MC2.Y)<1I/$ 9*.f,5$ 9W6=)>`NS*C-6)<. Ì Ë'DEC-+M+-CJ.)>1I/$ 9*.^I?T)q:)<1S:&=CMI/$k&=CMDE)m)>,/13(&=CMDE)?QOº Hf6=)>`NS*C-6)<. J?ËeDeC-+-+MCJ.)>17I?$ 9 .KI?T4*6=I:1)<..CM$ iZ)>n?)76=$c (5ËZDECM+-+-CJ.Y)<1I/$ 9*.>Qº N CX. , V ,?13GOi?6=I?S $ 9Z&=,?.Gd,5$ 96)<`/S CM6=)>. Ì Ë?Ë?DECM+-+MCz.Y)<1I/$ 9*.U&IZ17I?DE4*+-)&=)?Q|b.=.YS DeC-$*i'&( ,5&^,?+M+L&(*)b&=,/.YG:..0&3,56&,j&K&C-DE)fË [N9:)7&)>6DEC-$*) &=(*)H&C-DE)p,j&8P(*CX13(º N 8PC-+M+]17I?DE4*+-)&=)?Q QJÊLÀhÇJ²<Á”Êz±Uµ Œ^(*)H&I?&=,?+\S:&C-+MC-•>,5&C-I?$l9:S*)H&=Ie&(*)HTaI/6)>i?6=I?S*$ 9&=,/.YG:.>˜ H» F . H F

 H

.F  Q Œ^(*CX.PC-DE4*+MC-)>.&=( ,j&P&=(*)HTa63,?1 &=CMI/$I?T&=CMDE)m6=)7Dd,5C-$*C-$*ieTaI?6^&=(*) V ,?13GOi/6I/S*$ 9E&3,?.Gd&Id)7qO)<1S:&=)HC-.Pi/CMn/)7$ V c Ì  H» TF . F F

Œ^(*)>6)7TaI?6=)?[&=(*) V ,?13GOi?6=I?S $ 9W&=,/.YGyi?)&3. Ì DEC-+M+-CF.)>1I/$ 9{)7n?)>6c Ì ËkDEC-+M+-CF.)>1I/$ 9*.>QZŒ^(OS .>[\&( ) V ,?13GOi?6=I?S $ 9W&=,/.YGl8I?S +-9y&=,?G?)  Ì Ë?Ë *¼ Ì Ë  Ì ¾ . Ì Ë?Ë?ËfDEC-+M+-CJ.)>1I/$ 9*.&=IE17I?DE4*+-)&=)?Q ]­ 2À”´  B]µ –OS*4*4zI/.)ZC-$u‹q*,5DE4*+-) J*Q  [],5$yIjn/)76=(*)>,/9WI5T Ì Dd.)>1ZI?${,/1717I?S*$N&HI5TU)>n?)76=cl1I/$/&=)qO&m.Y8PCM&=13({C-.f&=I V )e&=,5G/)7$ C-$N&Id,?1>1I?S $/&6c{; 63.0&e&C-DE)k&=(*)kTaI/6)>i?6=I?S*$L9x&=,?.Gu6S $ .¼h,j&e&C-DE)ËN¾ [FCM&eC-$ 17S*6=.e,y17I?$N&)7qO&e.8PCA&313(*CM$ iyIjn?)>6( )>,?9uI?T Ì Dd.)>15QEŒ^(*CX.H( ,?. V )7)>$x.(*Ij8P$u,?.H,W.Y(L,?9:)<9y6=)>1&=,5$ i?+-)ZC-$xrC-i Q  Qd–OS V .)>`NS*)>$N&+-c_)>,/13(y&=CMDE)e&(*)ETaI/6)>i?6=I?S*$L9W&=,/.YG_6=S*$ .>[]CM& 4*6=)7)>DE4:&=.2&=(*) V ,/13GOi?6=I?S*$ 9'&=,/.YGZ,5$ 9EC-$ 1S 6=.I/$*)1I/$N&)qO&U.8PCA&313(]QFR$E17I?DE4*+-)&=CMI/$eI?T\)>,?13(eC-$ .0&3,5$ 17)I5Tz&( )PTaI?6=)7i/6I/S*$ 9'&=,?.G[ &=(*) V ,?13GOi/6I/S*$ 9l&3,?.G_6=S*$ .p,?$ 9yC-$ 17S*6=.',5$*I?&(*)>6p1I?$N&=)qO&p.Y8PCM&=13(]Qk}~CA&=({&=(*CX.mI V .)76=nj,j&=CMI/$][\&I_.C-De4 +MCMTacyI/S*6p1I/De4 S:&=,5&C-I?$ I?T&(*)E,?1&S ,?+17I?DE4*+-)&C-I?$y&C-DE)ZI?TFº b[8K)e17,5$yCMDd,5i/CM$ )p&(L,j&f&(*)e)q:)>17S:&C-I?$_&C-De)eI5TK)7n?)>6cTaI?6=)7i/6I/S*$ 9&3,?.GCX.CM$ 176)<,?.)>9 V c_&08Il1I/$/&=)qO&'.8PCA&313(y&=CMDE)>.¼aI/$*)9:S*)E&=IlCA&3.Y)>+ATP,5$ 9{&=(*)I5&=(*)76p9*S*)E&IW&=(*) V ,?13GOi/6I/S*$ 9y&=,?.G_6=S*$*$*C-$*i_,5Tƒ&)76m)<,?13(x&C-DE) CM&E17I?DE4*+-)&=)>.3¾ QsŒ^(OS .7[F&(*)$*)&E)Ž)>1&eI?Tb1I/$/&=)qO&d.Y8PCM&=13(*)<.Z1>,5$ V )CMDd,5i/CM$ )>9u&=I V )1>,5S .C-$*i_&=(*))7q:)>1S*&C-I?$s&=CMDE)I5T&=(*) TaI/6)>i?6=I?S*$ 9l&=,?.GW&IWC-$ 16=)>,/.Y) V c Jk17I?$N&)7qO&m.8PCM&=13({&C-De)<.7[JChQ )?Q'&I ( JkDECM+-+MCU–:)>1ZTa6I/D (5ËkDECM+-+MC–O)>1>.7QeŒ^(*C-.H(L,?.f4*CX1 &I/6CX,5+-+-c V )7)>$W.(*Ij8P$CM$lrC-i Q  Q  



 



















 







 









Context Switching Time



  Foreground  Back ground   01

51 52



   Foreground  Back  Foreground ground   

100

rCMi/S*6)  ˜FŒ,/.YGk–:13( )>9:S*+-)HTaI?6‹Fq*,5DE4*+-)0J*Q 

time in milli Secs

ÌI

Version 2 CSE, IIT Kharagpur

bIj8«S .CM$ iE‹Fq:4*6>Q J*Q  [:8K)Hi?)7&^&(*)H&=CMDE)m6=)>`NS*C-6)<9 V cd&=(*) V ,?13GOi/6I/S*$ 9d&3,?.Gd&Id17I?DE4*+-)&)/˜ Ì Ë/Ë?Ë . J5Ë9*  # 1  &65   Ì  Ì ( Ë?J Ë #%$k&(*)fTaI?+-+-Ij8PCM$*iZ&08Ie–O)<1 &C-I?$L.7[:8))7q*,5DECM$ )b&08IZC-DE4LI/6Y&3,5$N&^)7n?)>$N&Y@%9:6C-n?)>$k.=13(*)>9:S +M)>6=.>˜‹ r«¼”‹U,?6+-C-)>.Y&9)>,?9*+MC-$*)mrCM63.0& ¾ ,?$ 9yg y|’¼hg,j&=) lI/$*I5&=I?$*CX1'|b+Mi/I?6=CA&=(*D¾ Qp‹ rwCX.&(*)eI?4:&=CMDd,?+9:cO$ ,?DEC-1Z4*6=CMI/6CM&0cW6)<,5+M@h&=CMDE)p&3,?.Gl.13(*)<9:S*+-CM$ i,5+-i?I/6CM&( D ,?$ 9g _| CX.&(*)mI/4:&C-Dd,5+].Y&=,j&=C-1H4 6C-I?6=CA&0cd6=)>,?+A@B&C-DE)f&=,?.Gk.=13(*)>9*S*+MC-$*id,5+-i?I/6CM&( DQ ¥m£ žªJ¤ ž¥  7¦Wž £ªJ¤  ·›k© Wž 4 7¦  #%$k‹K,56=+MC-)>.Y& )>,?9*+MC-$*)rC-6=.Y&f¼h‹ fr¾F.=13(*)>9:S +MC-$*i [O,j&)7n/)76=cZ.=13(*)>9:S +MC-$*ip4zI?C-$N&U&( )b&3,?.GZ( ,onOC-$*iH&( ).(*I?6&)<.0&K9*)>,?9*+MC-$*)bCX.U&=,5G/)7$ S*4ZTaI?6F.=13(*)<9:S*+-CM$*iLQŒ^(*) V ,/.YCX1K4*6C-$ 17CM4*+-)^I5TL&(*CX.,5+-i?I/6CM&(*D CX.2n?)>6cHC-$N&S*CM&C-n?),?$ 9'.CMDE4*+-)&IHS*$ 9*)763.0&3,5$ 9\Q2Œ^(*)P.=13(*)>9:S +-, V CM+-CM&0c &=)>.Y&fTaI/6m‹ fr~C-.m,?+-.I.C-De4 +M)/Qe| &=,/.YG_.)&mCX.H.13(*)<9:S*+X, V +-)eS*$ 9:)>6p‹ frK[\CMT,?$ 9{I?$*+-clCATCA&p.=,j&=C-.Y; )>.&=(*)d1I?$L9:CA&=CMI/$y&=( ,j&H&=(*) &=I5&=,?+*4*6=I:1)>.=.I?6S:&=CM+-CM•<,j&=CMI/$9:S*)P&=IH&(*)P&3,?.GZ.)&UCX.F+-)>.=.2&=( ,5$ Ì Qr I?6U,m.)&UI?T\4L)>6C-I:9:C-1^6=)>,?+A@B&C-DE)^&=,/.YG:. <º F67YOº H27   7º /[/‹ r .=13(*)>9*S*+-, V C-+MCM&0ck16=CM&)76=C-I?$W17,5$ V )m)q:4*6=)>.=.Y)<9k,?.>˜ &» » F L» . » F z» Ì    x3¼ J */¾ 8P(*)>6) »2C-.b,on?)763,5i/)bS:&=CM+-C-•>,j&=CMI/$_9:S*)H&=IE&(*)H&3,?.Gkº F,?$ 9 {CX.^&( )m&I?&=,5+\$OS*D V )>6I5T&=,/.YG:.^C-$W&(*)H&3,?.G.)&9:S +-, V +M)/Q ‹ r“( ,?. V )>)7$_4 6Ijn/)7$&I V )',?$_I?4*&C-DE,?+JS*$ CM4*6=I:1)<..I?6b.=13(*)<9:S*+-CM$*i,5+-i?I?6=CM&(*D J BQŒ^(*CX.De)<,5$ .&( ,5&CMTU,.)&bI?T&=,?.G:. CX.bS $ .13( )>9:S*+X, V +M)eS*$ 9:)>6f‹ r[z&( )7$y$ IkI5&=(*)76H.=13(*)>9:S +MC-$*i,?+Mi/I?6=CA&=(*D 17,?$lTa)>,/.YC V +-cW.=13(*)<9:S*+-)p&( C-.&=,/.YGl.)&9*S*+-, V C-+MCM&0cH&)<.0&2TaI?6‹ rs¼h‹q:4*69p&( ,5&&=(*)^4L)>6C-I:9'I?T )>,/13(p&=,/.YGmCX.&(*)P.=,5DE)^,?.CA&3.29:)>,/9:+MC-$*)/Q¶bIj8K)>n?)76<[ C-$Z4 6=,/1 &CX17,?+/4 6I V +M)>Dd.J&=(*)^4L)>6C-I:9'I?TL,b&=,?.GmDd,ocp,j&2&C-DE)>. V )P9:CAŽ)76=)7$N&Ta6I/D CA&3.9:)>,/9:+-CM$*)/Q#%$E.S 13(Z1>,?.)>.>[o&(*)P.=13(*)>9:S +-, V CM+-CM&0c &=)>.Y&$*)>)>9*.b&I V )e13( ,5$*i/)>9\QP#™T L»  N»Y[L&(*)>$y)<,?13(W&=,/.YG$*)>)>9*. &o»U,5DEI/S*$N&bI5TU17I?DE4*S:&=CM$*ik&C-DE)')7n/)76=cDEC-$J¼ z» 7/»™¾9*S*6=,5&C-I?$ I?T&=CMDE)?QFŒ^( )76=)TaI?6=)?[O8)p17,?$6=)78P6=CA&=)m‹q:4*6˜ &<» » F 1  F¼; » 7  » ¾ Ì   uK¼ J  ¾ ¶bIj8K)>n?)76<[CAT L» N»[\CA&mCX.H4LIN..C V +M)Z&( ,5&p,.)&mI5TU&=,?.G:.fC-.H‹ fr .=13(*)>9*S*+-, V +-)?[\)7n/)7${8P(*)7${&(*)Z&=,/.YG_.)&HT”,?CM+X.&IWDE)7)&H&=(*) ‹FqO4 6 J*Q  QfŒ^(*)>6)7TaI?6=)?[‹q:4*06 J:Q  CX.H1I?$L.Y)>6nj,j&=CMn/)p8P(*)>$][ L» /»^,?$ 9lCX.$ I5&f,$*)<1)>.=.=,56=c17I?$ 9:CM&C-I?$J[ V S:&fI/$*+-c_,k.S 1C-)7$N& 17I?$ 9:CM&C-I?$TaI?6,Ei/CMn/)7$k&=,?.Gk.)&P&=I V )p‹ rt.=13(*)<9:S*+X, V +-)?Q ]­ 2À”´  B Jµ KI?$ .CX9:)76P&=(*)mTaI/+M+-Ij8PCM$ iE&(*6=)7)m4z)76=CMI:9:CX1m6=)>,?+A@B&C-DE)H&=,?.G:.^&I V )'.=13(*)>9*S*+M)<9WS .CM$*i‹ r“I?$_,ES $*CM4 6I:1)<..I?6<˜ º F. ¼ & F . Ì Ë [  F . J5ËN¾ [Oº H . ¼ & H6. (:[  H . (?Ë/¾[:Iº N . ¼ &N . Ì Ë*[ EN . ,(?¾ Q )&=)76=DeC-$*)m8P(*)7&(*)>6P&( )H&=,?.Gk.)&PCX..13( )>9:S*+X, V +M)/Q QJÊLÀhÇJ²<Á”Êz±Uµ Œ^(*)H&=I5&3,5+\S:&=CM+-CM•<,j&=CMI/$W9*S*)f&IE&( )H&(*6=)7)f&=,/.YG:.^CX.Pi?C-n?)>$ V c˜ N» F . H F

    NF  . Ë *  Œ^(*CX.PCX.^+M)<..&=( ,5$ Ì Q2Œ^( )76=)TaI?6=)?[O&=(*)H&=,/.YG.Y)7&^CX.‹ r“.13(*)<9:S*+X, V +-)?Q  | nj,56=C-,?$N&ZI5TH‹ fr .=13(*)>9:S +MC-$*ixC-. Áh±KÁ {TÇ U­ ]Á”² > Á”Æj®7² > –:13(*)<9:S*+-CM$*iLQ#%$ 1 Jr[F,5&e)>n?)76=ct.=13(*)>9:S +MC-$*i 4zI?C-$N&>[L,E+X,jq:CM&0cknj,5+-S*)mCX.b17I?DE4*S:&=)>9TaI?6)>n?)>6cd&3,?.GkC-$W&(*)Z.Yc:.Y&)7DW[ ,?$ 9&( )m&=,/.YG(L,onNC-$*iE&=(*)pDECM$ CMDZS*D’+X,jq:CM&0ckCX.P)q:)<1S:&=)>9 ; 63.Y&>Q J,jq:CM&0clI5T^,k&3,?.GlDE)>,?.S*6=)>.&( )E,?DEI?S*$N&mI5TK&C-De)e&=( ,j&p8I?S*+X9_6=)7Dd,?CM$xCATK&(*)e&3,?.G_C-.f&3,5G/)7$yS 4{TaI/6H)q:)>17S:&C-I?$x$*)qO&,/.YS 6)eI5TF&=(*) L )7qOC V C-+MCM&0c_,onj,5C-+-, V +-)'TaI/6H.=13(*)<9:S*+-CM$*iW,&=,/.YGQZŒ^( )eDd,?CM$x9:CAŽ)76=)7$L1) V )7&08K)>)7$ 1 Jr ,?$ 9‹ rC-.P&( ,5&S*$*+-CMG/)H‹ r[  Jrs&3,5G/)>.CM$N&=I1I?$L.YCX9:)763,j&=CMI/$&=(*)m)q:)>17S:&C-I?$&C-DE)mI5T,Z&=,?.GQ 





 



















































 





 

Ì*

Version 2 CSE, IIT Kharagpur

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[/8P(*)>6)  » CX.&=(*)b4z)76=C-IO9EI?T&(*)b&=,?.Geº » ,5$ 9EGECX.U,p17I?$ .Y&=,?$/&.=.YC-I?$J[N4 +MI?&=.I5T24*6=CMI/6CM&0cEno,?+MS )>.^I5T&=,/.YG:.^S*$ 9*)76g _|TaI/6^&=,?.G:.I5T9*CAŽ)76=)7$N&4z)76=CMI:9*.P1>,5$ V )m)>,/.YC-+-c I V &=,?CM$*)<9\QbŒ^(*)<.Y)Z4*+-I5&=.b( ,on?) V )>)7$y.(*Ij8P$lC-$yrCMiLQ Ì ËL¼”,N¾P,5$ 9lrC-i Q Ì Ë ¼ V ¾ Qb#™&f17,?$ V )'I V .Y)>6n/)>9Ta6I/D rC-i Q Ì Ë_¼h,/¾,?$ 9s¼ V ¾ &=( ,j&P&=(*)m4*6=CMI/6CM&0cdI5T,Z&=,?.GC-$ 16=)>,/.Y)<.+MC-$*)>,?6+-c8PCM&(&( )p,56=6C-nj,5+6=,5&)HI5T&(*)H&3,?.Gk,5$L9C-$On?)>6=.)7+-cd8PCA&=(WCA&3.P4L)>6C-I:9\Q QJ³ R´OÅÇÀh­:ÉÁhÀ”Á”² x¬K´O®>² 3BÊLÆ ¿ È_µ |b$dCMDE4zI?6&=,5$N&F4*6=I V +M)>D &(L,j&UCX.F,?9 9:6)<..)>9e9:S*6=C-$*iH&(*)9:)>.CMi/$EI5T],HS*$*C-4*6=IO17)>.=.YI/6Y@ V ,/.Y)<9 6=)>,?+A@B&C-DE).c:.0&=)7D CX.&IH13(*)>13Gm8P( )&( )76F,H.Y)7&I5T 4z)76=CMI:9:CX16)<,5+M@h&=CMDE)K&=,/.YG:.217,?$'Ta)<,?.C V +-c V ).13(*)<9:S*+-)>9'S*$ 9:)>6Fg _|pQN–:13(*)>9*S*+-,5@ V C-+-CA&0ceI5TJ,f&3,?.Ge.)&US $ 9:)76^g _|«17,5$ V )9*)&)>6DEC-$*)>9eTa6=I?D,mGO$*Ij8P+-)>9:i/)PI5T&( )b8I?63.0&@™1>,?.))7q:)>1S*&C-I?$E&C-DE)>.K,5$ 9E4z)76=CMI:9*.UI5T &=(*)b&3,?.G:.7Q2|«4z)76&C-$*)>$/&P`NS*)>.Y&C-I?$,j&K&(*CX.4LI/CM$N&^C-.K(*Ij8 17,?$k,e.Yc:.Y&)7D 9:)>n?)>+MI/4L)>6K9:)7&)76=DECM$ )b&=(*)8I?63.Y&Y@%17,?.)P)q:)<1S:&=CMI/$&C-DE) I?T\,m&=,?.G')>n?)>$ V )7TaI?6=)^&(*)f.Yc:.Y&)7D C-.K9:)7n/)7+-I?4z)>9\Q2Œ^(*)8KI/6=.Y&Y@%17,/.Y))q:)>17S:&C-I?$E&=CMDE)<.U,56=)PS .S ,5+-+-ce9:)7&)>6DEC-$*)>9E)7qO4z)76=C-De)>$N&=,5+-+-c I/6^&(*6=I?S*i/(.C-D'S*+X,j&=CMI/$.Y&S 9*CM)<.7Q Œ^(*)UTaI?+-+MIj8PC-$*ib,?6)K.YI/De)UCMDE4zI?6&=,5$N&216=CA&=)76=C-,^&=( ,j&217,?$ V )KS .)>9H&=I13(*)<13Gb&=(*).=13(*)<9:S*+X, V C-+MCM&0cfI5T ,.)&I5T:&=,/.YG:..Y)7&S*$L9:)76g _|pQ 















J5Ë

Version 2 CSE, IIT Kharagpur

Priority

Priority

Rate

Period

(a)

(b)

rCMi/S*6) Ì Ë*˜+F6=CMI/6CM&0c|b.=.CMi/$*DE)7$N&^&IŒ,?.GO.C-$lg _ |

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Ì 3¼ J   Ì ¾ 7  » » F » F » Ì –OC-DECM+X,56=+-c?[OTaI?6P$ . J'8)Hi?)7& H H » F » J K¼ J  Ì ¾ 7  » F » vË * J8# r I?6P$ . e8)fi/)& N N » TF » LK¼ J  Ì ¾ 7  » F » vË  * JÌ 







































2











































Version 2 CSE, IIT Kharagpur

}~(*)>$W$ 8)Hi?)&

  »

«¼KJ    Ì ¾ 7  » F » F z» .  Ë ‹Uno,?+MSL,j&C-I?$'I5TL‹FqO4 6>Q J*Q Ì Ë8P(*)>$'$ C-$On?I/+Mn/)>.J,?$pC-$ 9:)&=)76=DECM$ ,5&))q:4*6=)>.=.YC-I?$'I5T:&( )K&0cO4z) Ë*Q cm,54 4*+McOC-$*i ¶I/.4*CM&=,?+ . 6=S*+-)8)P17,?$'n?)>6CMTacm&=( ,j&2&( )P6C-i?(N&2( ,5$ 9e.C-9:)^I?T &( )P)q:4*6=)>.=.YC-I?$Z)7nj,5+-S ,j&=)>.&I J . Ë Q  J:Qr*6=I?DÄ&=(*), V Ijn?)^1I/De4 S:&=,5&C-I?$ .>[ CM&mC-.H1+-)>,?6f&( ,5&f&=(*)eDE,5q:CMDZS*D  S:&=CM+-CM•<,j&=CMI/$y&=( ,j&m17,?$ V )E,/13(*C-)7n?)<9_S*$L9:)76mg _| CX. Ì Q'Œ^(*CX.HCX.f,/13(*C-)7n?)<9_8P(*)>$_&=(*)76=) CX.mI?$ +Mcy,W.CM$ i?+-)Z&3,?.G_C-${&=(*).cO.Y&)>DWQ|b.H&=(*)d$OS*D V )76pI?TU&3,?.GO.HC-$ 176)<,?.)>.>[\&(*),/13(*C-)7nj, V +-)  S:&=CM+-CM•<,j&=CMI/${T”,?+M+X.m,5$L9x,/.  &(*)p,/13(*C-)7nj, V +-)mS:&=CM+-CM•<,j&=CMI/$W.Y&=, V C-+MC-•7)<.,j&+-I?i J:[*8P(*CX13(WC-.,54*4 6Ioq:C-DE,5&)>+McË Q  J:QFŒ^( C-.bC-.P4 C-1&I?6=CX,5+-+Mck.(*Ij8P$WC-$WrC-i Ì/Ì Q2}{)m$ Ij8«C-+M+-S .Y&63,j&)f&=(*)p,54*4 +MCX17, V CM+-CM&0cI5T&(*)pg _| .13( )>9:S*+X, V CM+-CA&0c16=CA&=)76=C-,'&(*6=I?S i?(W,'Ta)>8)7q*,5DE4*+-)>.>Q

























1 Σ ui 0.692

(1,0)

Number of tasks

r C-i?S*6=) Ì?Ì ˜|13(*CM)>nj, V +-) P&=CM+-CM•<,j&=CMI/$8PCM&(&=(*) S*D V )>6I5T2Œ,/.YG:.^S*$L9:)76g y| ]­ 2À”´  B ]µ K(*)>13G8P(*)&=(*)76b&(*)pTaI?+-+MIj8PC-$*i.)&I5T4z)76=CMI:9:CX1m6=)>,?+A@B&C-DE)H&=,?.G:.PCX..13( )>9:S*+X, V +M)mS*$L9:)76fg _| I/$_,ES*$*C-4*6I?@ 17)>.=.YI/6>˜º F . ¼ & F . J5Ë*[  F . Ì Ë?ËN¾ [:º H . ¼ & H6. ?Ë*[  H . Ì (5ËN¾ [OOº N . ¼ &N . ?Ë*[ EN . J5Ë?ËN¾ Q S:&=CM+-CM•<,j&=CMI/$W,/13(*C-)7n?)<9k9:S )H&IE&( )H&(*6=)7)Hi?C-n?)>$k&3,?.G:.7Q QJÊLÀhÇJ²<Á”Êz±Uµ ])7&S .; 63.0&b17I?DE4*S:&=)H&(*)H&=I5&=,?+  N J5Ë  /Ë  /Ë . Ë  » . Ë/Ë Ì (?Ë J?Ë?Ë Ì » F Œ^(*CX.C-.^+-)>.=.K&( ,?$ Ì [O&(*)>6)7TaI?6=)&( )f$ )>1)<..=,56=cd1I/$ 9:CM&C-I?$TaI/6P.13(*)<9:S*+X, V C-+-CA&0cdI?TJ&( )&3,?.G:.KCX.^.,5&CX.0;L)>9\Q Ij813(*)>13GOC-$*i'TaI/6 &=(*)m.YS 17CM)>$ 1ck17I?$ 9*CA&=CMI/$][O&(*)f&3,?.Gk.Y)7&CX.P.=13(*)>9:S +-, V +M)fS*$ 9*)76g _|ÄCM'T JCMSW,5$L9 ,ocN+X,5$L9 .1I/$ 9:CM&C-I?$i?C-n?)>$ V ck‹q:4*69\Q K(*)>13GOC-$*ifTaI?6F.=,j&=C-.YT”,?1&C-I?$eI5Tz‹Fq:4*6>6Q J:Q Ì Ë [j&(*)^Dd,jq:C-D'S*D ,?13(*C-)7nj, V +M)S:&=CM+-CM•<,j&=CMI/$eCX.2i?C-n?)>$ V c˜ LK¼ J   Ì ¾ . Ë  * Œ^(*N )Z&I?&=,?+S*&C-+MC-•>,5&C-I?$y(L,?.f,5+-6)<,?9:c V )>)7${TaI?S*$L9l&I V )eË Q  Q bIj8 .S V .Y&CM&S:&=CM$*i&( )>.)ZC-$_&=(*) ]C-S{,5$L9 J,ocO+X,5$ )9 .16=CA&=)76=CMI/$]˜ z»  3¼ J   Ì ¾[*8K)Hi/)&Ë  vË  **Q ‹FqO» 4 T6>F Q J:Q Ì Ëu8P(* CX13(wC-.,s.YS 17CM)>$/&1I?$L9:CA&=CMI/$wTaI?6kg _| .=13(*)>9:S +-, V CM+-CM&0cCX.k.=,j&CX.Y; )>9\QČ^(*)>6)7TaI?6=)?[K&(*)l&3,?.G“.)&kCX.kg _| .=13(*)>9*S*+-, V +-)?Q  ]­ 2À”´  B Jµ K(*)>13G_8P( )&( )76m&=(*)dTaI?+-+-Ij8PCM$*il.)&pI?TK&=(*6)>)d4L)>6C-I:9:CX1Z6=)>,?+A@B&C-DE)E&=,/.YG:.HCX.p.13( )>9:S*+X, V +M)ES*$ 9*)76'g y| I?$u, S*$ CM4*6=I:1)<..I?6<˜Gº F . ¼ &F . J?Ë*[ IF . Ì Ë/Ë/¾[NOº H . ¼ &H . /Ë*[ EH . Ì (5Ë/¾[Oº N . ¼ & N .  Ë*[  N . J?Ë?Ë/¾Q QJÊLÀhÇJ²<Á”Êz±Uµ J)&S .; 63.Y&b1I?DE4*S*&)f&(*)H&=I5&3,5+  S*&C-+MC-•>,5&C-I?$l9:S*)H&=IZ&=(*)mi?C-n?)>$k&3,?.Gk.)&>˜ N J5Ë ?Ë Ë z » . Ë?Ë  (?Ë  J5 Ë/Ë . Ë *,( Ì Ì » F bIj8w13( )>13GOCM$ iHTaI?6 JCMS,5$ 9 ,ocN+X,5$L9e176CM&)>6C-I?$J˜ N» F z» vË  *'UN.CM$L1)bË *,( w Ë  * [5&( )&=,/.YGE.)&UCX.F$*I?&g y|«.=13(*)>9:S +-, V +M)/Q 



































JJ

Version 2 CSE, IIT Kharagpur



JCMSW,5$ 9! ,ocN+X,5$L9d&)<.0&'¼h‹q:4*6$ .Y)/Q

1 +C))5:  5 4)  $ ( &!) 4%):0 > ) %;<:3$  = 6 T8&4)*)  <  %;<7.   1 %& > )#. > (?  ,$ $  4) 4%'3 7   +6))5:; 59+ )  > 9$ ( &,) 4% ),0 > ) %"< <:3E ;;),0 > >*.   14  %'&(> )#. > (?

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T1

T2 10

T1 30

T2 40

T1 60

T2 70

90

time in msec

(a) T1 is in phase with T2

T2

T1 20

T2 30

T1 50

T2 60

80

time in msec

(b) T1 has a 20 msec phase with respect to T2

rCMi/S*6=) Ì J:˜2}xI?63.0& ,?.)mgP)>.4zI?$ .)mŒ^CMDE)HTaI?6,EŒ,?.GRf1717S*63.K}~( )7$W#™&C-.CM$F( ,/.Y)H8PCM&(W#™&=.b¶C-i?(*)>6 F6=CMI/6CM&0cdŒ,/.YG:. | TaI/6Dd,5+4 6IOI5TI5T&(*CX.pŒ^(*)>I?6=)7D CX. V )7c?I/$ 9{&(*).=1I/4L)EI?TK&=(*C-. V IOI?GQ¶bIj8K)>n?)76<[\8)d4*6IjnOCX9:)d,5$xCM$N&=S*CA&=CMn/)d6)<,?.I?$*C-$*i ,/.&=I8P(OclŒ^(*)7I/6)>D J:Q D'SL.0& V )Z&6=S*)/QHr I?6H,dTaI/6Dd,5+4*6=IOI5TFI5TU&(*) ])>(*I:1•7GOc' z6)<.YS*+M&=.b&( )Z6=)>,/9:)76fC-.6=)Ta)76=6=)>9W&=I  ™Qf#%$*@ &=S*CA&=CMn/)7+-c?[j8)K1>,5$pS*$L9:)763.0&3,5$ 9m&(*CX.J6=)>.S*+M&JTa6=I?D &( )UTaI/+M+-Ij8PCM$ i6)<,?.I?$*C-$*iN.7Q]rC-6=.Y&+-)&2S .J&6=c&=IbS*$L9:)763.0&3,5$ 9m&(*)KTaI?+-+MIj8PC-$*iPT”,/1 &
  -  . 1 )0! !+  O)=)5:211&(-;  0 9 # " ) 2-# $ , <  8       0 ): ?

ŒIf.)7)8P(Ocm&( C-..Y&=,5&)7DE)>$/&2D'SL.0& V )K&=6S )?[?17I?$ .C-9*)76&(*)TaI/+M+-Ij8PCM$ i.Y&=,j&=)7DE)7$N&n?)>62,(*C-i?(*)>624*6C-I?6=CM&0c &3,?.GdC-.^6=)>,/9:c?[O&=(*)H+MIj8)764*6=CMI/6CM&0cE&=,/.YG:.^17,?$$*I?&P)q:)>17S:&)m,5$L9k( ,on/)b&=IE8,?CA&.&=( ,j&<[ ,Z+MIj8)76^4*6=CMI/6CM&0ce&=,/.YG8PC-+-+ ( ,on/)f&Id8^,5CM&PTaI?6^&( )p)7$N&C-6=)p9:S*63,j&C-I?$I5T2)q:)<1S:&=CMI/$I?T)<,?13((*C-i?(*)>64*6=CMI/6CM&0cd&=,/.YG&( ,5&,56=C-.)>.P9*S*6C-$*ie&( )p)q:)>17S:&C-I?$WI?T&=(*) +-Ij8K)>6b4*6=C-I?6=CA&0c&=,/.YGQ _I?6=)p$OS*D V )76fI5TUC-$ .Y&=,?$ 1)<.I5T,d(*C-i?(*)>64*6=C-I?6=CA&0c&=,/.YGW8PC-+M+I:171S 6>[z8P(*)7$x,d&3,?.GCX.fCM$_4 ( ,?.)'8PCM&({CA&<[ 

J2

Version 2 CSE, IIT Kharagpur

P8 (*)>$uCA&'CX.mC-$u4*( ,?.)E8PCA&=(sCA&Z6=,5&(*)>6H&( ,?$xI/S:&pI5T^4*( ,/.Y)E8PCM&(sCM&>QŒ^(*C-.p( ,?. V )7)>$sCM+-+MSL.0&=6=,5&)>9y&( 6I/S*i?(s,W.YC-DE4*+-))q*,5DE4*+-) C-$trCMiLQ Ì J:Q#%$trCMiLQ Ì J*¼h,/¾ [2,l(*C-i?( )76p4*6=C-I?6=CA&0cl&3,?.GxŒ Ì . ¼ Ì Ë*[ ?Ë/¾CX.pC-$s4*( ,/.Y)d8PCM&(,W+-Ij8)76p4*6=C-I?6=CA&0cl&3,?.GxŒJ . ¼ /Ë*[ Ì J5ËN¾ [ &=(*)d6)<.Y4zI?$L.Y)e&C-DE)dI5T^ŒJC-.  ËDd.)>1?Q¶Ij8)7n?)>6>[\C-$srCMiLQ Ì J*¼ V ¾ [J8P(*)7$sŒ Ì ( ,?.m, J?ËkDd.)>1E4*(L,?.)?[\&=(*)d6)<.Y4zI?$L.Y)e&C-DE)dI5T Œ J V )<1I/De)<. *?Ë*QŒ^(*)76=)TaI/6)/[jCAT,&3,?.GmDE)7)&3.2CA&3.; 63.0&9:)>,/9:+-CM$*)^S*$L9:)762•7)>6If4*( ,/.YC-$*iL[o&(*)>$'&=(*)7cpCM&8PCM+-+ODE)7)&F,?+M+:CM&=.29:)<,?9:+-CM$ )>.>Q 

deadline for T1 T1 20

100 (a) T1 meets its first deadline

deadline for T2 T1

T2 20

50

150 (b) T2 meets its first deadline

deadline for T3

T1

T3

T2 20

50

T1 100

T2 120

T3 150

190

(c) T3 meets its first deadline

r C-i?S*6=) Ì  ˜ K( )>13GOCM$ i J)7(*I:17•7GOc' . K6=CA&=)76=CMI/$kTaI/6Œ,/.YG:.^I5T2‹q*,5DE4*+-)J:Q  ]­ 2À”´1 B Jµ K(*)<13G8P(*)7&(*)>6^&(*)H&3,?.Gk.Y)7&I5T2‹q*,5DE4*+-)J:Q *ZCX.P,?1&S ,?+M+-ck.=13(*)>9*S*+-, V +-)fS $ 9:)76g _|pQ QJÊLÀhÇJ²<Á”Êz±Uµ Œ^(*I/S*i?(x&=(*)d6)<.YS +A&3.mI5T ]C-Ss,5$ 9 ,ocN+X,5$L9 .f&=)>.Y&'8)76=)d$*)7iN,j&C-n?)/[J,?.p4L)>6m&(*)k6)<.YS*+M&=.pI5TP‹q*,5DE4*+-) J:Q *8K)1>,5$ ,?4*4*+-cd&(*) J)7(*I:17•7GOcd&)<.0&b,?$ 9I V .)76=n?)&(*)HTaI/+M+-Ij8PC-$*i ˜ r I?6K&(*)&=,/.YGeºGFo+˜ &F OF^(*I/+-9*..CM$ 17=) J?2Ë 1 &65 Ì Ë?2Ë 1 &65jQŒ^( )76=)TaI?6=)?[/CA&^8I?S*+X9dDE)7)7&CM&=.U; 63.0&^9:)>,/9:+MC-$*)E¼aCM&^9*IN)<.U$ I5&K(L,on?) ,?$Nc&=,?.G:.&( ,5&P( ,on?)H(*C-i?(*)>6P4*6=CMI/6CM&0c*¾ Q r I?6P&=(*)m&=,/.YGº H ˜Uº F CX.PCM&=.( CMi/(*)764 6C-I?6=CA&0c&=,?.G,?$ 9l1I/$ .C-9:)>6C-$*idËE4*( ,/.YC-$*iL[*CA&8KI/S*+-9I:1717S*6I/$ 1) V )TaI/6)m&=(*)'9:)<,?9:+-C-$*)mI5T º H QFŒ^( )76=)TaI?6=)?[\¼ & F  & H ¾  H (*I/+-9 ..YC-$ 170) J?Ë  /Ë . (59Ë 1  &65 Ì (?2Ë 1 & 55QŒ^(*)>6)7TaI?6=)?[:º H DE)7)&3.PCM&=.^; 63.Y&P9*)>,?9*+MC-$*)?Q r I?6F&=(*)&3,?.GZOº N?˜^3¼ J2& F  J2& H  &N7¾ ENb(*I?+X9*.>[N.CM$L1=) J!6J?Ë  J!/Ë   Ë . Ì  2Ë 1 &65 J5Ë?9Ë 1  &65jQ}{)( ,on?)b17I?$ .CX9:)76=)>9 J%! &F2,5$ " 9 J ! &HF.CM$L1)FGº F2,?$ 9mOº HI:1717S*6]&08PCX1)F8PCM&( CM$'&(*)U; 6=.Y&9:)<,?9:+-C-$*)FI5T*º N Q2Œ^(*)>6)7TaI?6=)?[<º N De)>)&3.JCM&=.]; 6=.Y&9*)>,?9*+MC-$*)?Q2–OI [<&=(*) i/CMn/)7$d&3,?.GE.)&^CX..=13(*)<9:S*+X, V +-)bS*$L9:)76Pg _|'Q*Œ^(*)H.13(*)<9:S*+X, V C-+-CA&0ce&)<.0&TaI?6^º N (L,?.4*CX1 &I/6CX,5+-+-c V )7)>$.(*Ij8P$C-$rC-i Q Ì  Q–:CM$ 17) ,?+M+O&=(*)&=,?.G:.De)>)&&=(*)7C-62; 63.0&F9:)<,?9:+-CM$ )>.S*$ 9:)>6•7)>6If4*( ,/.YC-$*i [5&(*)>c',56=)g _|~.13( )>9:S*+X, V +M)^,?1>1I?639:C-$*i&I ])7( IO17•7GO'c .26=)>.S*+M&=.>Q 

















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J8#

Version 2 CSE, IIT Kharagpur

Ti(1)

T1(1)

T1(2)

T1(3)

r CMi/S*6) Ì # ˜2#%$ .Y&=,5$L1)>.^I?Tº F Rn?)>6P,E–:CM$*i/+M)m#%$L.0&3,5$ 17)fI?Tº » &3,?.Gkº » QbR V .)76=n?)f&(L,j&b&=(*)m&3,?.Gkº F ( ,/.^&(*)'(*CMi/(*)>.Y&b4*6=C-I?6=CA&0c,5$ 9&=,/.YGº » ( ,/.^&(*)'+M)<,?.Y&b4*6=CMI/6CM&0c?QŒ^(*C-.b4*6=CMI/6CM&0ckI/6=9:)>6C-$*i 1>,5$ V )H,?.=.YS*DE)<98PCM&(*I/S:&^,?$Oce+-I/.=.KI5TJi/)7$*)>6=,?+MC-•>,5&C-I?$k.CM$L1)&=(*)H6)<`NS*CM6=)>94*6C-I?6=CM&0ceI/6=9*)76=CM$*iZ,5DEI?$*ie,5$,56 V CA&=6=,?6cE1I/+M+-)>1&C-I?$ I?TK&3,?.GO.H1>,5$s,5+-8,oc:. V ),/13(*C-)7n?)<9 V cy,W.CMDE4*+-)d6=)7$ ,?DeC-$*iWI5TK&(*)E&3,?.G:.7Q KI?$L.YCX9:)76H&=( ,j&m&=(*)E&=,/.YGyº»P,?66=C-n?)>.f,j&m&=(*)E&C-DE) C-$ .Y&=,5$N&mË*Q KI/$ .YCX9:)>6&=(*)e)q*,5DE4*+-)e.(*Ij8P${CM$xrC-i Q Ì # 0Q S*6=CM$*i&( )Z;L6=.Y&fC-$ .Y&=,?$ 1)eI5TF&=(*)e&=,/.YGWº\»Y[&=(*6=)7)eCM$ .Y&=,?$ 1)<.fI5TF&=(*) &3,?.GºGF(L,on?)HI:171S 66=)>9\Q‹K,?13(&=CMDE)mGº FI:1717S*6=.>[Oº\»2( ,?.&=Id8,?CA&.CM$ 17)Hº F(L,?.^(*C-i?(*)>6P4*6=CMI/6CM&0cE&( ,?$Wº»YQ J)&bS .b$*Ij8 9:)&=)76=DECM$*)p&(*)Z)q*,?1&b$OS*D V )76I5T2&C-DE)>.P&=( ,j&fGº FfI:171S 6=.P8PCM&( CM$y,k.YC-$*i?+-)pC-$ .Y&=,5$L1)pI?T2º\»YQŒ^(*C-.CX.i?C-n?)>$ V c QU–OC-$ 17)Gº FI .K)q:)>17S:&C-I?$k&C-De)HCX. &Fj[O&(*)>$&( )b&=I5&=,?+)q:)>17S:&C-I?$k&C-De)f6=)>`NS*C-6)<99:S*)&IZ&=,/.YGEGº F V )TaI/6)&( )f9:)<,?9:+-C-$*)I?TJº\» CX.    &4FjQŒ^( C-.U)q:4*6=)>.=.YC-I?$d17,?$E)>,?.C-+Mc V )i?)7$ )763,5+-CM•>)>9'&I'17I?$ .C-9*)76&=(*))q:)<1S:&=CMI/$E&C-De)<.U,?+M+:&3,?.G:.( ,onOCM$ im(*CMi/(*)76U4*6C-I?6=CM&0c &=( ,5$lº\»P¼aCBQ )?Qº F87YOº H27   7Yº\» OF¾QŒ^( )76=)TaI?6=)?[*&(*)m&=CMDE)mTaI/68P(*CX13(Wº»F8PCM+-+J( ,on/)f&I8^,5CM&b9:S*)m&=I,5+-+]CM&=.( CMi/(*)764 6C-I?6=CA&0c&=,?.G:. 1>,5$ V )m)7qO4 6)<..)>9,?.>˜ »  F  » &

 
‹FqO4 6)<..CMI/$ J*Q Ì/Ì i/CMn/)>.&=(*)^&I?&=,5+*&C-De)b6)<`/S CM6=)>9'&Im)7qO)<1S:&=)Pº\7» .(*C-i?(*)>64*6=CMI/6CM&0cm&=,/.YG:.TaI/6F8P(*CX13(eº\»\8I?S +-9Z( ,on?)&=Im8,?CA&
  3¼ J Ì J/¾    F

Œ^( ,5&CX.>[:CAT&(*)m.S*D I?TJ&( )f)7q:)>1S*&C-I?$&C-De)<.^I5T2,5+-+z( CMi/(*)76P4*6=C-I?6=CA&0ce&=,/.YG:.I:1717S*66=C-$*i V )TaI/6)fº\7» .K; 63.0&9:)<,?9:+-CM$ )?[*,5$L9&=(*) )7qO)<1S:&=CMI/$u&C-DE)I5T^&( )&=,/.YG{CA&3.Y)>+ATCX.p+-)>.=.H&=( ,5$sCM&=.Z4L)>6C-I:9 L»Y[&=(*)7$sº\»b8KI/S*+-9u17I?DE4*+-)&) V )7TaI?6=)CA&3.m; 63.Y&Z9:)>,/9:+-CM$*)/ Q I5&=) &=( ,j&C-$l‹q:4*6`NS ,?+&(*)>CM6P6=)>.4L)<1 &=CMn/)H9*)>,?9*+MC-$*)>.>[*CBQ )/Q L» . N»0QF#™T z» N»Y[*&(*)>$&=(*)p‹q:4*6)>9DEI:9:CA;z17,j&=CMI/$ .P,?.TaI/+M+-Ij8.>Q »OF /» &o'»  & /»

  K¼ J Ì N¾   F

bI5&)&( ,5&Z)7n/)7$uCMTb‹q:4*66)CX.'.I?DE)4zI/.=.C V C-+-CA&0cy&( ,5&'&( )d&=,/.YGu.Y)7&'Dd,ocx.Y&C-+-+ V ).=13(*)>9:S +-, V +M)/Q Œ^(*CX.DEC-i?(N&(L,54*4z)7$ V )>1>,5S .)pC-$y‹q:4*6<Q J:Q Ì d8)p( ,on/)p1I/$ .YCX9:)>6)<9•>)76=I4*( ,/.YC-$*ik,?DEI?$*i,?+M+J&(*)'&=,?.G:.7[z8P(*CX13(lC-.&(*)Z8KI/6=.Y& 1>,?.)?QF#%$_,ei?C-n?)>$W4*6I V +-)7DW[ .I?DE)H&=,/.YG:.^Dd,ock( ,on/)H$*I?$:@™•7)>6Ie4*( ,/.YC-$*iLQKŒ^(*)>6)7TaI?6=)?[:)7n/)7$W8P(*)>$W,e&3,?.G.)&$ ,?66=Ij8P+-cET”,5C-+-.^&=I DE)7)7&b‹q:4*!6 J*Q Ì *[N&(*)>6)HCX..I?DE)m13( ,5$ 17)f&( ,5&CA&Dd,ocC-$T”,?1 & V )p.=13(*)>9:S +-, V +M)HS*$L9:)76g y|pQ ŒIdS*$ 9:)>6=.Y&=,?$ 98P(Nc&(*CX.PCX..YIL[ 17I?$ .C-9*)76U,p&=,?.GE.Y)7&U8P(*)76=)I/$*)4 ,?6Y&=C-17S*+-,?6&=,/.YGeº\»\T”,?CM+X.K‹Fq:4*6>6Q J:Q Ì  [?Dd,5GOC-$*im&( )&=,/.YGE.)&KS $ .13( )>9:S*+X, V +M)/QŒ^(*)b&=,/.YGeDEC-.=.Y)<. CM&=.9*)>,?9*+MC-$*)p8P(*)>$WCA&bCX.PC-$W4*( ,?.)m8PCM&(_,5+-+]CM&=.P( CMi/(*)764 6C-I?6=CA&0cE&3,?.GQK¶Ij8)7n/)76<[O8P(*)7$W&=(*)H&=,/.YG( ,?.^$*I/$:@™•7)76=IE4*( ,/.YC-$*id8PCM&( ,5&p+M)<,?.Y&Z.YI/DE)dI5T^CA&3.p(*C-i?(*)>6p4*6=CMI/6CM&0cl&3,?.GO.>[]&=(*)d&3,?.GyDECMi/(N&Z,?1&S ,?+M+-cyDE)7)7&'CM&=.m;L6=.Y&'9:)>,/9:+-CM$*)17I?$N&63,56=cl&=I_,5$Ocy$*)>i/,5&C-n?)























J9(

Version 2 CSE, IIT Kharagpur

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9

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priority

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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›     –:13(*)<9:S*+-CM$*i'I5T]6=)>,5+M@B&C-De)&=,?.G:.FI/$,pS*$*C-4*6=I:1)>.=.I?6F8^,?.F,?$k,56=)>,HI?T]CM$N&=)7$ .)6=)>.)>,56313(ECM$&(*) Ì   Ë?.F,?$ 9e&( )S*$L9:)76=+McOC-$*i &(*)>I?6=cdC-.$*Ij8«8)7+-+A@%9:)>n?)7+-I?4z)>9]Q $*C-4*6=I:1)>.=.I?66=)>,5+M@B&C-De)&3,?.GZ.=13(*)>9:S +MC-$*ip,5+-i?I/6CM&( DE.U17,5$ V 6=I/,/9:+-c V )1+X,?.=.YCM; )>9eC-$N&I'17+MI:13GN@™9*6C-n?)>$][j)7n/)7$N&Y@%9:6=CMn/)7$][?,?$ 9 (Oc V 6CX9,?+Mi/I?6=CA&=(*Dd.7Q #%$_1+-I:13G/@%9:6=CMn/)7$W.13( )>9:S*+-)763.>[*&( )'.13( )>9:S*+-C-$*i4zI?C-$/&3.b,?6)p9:)7; $*)<9 V c&( )'C-$/&=)76=6S 4:&=.Pi/)7$*)>6=,5&)<9 V c&(*)e.Yc:.Y&)>D 17+MI:13GQ #%DE4LI/6Y&3,5$N&1+-IO13GN@%9:6=CMn/)7$W.13(*)<9:S*+-)763.P,56=)f&=, V +M)7@ V ,/.Y)<9W,5$ 9W1c:17+MCX15Q KcO17+MCX1m.=13(*)>9:S +M)>6=.P,?6) V )7C-$*i4*6=)>9:I/DECM$ ,?$N&+-cS .)>9 CM$W.DE,?+M+J)7D V )>9 9:)>9W,54 4*+MCX17,5&C-I?$ .^9:S )H&IE&( )7C-6.YC-DE4*+MCX1CM&0c,?$ 9+MIj86S*$&=CMDE)mIjn?)>6( )>,?9]Q |DEI?$ im&(*)f+X,56=i?)$OS*D V )>6KI?T]6)<.YS*+M&=.U&(L,j&,?6),ono,?CM+X, V +M)C-$k)>n?)>$/&@™9*6C-n?)>$E.=13(*)<9:S*+-CM$*iZI5T]6=)>,?+A@B&C-DE)&=,/.YG:.UI?$k,'S*$*C-4*6I?@ 1)<..I?6<[N&08KIE,5+-i?I/6CM&(*Dd.P,?6)fDEI/.Y&b.YC-i?$*CM;L1>,5$N&>˜ ‹ fr«¼h‹U,?6+-CM)<.0#& )>,?9*+MC-$*)prCM63.Y&3¾ ˜FŒ^( C-.PCX.P,5$WI/4:&C-Dd,5+]9:cO$ ,?DeCX1f4*6=CMI/6CM&0ck.=13(*)>9:S +MC-$*id,5+-i?I/6CM&(*DWQ g _| ¼hg,j&=) lI?$ I5&I/$*CX1f|b$ ,5+-c:.YCX.3¾ ˜Œ^(*CX.PC-.P,?$WI?4:&=CMDd,5+].Y&=,5&CX1H4*6=CMI/6CM&0c.13( )>9:S*+-C-$*id,5+-i?I?6=CM&(*DWQ Œ^(*I?S i?(W‹ frCX.,5$WI/4:&C-Dd,5+]6=)>,?+A@B&C-DE)H&=,?.Gk.=13(*)>9*S*+MC-$*i,?+Mi/I?6=CA&=(*D I?$l,ES*$ CM4*6=I:1)<..I?6<[:CA&f.YS:Ž)763.^Ta6=I?D ,eTa)78Ä.Y(*I/6Y&@ 1I/DeC-$*iN.7Q^#™&f1>,5$*$ I5&bi/S ,563,5$N&)>)H&( ,5&b&=(*)Z176CM&CX17,?+\&=,/.YG:.DE)>)&b&=(*)7C-6f6)<.Y4z)>1&C-n?)p9:)<,?9:+-CM$ )>.S $ 9:)76&63,5$L.YC-)7$N&bIjn/)76=+MIN,?9\Q )>.C-9:)<.7[jC-DE4*+-)7DE)7$N&=,5&C-I?$eI5T 6=)>.I?S 6=17)K.( ,56=CM$ i,5DEI/$*i6=)>,?+A@B&C-DE)U&3,?.G:.CX.)qO&6=)7DE)>+Mcp9:C 1S*+M&>QŒ^(*)>6)7TaI?6=)?[5‹ r@ V ,?.)>9 ,5+-i?I/6CM&(*Dd.^,56=)f6=,?6)>+McdS .)>9C-$W4*6=,/1 &=C-17)H,?$ 9g _|P@ V ,/.Y)<9.=13(*)>9*S*+MC-$*id,5+-i?I/6CM&( DE.^(L,on?) V )<1I?DE)H4zI?4*S*+X,569d&(*)fDd,5C-$6)<.YS +A&3.K17I?$ 17)76=$*CM$ ip(*Ij8w&Ie9:)&=)76=DECM$*)&( ).=13(*)>9:S +-, V CM+-CM&0ceI?TJ,'.)&^I5T\&3,?.GO.US*$ 9:)>6P‹ frx,?$ 9 g _|'Q }{)H,5+X.IZ9:CX.17S .=.Y)<9d(*Ij8w&IeIjn?)>6=17I?DE)b.I?DE)b4 6I V +M)>Dd.F&=( ,j&,56=CX.Y)8P( CM+-)f9*)7n?)>+MI/4*C-$*iZ,Z4*63,?1 &=C-1>,5+z6)<,5+M@h&=CMDE)f.Yc:.Y&)7DWQ2#%$ 4 ,56&CX1S +-,?6>[N8K)f9:CX.17S ..)>9k(*Ij8w&Ie( ,?$ 9:+-)f.CA&=S ,j&=CMI/$ .^.YSL13(k,?.8P(*)>$&=,/.YGE4*6=C-I?6=CA&=CM)<.K9:CMŽ)76Ta6I/D&=,/.YGd16=CM&CX17,5+-CM&C-)>.>[O,?$ 9 ,?.=.YC-i?$ CM$*i'4*6C-I?6=CM&C-)>.K&IZ&=,/.YG:.KTaI?6Pg _| .=13(*)>9:S +MC-$*ie8P(*)7$&(*)H4*6=C-I?6=CA&0cE+-)7n?)>+-..S*4*4zI?6&)>9 V cE&(*)HI/4L)>6=,5&C-$*iZ.c:.0&=)7D CX. +M)<..&=( ,5$&( )m$NS D V )76I?TJ&=,/.YG:.&=I V )p.=13(*)>9*S*+M)<9\Q ;





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§Ÿ› › 



Version 2 CSE, IIT Kharagpur

Ì Qb–N&=,5&)m8P(*)7&(*)>6P&( )mTaI?+-+MIj8PC-$*id,?.=.)76&C-I?$ .^,56=) ¬ 6=S*)pI?6 >P,?+-.)?QU}{6=CA&=)mI?$*)mI/6^&08KI.)7$N&=)7$ 17)>.&IH—0SL.0&=CATacWc?I/S*6P13(*I?CX1)mC-$ )>,/13(17,/.Y)/Q ¼”,N¾k|bn?)763,5i/)b6=)>.4LI/$ .)b&=CMDE)HC-.^,?$C-DE4LI/6Y&3,5$N&4z)76TaI?6=Dd,5$ 17)DE)7&6=C-1bTaI?6P6=)>,?+A@B&C-DE)bI?4z)763,j&=CM$ ie.c:.0&=)7Dd.( ,5$ 9*+MC-$*iE6S $:@ $*C-$*idI?T( ,?6=96)<,5+M@h&=CMDE)f&=,/.YG:.>Q ¼ V ¾ b$*+MC-G?)H&3, V +-)@%9:6=CMn/)7$.13(*)<9:S*+-)763.7[*17c:1+-C-1H.=13(*)>9:S +M)>6=.P9:IE$*I?&6)<`NS*CM6=)&I.Y&I?6=)H,Z4 6)<1I?DE4*S*&)>9W.=13(*)>9*S*+M)/Q ¼”1>¾}~(*)>$yg _| C-.bS .Y)<9TaI?6f.13(*)<9:S*+-CM$ i,.Y)7&bI5T( ,?6=96=)>,?+A@B&C-DE)p4L)>6C-I:9:CX1H&=,/.YG:.>[:&(*)ZS*4*4z)76 V I/S*$ 9WI?$_,/13(*C-)7nj, V +-) S:&=CM+-C-•>,j&=CMI/$WCMDE4*6=Ijn?)<.,/.&(*)m$OS*D V )>6PCM$&3,?.GO.^C-$&(*)p.c:.0&=)7D V )7C-$*i9:)>n?)>+MI/4L)<9kC-$ 16=)>,/.Y)<.7Q ¼”9 ¾#™T,E.Y)7&^I?T4z)76=CMI:9:CX1f6)<,5+M@h&=CMDE)f&=,/.YG:.KT”,5C-+X*. ])7( IO17•7GO'c .&=)>.Y&>[O&=(*)7$WCM&b17,5$W.=,jTa)7+-c V )p1I?$L1+-S 9:)>9k&(L,j&P&( C-.&3,?.Gk.Y)7& 1>,5$$*I5& V )HTa)>,/.YC V +-ck.13( )>9:S*+-)>9S*$L9:)76g y|pQ ¼a)<¾| &=CMDE)@%.+MCX1)<96=I?S $ 9O@™6I V CM$.13( )>9:S*+-)76PSL.Y)<.^4*6)>)7DE4:&=CMn/)H.=13(*)<9:S*+-CM$*iLQ ¼ƒT=¾WŒ^(*)DECM$*C-D'S D 4z)76=C-IO9eTaI/6K8P( C-13(,H&3, V +-)@%9:6=CMn/)7$.=13(*)>9*S*+M)>6U.=13(*)>9:S +MC-$*ip$4z)76=C-IO9*C-1^&3,?.G:.F$*)>)>9*.F&I'4 6)7@™.Y&I/6)^&=(*) .=13(*)>9*S*+M)HCX.Pi?C-n?)>$ V cDd,5q IF 7 EH97    N[:8P(*)76=) L»2CX.^&(*)m4z)76=C-IO9I5T&( )H&=,?.Gº\»0Q ¼aiO¾kg y| C-.b,5$WI?4:&=CMDd,?+J.0&3,j&=C-1f4*6=CMI/6CM&0ck.=13(*)>9*S*+MC-$*i,?+Mi/I?6=CA&=(*D &I.=13(*)<9:S*+-)H,d.)&I?T4z)76=C-IO9*C-1H6=)>,?+A@B&C-DE)&3,?.G:.PI?$W, $*I/$:@™4*6)>)7DE4:&=CMn/)fI/4L)>6=,5&C-$*iE.Yc:.Y&)7DWQ ¼a(L¾|1c:1+-CX1y.=13(*)>9:S +M)>6kCX.DEI/6)_4 6I?;L1C-)7$N&k&=( ,5$,t4*S*6=)_&3, V +-)@%9:6=CMn/)7$ .=13(*)>9*S*+M)>6kTaI/6.=13(*)>9*S*+MC-$*iv,.)&I?T'( ,?6=9 6=)>,?+A@B&C-DE)&3,?.G:.7Q ¼”CX¾g y|CX.^,5$I?4:&=CMDd,5+].Y&=,5&CX1b4 6C-I?6=CA&0cd.=13(*)>9*S*+M)>6^CM$k&(*)Hi?)>$*)763,5+17,/.Y)8P(*)76=)&(*)f&=,/.YGd4z)76=C-IO9 .,?$ 9k9*)>,?9*+MC-$*)>.I?T, .)&I?T( ,?6=96)<,5+M@h&=CMDE)H4z)76=CMI:9:CX1f&=,/.YG:.Dd,ock9:CMŽz)>6>Q ¼ —¾W–O)>+AT\.S .Y4z)7$L.YC-I?$eI5Tz&=,/.YG:.2C-De4L,?1 &3.&(*)P8I?63.Y&21>,?.)6=)>.4zI?$ .)U&=CMDE)<.I5Tz&(*)^C-$ 9:C-nOC-9:SL,5+:&3,?.GO.2DZS 13(eDeI/6)P,/9:n?)>6=.)7+-c 8P(*)>$y4*6=)7)>DE4:&C-I?$_I?TF&=,/.YG:.bC-.H.S*4*4zI?6&)>9 V cW&=(*)'I/4L)>6=,5&C-$*i.c:.0&=)7D 1I/DE4 ,56=)>9W&=I&=(*)e1>,?.)'8P(*)>$_4*6=)7)>De4*&C-I?$ CX.P$*I?&P.S*4*4zI?6&)<9\Q ¼”G:¾|“.YS*CM&=, V +-); i?S 6)FI?T*DE)76=CA&]&=I17I?DE4 ,56=)2&( )U4z)76TaI?6=DE,?$ 1)I?T*9:CAŽ)76=)7$N&( ,5639H6)<,5+M@h&=CMDE)&3,?.Gf.=13(*)<9:S*+-CM$*i,?+Mi/I?6=CA&=(*Dd. 1>,5$ V )H&=(*)p,on?)>6=,?i?)&3,?.G6=)>.4LI/$ .Y)f&=CMDE)>.6)<.YS*+M&C-$*ieTa6=I?D )<,?13(W,5+-i?I?6=CM&(*DWQ ¼”+X¾}~(*)>$l,e.)&PI?T4z)76=CMI:9:CX1f6)<,5+M@h&=CMDE)&=,?.G:.C-. V )7C-$*i.=13(*)>9:S +M)<9S .CM$ iEg _|p[*CM&17,?$$*I?& V )H&=(*)m17,?.)f&( ,5&,Z+-Ij8)76 4*6=C-I?6=CA&0cE&3,?.GkDE)7)7&=.^CM&=.9:)<,?9:+-C-$*)?[*8P( )76=)>,?.^.I?DE)H(*C-i?(*)>6P4*6C-I?6=CM&0cE&=,?.Gk9:IO)<.^$*I5&.Y#& )>,/9:+-CM$*)mrC-6=.Y&3¾^,?+Mi/I?6=CA&=(*D 4zI/.=.Y)<..)>.i?IOI:9k&63,5$ .CM)>$N&^Ijn/)76=+MIN,?9d(L,5$ 9:+-C-$*id17,54L, V C-+MCM&0c?Q ¼a$L¾| &=CMDE)@%.+MCX1)<96=I?S $ 9k6=I V CM$W.13( )>9:S*+-)76PCX.P,5$)q*,?De4 +M)mI?T,e$*I?$*@B4*6=)7)>DE4:&C-n?)m.=13(*)>9:S +M)>6>Q ¼aIO¾kg y| ¼”g,5&) _I?$*I?&I?$ C-1d|b$ ,5+-c:.YCX.=¾fCX.p,?$xI/4:&C-Dd,5+U,?+Mi/I?6=CA&=(*D TaI/6'.=13(*)>9:S +MC-$*i_.Y&=,j&=C-1E4 6C-I?6=CA&0cy$*I?$:@™4*6=)7)>De4*&C-n?) 4z)76=CMI:9:CX1H6=)>,5+M@B&C-De)&=,/.YG:.^( ,onOC-$*iE( ,56399:)>,/9:+MC-$*)<.7Q ¼a4L¾‹ r ,?+Mi/I?6=CA&=(*D CX.m,5$xI?4:&=CMDd,5+F,?+Mi/I?6=CA&=(*D TaI?6p.=13(*)>9*S*+MC-$*i( ,5639{6)<,5+M@h&=CMDE)Z&=,?.G:.fI?$u,kS*$ CM4*6=I:1)<..I?6H8P(*)>$y&=(*) &3,?.Gk.Y)7&C-.P,EDECMqO&S*6=)HI5T24L)>6C-I:9:C-1H,?$ 9W,54z)76=CMI:9:CX1&=,?.G:.7Q ¼h`O¾W–OS 4*4LIN.Y)8K)( ,on?),Z.Y)7&UI5T]6=)>,?+A@B&C-DE)C-$ 9:)74z)7$L9:)7$N&&=,?.G:.&=I V )b6=S*$I?$,pS*$*C-4*6I:17)>.=.YI/6FS*$ 9:)>6g _|.=13(*)<9:S*+-CM$*iLQ }x)dI V .)76=n?)'&( ,5&',&=,?.GlI5T^4z)76=CMI:9 Ì ËDeC-+-+MCK–:)>1e( ,?.HDE)7&pCA&3.m9:)>,/9:+MC-$*)/QEr 6I/D &( C-.HI V .Y)>6nj,5&C-I?$][\CM&p8KI/S*+X9 V ) .=,jTa)f&I17I?$ 17+MSL9:)f&( ,5&b,'&3,?.G( ,onOC-$*iE4L)>6C-I:9kI?+T (ZDECM+-+-C–O)>1H8I?S +-99:);L$*CA&=)7+-ck( ,on/)DE)7&CA&3.9:)>,/9:+-CM$*)<.7[ ¼a6 ¾ Kc:1+-CX1m.13(*)<9:S*+-)763.^,56=)fDEI/6)f4*6=I5;L17CM)>$/&^&=( ,5$&=, V +-)@%9:6C-n?)>$.=13(*)<9:S*+-)763.7Q ¼h.=¾#%$,m$ I?$*4*6=)7)>DE4:&=, V +M)PI/4L)>6=,5&C-$*ip.cO.Y&)>D )>De4 +MIjcOC-$*ipg _| .=13(*)<9:S*+-CM$*i'I5T\,p.Y)7&UI5T6=)>,5+M@B&C-De)P4z)76=C-IO9*C-1^&3,?.G:.7[N.Y)>+AT .S .4L)>$ .YC-I?$lI?TF,d(*C-i?(*)>64*6=CMI/6CM&0c&3,?.Gx¼h9:S*)p&=I# ?R )&315Q ¾DE,ocC-$ 16=)>,/.Y)f&( )'6=)>.4LI/$ .Y)H&=CMDE)ZI5TF,d+-Ij8K)>6P4*6C-I?6=CM&0c &3,?.GzQ ¼ƒ&3¾Œ^(*)m8I?63.Y&Y@%17,?.)b6=)>.4zI?$ .)f&C-De)HTaI/6,Z&=,?.GI:1717S*63.^8P(*)7$CA&CX.PI/S:&PI5T24*( ,?.)H8PCA&=(WCA&3.P(*C-i?(*)>6P4*6C-I?6=CM&0cE&=,?.G:.7Q ¼aSL¾|~.4LI/6=,/9:CX1K.)76=n?)>6SL.Y)<9'&=IH( ,5$ 9*+M)P,54z)76=C-IO9*C-1^,5$L9e.4zI?63,?9:CX1F&3,?.G:.TaI?6Fg _|~.=13(*)<9:S*+-CM$*im,/13(*CM)>n?)<.( CMi/(*)76F.=13(*)>9*S:@ +X, V +-)mS:&=CM+-CM•<,j&=CMI/$17I?DE4 ,56=)>9k&Id,E9:)7Ta)76=6=, V +M)m.)76=n?)>6>Q ¼”n:¾}~(*C-+-)S .CM$*im,m17cO17+MCX1^.13( )>9:S*+-)762&=Ip.13( )>9:S*+-),H.Y)7&FI5T6)<,5+M@h&=CMDE)&=,/.YG:.2I/$E,fS*$ CM4*6=I:1)<..I?6<[j8P(*)7$d,m.S*CM&=, V +M)^Ta63,5DE) .CM•>)'.=,j&CX.YTacNC-$*i,?+M+J&(*)m&=(*6=)7)p6=)>`NS*C-6)<9W1I?$L.0&=6=,?CM$N&=.( ,?. V )7)>$lTaI?S $ 9\[zCA&bCX.i/S ,563,5$N&=)7)>9k&(L,j&b&=(*)m&3,?.G.)&8I?S +-9 V )HTa)>,/.YC V +-ck.13( )>9:S*+-)>9 V cd&=(*)p1c:1+-CX1f.=13(*)<9:S*+-)76








Version 2 CSE, IIT Kharagpur

¼a8b¾}~(*)>$EDeI/6)K&(L,5$EI?$*)Ta63,5DE)^.,5&CX.0Tac',?+M+O&=(*)1I/$ .Y&63,5C-$/&3.JI/$'Ta63,5DE)P.CM•>)8P( CM+-)P.13(*)<9:S*+-CM$ iH,H.)&I5Tz( ,5639Z6)<,5+M@h&=CMDE) 4z)76=CMI:9:CX1f&=,/.YG:.S .CM$ i,e17c:1+-C-1H.=13(*)>9:S +M)>6>[O&=(*)m+-,?6i/)>.Y&I?T&=(*)>.)fTa6=,?DE)p.YC-•7)<.P.Y(*I/S*+X9 V )p13(*I/.)7$JQ ¼aq*¾ fIOI:96)<,5+M@h&=CMDE)f&=,/.YGk.=13(*)<9:S*+-CM$*id,?+Mi/I?6=CA&=(*Dd.K)>$ .S*6)fT”,?CM6=$*)>.=.K&Id6=)>,?+A@B&C-DE)&3,?.G:.K8P( CM+-)p.13( )>9:S*+-C-$*i Q J*Qb–N&=,5&)m8P(*)7&(*)>6P&( )mTaI?+-+MIj8PC-$*id,?.=.)76&C-I?$ .^,56=) ¬ 6=S*)pI?6 >P,?+-.)?QU}{6=CA&=)mI?$*)mI/6^&08KI.)7$N&=)7$ 17)>.&IH—0SL.0&=CATacWc?I/S*6P13(*I?CX1)mC-$ )>,/13(17,/.Y)/Q ¼”,N¾#%$y&=, V +M)7@™9:6=C-n?)7$_.=13(*)<9:S*+-CM$*ikI5T&( 6)>)'4z)76=CMI:9:CX1m&3,?.GO.bº F [ º H [ Oº N?[L&=(*)Z.=13(*)<9:S*+-CM$*ik&=, V +M)ZD'S .Y&(L,on?)'.=13(*)>9*S*+M)<.TaI/6 ,?+M+\&3,?.G:.9*6=,o8P$S*4&Ie&( )H&C-De)mC-$N&)>6nj,5+ Ë*[ Dd,jq]¼a4 Ì [ E4 J:[ 4 /¾ B[:8P( )76=)f4 C]C-.&=(*)m4z)76=CMI:9I5T&(*)H&3,?.Gº » Q ¼ V ¾}~(*)>$~,x.)&dI?T( ,?6=9“6)<,5+M@h&=CMDE)4L)>6C-I:9:CX1k&3,?.G:.d,56=) V )7C-$*it.=13(*)>9:S +M)<9S .C-$*ix,u17cO17+MCX1W.13( )>9:S*+-)76<[UCMTm,x17)76&=,?CM$ Ta63,5DE).C-•7)bCX.UTaI?S*$ 9d&I V )f$*I5&^.YS*CM&=, V +-)?[N&(*)>$k,5$OceTa63,5DE)b.CM•>).DE,?+M+-)76U&( ,?$&=(*C-.K8I?S*+X9d$*I?&,5+X.I V )H.S*CA&3, V +-)TaI/6 .=13(*)>9*S*+MC-$*ie&( )H&=,?.G:.7Q ¼”1>¾}~(*)>$x,.)&HI5TK(L,5639_6=)>,5+M@B&C-De)Z4L)>6C-I:9:CX1p&3,?.GO.H,?6) V )>CM$*il.=13(*)>9*S*+M)<9_S .C-$*i,1c:17+MCX1Z.=13(*)<9:S*+-)76<[\CATK,17,?$ 9:CX9*,j&=) Ta63,5DE)E.YC-•7)Z)q*1)>)>9*.b&(*)E)7qO)<1S:&=CMI/$l&C-DE)EI5TU)>n?)76=c&=,?.Gl,5$ 9{.`NS ,?6)>+Mcl9:C-nOC-9*)>.b&(*)EDd,j—0I/61c:17+M)/[&(*)>$yCM&H8I?S +-9 V )p,E.YS*CM&=, V +-)HTa6=,?De)m.CM•>)f&I.=13(*)>9:S +M)f&( )mi?C-n?)7$W.)&PI5T&=,/.YG:.>Q ¼”9 ¾rC-$ 9:C-$*i,5$I?4:&=CMDd,?+].13(*)<9:S*+-)fTaI?6,E.Y)7&I5TCM$ 9*)74z)7$ 9:)>$N&4L)>6C-I:9:C-1H(L,56396=)>,5+M@B&C-De)f&3,?.GO.^8PCM&( I?S:&b,?$Ncd6=)>.I?S 6=17)@ .( ,56=C-$*ie17I?$ .Y&63,5C-$N&=.^S*$L9:)76.Y&=,5&CX1H4*6C-I?6=CM&0c1I?$L9:CA&=CMI/$ .^CX.P,5 $ F@™17I?DE4*+-)&)H4*6=I V +M)>DQ ¼a)<¾Œ^(*)p‹ r“,5+-i?I/6CM&(*D CX.PI/4:&C-Dd,5+\TaI/6.13(*)<9:S*+-CM$ ie6=)>,?+A@B&C-DE)f&=,?.G:.^CM$l,eS*$*C-4*6=I:1)>.=.I?6^CM$l,e$*I/$:@B4 6)>)7DE4:&C-n?)H)>$NnOCM@ 6=I?$*DE)>$/&$sg _| C-.HSL.Y)<9_&=IW.=13(*)<9:S*+-)e,W.)&mI5TU(L,5639y6=)>,?+A@B&C-DE)Z4z)76=CMI:9:CX1Z&=,/.YG:.fCM$u,S*$*C-4*6I:17)>.=.YI/6f)>$OnNC-6=I?$*DE)7$N&<[\CAT &=(*)4 6I:1)<..I?6 V )<1I?DE)<.UIjn/)76=+MIN,?9:)<9Z,?$NcZ&C-DE)f9*S*6C-$*ie.Yc:.Y&)7D)7qO)<1S:&=CMI/$9:S*)&IZIjn?)76=6=S*$ V ce&=(*)b+-Ij8K)<.0&K4*6C-I?6=CM&0c &3,?.Gz[:CM&8I?S*+X9 V )mn?)>6c9:C 17S*+A&P&=Id4*6=)>9:CX1 &P8P(*CX13(&=,?.G8I?S*+X9DeCX.=.^CA&3.9:)>,/9:+-CM$*)/Q ¼aiO¾}~(*C-+-)p.13(*)<9:S*+-CM$ ie,e.Y)7&PI5T6=)>,?+A@B&C-DE)4z)76=C-IO9*C-1b&=,/.YG:.8P(*I/.)f&=,/.YGd4z)76=CMI:9*.^,?6)f( ,56=DEI?$*CX17,?+M+-cE6)>+-,5&)<9\[O&(*)HS*4 4L)>6 V I?S*$L9I/$k&=(*)p,?13( CM)>no, V +-)  ÄS:&C-+-CM•<,j&C-I?$CX.^&(*)p.=,5DE)fTaI?6 V I5&=(l‹ fr,5$L9g y| ,5+-i?I?6=CM&(*Dd. ¼a(L¾#%$s,W$*I/$:@B4 6)>)7DE4:&C-n?)E)>n?)7$N&@™9:6=C-n?)7$y&=,/.YGx.13(*)<9:S*+-)76<[J.=13(*)>9:S +MC-$*i_9:)<1CX.YC-I?$ .m,?6)EDd,?9*)dI?$*+-c{,5&f&=(*)k,?66=CMnj,?+2,?$ 9 17I?DE4*+-)&C-I?$WI?TJ&=,/.YG:.>Q ¼”CX¾Œ^(*)pTaI?+-+MIj8PC-$*idCX.^&(*)Z1I?6=6=)>1 &,?663,5$*i/)7DE)7$N&PI5T2&=(*)m&( 6)>)mDE,5—0I?6b17+-,/..)>.^I?T6)<,5+M@h&=CMDE)m.13( )>9:S*+-C-$*ik,5+-i?I/6CM&( DE.^C-$ ,/.17)7$ 9:C-$*iEI?639:)>6I?T&=(*)7C-6P6=S*$:@B&C-De)mIjn/)76=(*)>,/9*.>Q .Y&=,5&CX1H4*6C-I?6=CM&0cd4*6)>)7DE4:&=CMn/)H.=13(*)<9:S*+-CM$*id,?+Mi/I?6=CA&=(*Dd. &3, V +-)@%9:6=CMn/)7$W,5+-i?I?6=CM&(*Dd. 9:cO$ ,?DEC-1H4*6=C-I?6=CA&0c,?+Mi/I?6=CA&=(*Dd. ¼ —¾Œ^(*)K‹ fr.=13(*)<9:S*+-CM$*i,?+Mi/I?6=CA&=(*D«$ )7)>9 .\&ITa6)<`/S )7$N&+-c)7q*,5DECM$ )2&(*)K6)<,?9:cf`/S )7S*)FI?TO&(*)&3,?.G:.J,j&6)>i?S*+X,56\C-$N&)76=nj,5+X. &=I9:)&=)76=DeC-$*)H8P(*CX13(&=,?.Gk.(*I?S +-9.0&3,56&P6S*$ $*CM$ id$*)qO&>Q ¼”G:¾#%$g _| .13(*)<9:S*+-CM$ i [NCATJc/I?SdI V .Y)>6n/)P&( )f.)>`NS*)>$ 1)C-$k8P( C-13(k,Z.Y)7&I?T]4L)>6C-I:9:CX16=)>,5+M@B&C-De)b&=,/.YG:. <º F [Oº H [QMQ-QŒ^$ ,?6)d&3,5G/)7$tS*4tTaI?6')7q:)>1S*&C-I?$][2&=(*)&3,?.Gx)7q:)>1S*&C-I?$s4 ,5&Y&)>6$8KI/S*+-9t6)>4L)<,j&')>n?)76=c  t;¼  F 7  H 7   7  ¾pC-$N&)76=nj,5+B[ 8P(*)>6)  » C-.^&=(*)m4z)76=CMI:9I5T&(*)H&3,?.Gº » Q ¼”+X¾}~(*C-+-).13(*)<9:S*+-CM$ i,W.)&mI5TCM$ 9*)74z)7$ 9:)>$N&p( ,5639y6=)>,?+A@B&C-DE)Z4z)76=C-IO9*C-1Z&=,/.YG:.fI?$u,S*$*C-4*6I:17)>.=.YI/6>[]g y| 17,5$ V ),/. 4*6=I5;z1C-)7$N&,?.P‹ frtS $ 9:)76.I?DE)m1I/$ .Y&63,5C-$/&3.I?$&(*)H&3,?.Gk.)&>Q ¼”Dd¾r I?6d.13( )>9:S*+-C-$*i{6=)>,?+A@B&C-DE)&3,?.GO.eC-$4*63,?1&CX17,?+S $*CM4 6I:1)<..I?6 V ,?.)>96)<,5+M@h&=CMDE).Yc:.Y&)7Dd.>[U.S V @™I?4:&=CMDd,5+^(*)>S*6=C-.Y&CX1 .=13(*)>9*S*+MC-$*iH,5+-i?I/6CM&(*Dd.,56=)U$*I/6Dd,?+M+-cHS .Y)<9',/.JI/4:&C-Dd,5+:.13( )>9:S*+-C-$*if,?+Mi/I?6=CA&=(*Dd.,?6)1I/De4 S:&=,5&C-I?$ ,?+M+-cHCM$N&63,?1&=, V +M)/Q ¼a$L¾Œ^(*)mg y| .=13(*)<9:S*+X, V C-+MCM&0cEI5T2,'.)&PI5TJ4z)76=C-IO9*C-16)<,5+M@h&=CMDE)&=,/.YG:.K8KI/S*+-9 V )H(*C-i?( )76^CAT&(*)>CM6^4z)76=CMI:9*.^,?6)( ,56=DEI?$*CM@ 1>,5+-+Mc6=)7+X,j&=)>91I/De4L,56=)>9k&Ie&(*)p1>,?.)H8P(*)76=)f&(*)>CM6P4z)76=CMI:9*.b,56=)$*I?&6)>+-,5&)<9\Q ¼aIO¾kg y| .(*I/S*+-9 V )f4*6=)Ta)76=6=)>9dIjn/)76F&=(*)f&C-DE)@%.Y+-C-17)>9d6=I?S $ 9O@™6I V CM$,5+-i?I?6=CM&(*D TaI?6P.=13(*)>9:S +MC-$*iZ,Z.Y)7&I?T.I5Tƒ&^6)<,5+M@h&=CMDE) &3,?.GO.^I/$,eS*$ CM4*6=I:1)<..I?6nj, V +-)dS:&C-+-CM•<,j&C-I?$I5T,_.)&'I?T^(L,5639u6)<,5+M@h&=CMDE)d4z)76=CMI:9:CX1E&=,/.YG:.'8KI/S*+-9s9*6I/4x8P( )7$s&3,?.G 4z)76=CMI:9*.P,?6)HDZS*+A&=CM4 +M)<.PI5T2)>,/13(kI5&=(*)7617I?DE4 ,?6)<9&=IZ&=(*)p17,/.Y)H8P(*)>$&(*)>c,56=)f$*I5&




; ; ;

2





9*

Version 2 CSE, IIT Kharagpur

¼h`O¾g y| .=13(*)<9:S*+-CM$*iI5T^,.)&HI5T6)<,5+M@h&=CMDE)Z4L)>6C-I:9:C-1Z&3,?.GO.S .CM$ ik&=(*) ]C-Sx,?$ 9 ,ocN+X,5$L9y16=CM&)76=C-I?$_DEC-i?(N&f4 6I:9:S 17) C-$:Ta)>,/.YC V +-)p.13( )>9:S*+-)>.^8P( )7$&(*)f&3,?.Gk4z)76=CMI:9*.P,?6)H9:CMŽz)>6)>$/&PTa6=I?D &(*)H&3,?.Gk9:)>,/9:+-CM$*)<.7Q ¼a6 ¾k|..S*DE)d&(L,j&Z,&3,?.G{.)&Z1>,5$ V ).13(*)<9:S*+-)>9uS*$L9:)76'g _|·,?$ 9x)7n?)>6cy&=,/.YG{De)>)&3.pCM&=.p17I?6=6)<.Y4zI?$ 9*CM$*il9:)<,?9:+-C-$*) 8P(*)>$k$*I'&=,/.YGE.Y)>+AT.YS .4z)7$ 9*.>Q2#%$d&( C-.^.cO.Y&)>DW[O,p&=,/.YGeDECMi/(/&^DEC-.=.KCA&3.K9:)<,?9:+-C-$*)b8P(*)>$,'(*CMi/(*)76K4*6=CMI/6CM&0c'&3,?.Gd.Y)>+AT .S .4L)>$ 9*.^TaI/6.YI/De)f; $ CA&=)p9:S*63,j&C-I?$JQ ¼h.=¾#%$l,e$*I?$ 4*6)>)7DE4:&=CMn/)H.=13(*)<9:S*+-)76<[*.13(*)<9:S*+-CM$ iE9*)>1CX.CMI/$ .P,56=)Dd,/9:)mI?$*+-ck,5&&=(*)p,56=6=CMnj,5+],?$ 91I?DE4*+-)&=CMI/$I?T&3,?.G:.7Q ¼ƒ&3¾| &=CMDE)p.+MCX1)<96=I?S $ 9k6=I V CM$W.13( )>9:S*+-)76PSL.Y)<.^4*6)>)7DE4:&=CMn/)H.=13(*)<9:S*+-CM$*iLQ ¼aSL¾#™&CX.P$*I?&^4zI/.=.C V +-)f&Id( ,on/)f,d.c:.0&=)7D &=( ,j&CX.P.=,jTa)m,5$ 9W,5&&=(*)p.=,5DE)f&C-De)mCX.PS*$ 6)>+MCX, V +-)?Q ¼”n:¾ U6)<1)>9*)7$ 17)I/6=9:)>6C-$*ie,5DEI?$*iE,E.Y)7&PI5TJ&3,?.G:.KCX.^)>.=.)7$N&CX,5+-+Mc9*)&)>6DEC-$*)>9 V cE&(*)p9 ,j&=,e9:)>4L)>$ 9:)7$L1c,?DEI?$*i'&(*)>DQ ¼a8b¾}~(*)>$y,k.)&I5TF4z)76=CMI:9:CX1m6=)>,5+M@B&C-De)p&=,?.G:.b,?6) V )7C-$*ik.=13(*)<9:S*+-)>9lI/$_,dS*$*C-4*6=IO17)>.=.YI/6S .CM$ ig _| .13( )>9:S*+-C-$*i [z,5+-+ &3,?.GO.^8I?S +-9.Y( Ij8.CMDEC-+-,?61I?DE4*+-)&=CMI/$k&=CMDE)^—0CM&Y&=)766=.Y&=,?$ 9 V cuŠ7Q¹ V*„3… R:‰ ƒ¸O†  ‡jƒ¸LˆI5TK,&=,?.G_.=13(*)>9:S +MC-$*iW,5+-i?I?6=CM&(*

D w¶Ij8 ,56=)p&( )E.=13(*)<9:S*+-CM$*i4LI/CM$N&3. 9:)&=)76=DECM$*)<9C-${¼”CX¾1+-I:13GN@™9:6=C-n?)7$J[z¼”CMCƒ¾K)7n?)>$N&Y@%9:6C-n?)>$][¼aC-CMCƒ¾K(Nc V 6=CX9.13( )>9:S*+-)763. W¶Ij8 8PC-+M+c/I?S*6^9:);L$*CA&=CMI/$I?T.=13(*)>9:S +MC-$*i 4LI/CM$N&3.bTaI/6b&=(*)e&(*6=)7)E1+X,?.=.)>.I5TK.=13(*)>9*S*+M)>6=.f13( ,?$*i?)Z8P(*)7$v¼”,/¾f.)7+MTƒ@™.S .4L)>$ .CMI/$_I5TF&3,?.G:.7[],?$ 9¼ V ¾f1I?$N&=)qO&H.8PCM&=13(*C-$*i Ijn?)>6(*)<,?9*.KI5T&=,/.YG:.P,56=)f&=,?G?)7$C-$/&=Id,?1717I?S*$N&,?+A@B&C-DE)U&3,?.G:.$ )7)>9'&IH.=,j&CX.YTacHTaI?6Fg _|“&=I V )P,5$'I/4:&C-Dd,5+*.=13(*)>9*S*+M)>6 TaI?6^&=(*)p.Y)7&PI5T&=,?.G:. (*QgP)>,?+A@B&C-DE)f&=,?.G:.P,56=)$ I?6=DE,?+M+-ck17+-,/..CA;L)>9CM$N&IE4z)76=CMI:9:CX15[L,54z)76=CMI:9:CX15[*,?$ 9.4zI?63,?9:CX1b6=)>,?+A@B&C-DE)f&=,?.GQ ¼”,N¾}~( ,5&',56=)'&=(*) V ,/.YCX1E16=CA&=)76=C-, V ,?.)>9{I?$x8P(*CX13(x,6=)>,?+A@B&C-DE)'&3,?.Gy17,?$ V )9:)&=)76=DECM$*)<9_&=I V )>+MI/$*i&IWI?$*)EI?TK&=(*) &=(*6)>)p17,5&)7i/I?6=CM)<. ¼ V ¾#09:)>$N&CMTac.I?DE)m13( ,563,?1&)76=CX.0&=C-1>.U&=( ,j&,?6)HS*$*CX`NS*)H&IE)<,?13(I5T&(*)H&=(*6)>)m17,j&=)7i/I?6=CM)<.KI?TJ&3,?.GO.>Q ¼”1>¾ fC-n?)H)q*,?De4 +M)<.^I5T&=,?.G:.^CM$W4*63,?1&CX17,?+.c:.0&=)7Dd.^8P(*CX13( V )>+MI/$*ie&IE)>,/13(I?T&=(*)H&(*6=)7)m1>,j&)>i?I/6C-)>.>Q

QP}~( ,j&U9*Ifc/I?SeS*$ 9:)>6=.Y&=,?$ 9 V c',5$EI?4*&C-DE,?+ .=13(*)>9:S +MC-$*im,5+-i?I/6CM&(* D k#0.CM&&6=S*)^&( ,5&&(*)^&=CMDE)b17I?DE4*+-)q:CM&0cpI5T\,5$eI?4:&=CMDd,?+ .13( )>9:S*+-C-$*i,?+Mi/I?6=CA&=(*D·TaI/6f.=13(*)<9:S*+-CM$*i,k.)&HI5TF6=)>,?+A@B&C-DE)p&=,/.YG:.CM$x,S*$ CM4*6=I:1)<..I?6C-.4*6=I?(*C V CM&C-n?)7+-cW)q:4z)7$ .CMn/)p&=I V ) I5T2,5$Oc4*63,?1&CX17,5+S .) _‹q:4*+X,5C-$c/I?S*6,?$ .Y8)76.)H—0CM&Y&)>6=.Z17,?S .Y)<9 ¶bIj8 1>,5$ &(*)>c V )mIjn?)7631I/DE) * Qb–OS*4*4zI/.)d,l.Y)7&pI5TK&(*6=)7)d4z)76=CMI:9:CX1e&=,/.YG:.HCX.f&I V ).13( )>9:S*+-)>9xS .C-$*il,l1c:1+-C-1E.=13(*)>9*S*+M)>6mI?$s,S*$*C-4*6=IO17)>.=.YI/6>Qd|..S*DE) &( ,5&&=(*)  S:&C-+-CM•<,j&C-I?$x9:S*)e&I&( )'&=(*6)>)'&=(*6=)7)Z&=,?.G:.fC-.f+-)>.=.b&=( ,5$ Ì Qp|b+-.I [],/..S*DE)'&=( ,j&fTaI?6H)<,?13(_I?TU&( )'&=(*6)>) &=,/.YG:.>[&(*)_9*)>,?9*+MC-$*)>.e)<`/SL,5+X.e&(*)W6=)>.4L)<1 &C-n?)W4z)76=CMI:9*.>Q~–OS*4*4zI/.)&( ,5&E8)l,56=), V +-)&=Ix; $ 9v,5$w,54*4*6=I?4 6CX,j&=)kTa63,5DE) .YC-•7)W¼a8PCM&( I?S:&p( ,onOC-$*i&Il.4*+MCM&p,5$Oc_I?TU&( )E&=,?.G:.=¾&(L,j&'.=,j&CX.Y; )>.&( )E&(*6=)7)d1I/$ .Y&63,5C-$/&3.fI5TKDEC-$*C-DeC-•>,5&C-I?$xI5T^1I?$N&=)qO& .Y8PCM&=13( )>.>[*DECM$*C-DECM•<,j&=CMI/$I?T2.=13(*)<9:S*+-)H&=, V +M)p.CM•>)?[ ,?$ 9l.=,j&CX.YT”,?1 &=CMI/$I?T29*)>,?9*+MC-$*)>.>Q IO)>.^&=(*CX.PCMDE4*+-c&( ,5&bCA&CX.P4zI/.=.YC V +-) &I,/..)76&^&( ,5&b8K)p1>,5$WTa)<,?.C V +-c.=13(*)<9:S*+-)H&(*)m&=(*6=)7)H&=,/.YG:.PSL.YC-$*iE&( )'1c:1+-CX1m.13(*)<9:S*+-)76 _#™Tc/I?Sl,5$ .8)76, d6=Dd,j&C-n?)>+Mc/[ &(*)>$l4*6=Ijn?)mc?I/S*6b,?$ .Y8)766P$*)7iN,j&=CMn/)7+-c?[*&=(*)7${.Y(*Ij8 ,?$l)q*,5DE4*+-)pC-$Nn/I?+-nOCM$*iE&=(*6=)7)m&3,?.GO.P&( ,5&f9:CX.Y4 6Ijn/)>. &(*)p,/..)76&C-I?$JQ  Q K+-,/..CMTacE&(*)m)7qOCX.Y&C-$*ie,?+Mi/I?6=CA&=(*Dd.KTaI?6.=13(*)>9:S +MC-$*ie6)<,5+M@h&=CMDE)f&=,/.YG:.CM$N&=IE,ZTa)>8 V 6=I/,/9k1+X,?.=.)>.>Q‹FqO4 +-,?CM$&(*)HC-DE4LI/6Y&3,5$N& Ta)>,5&S*6=)>.^I?TJ&( )>.) V 6IN,?91+X,?.=.Y)<.^I5TJ&3,?.Gk.=13(*)>9:S +MC-$*id,5+-i?I/6CM&(*Dd.>Q Ì Ë*Q KI?$ .CX9:)76^,'6=)>,?+A@B&C-DE).c:.0&=)7D 8P(*CX13(17I?$ .C-.Y&=.KI5T&(*6=)7)&=,/.YG:.KGº Fo[OOº H5[*,5$L9º N [O8P(*CX13(k( ,on/) V )7)>$13( ,?6=,/1 &=)76=CM•>)>9EC-$k&=(*) TaI?+-+MIj8PC-$*iZ&=, V +M)/Q 





Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

b)7&)>6DEC-$*)d8P(*)&=(*)76'&(*)&=,?.G{.)&pCX.p.=13(*)>9*S*+-, V +-)dI?$t,WS*$*C-4*6=I:1)>.=.I?6HS .CM$*i_‹ rQ2–O(*Ij8,5+-+FCM$N&=)76=De)<9:CX,j&).Y&)>4 .pC-$ c?I/S*6P1I?DE4*S*&=,j&=CMI/$]Q J9*Q#b)7&)>6DEC-$*)'8P(*)7&(*)>6f&(*)ZTaI?+-+MIj8PC-$*ik.)&HI5TF4z)76=C-IO9*C-1Z6)<,5+M@h&=CMDE)m&3,?.G:.bCX..=13(*)<9:S*+X, V +-)'I/${,S $*CM4 6I:1)<..I?6bS .YC-$*ig _|pQ –O(*Ij8 &(*)dC-$N&)76=DE)>9:CX,j&=)d.0&=)74 .HC-$uc?I?S 6p1I?DE4*S*&=,j&=CMI/$]QE#0.pg _|I/4:&C-Dd,5+8P(*)>$x&(*)E&3,?.Gy9:)>,/9:+-CM$*)<.p9:CMŽz)>6HTa6I/D &=(*) &=,/.YG4z)76=C-IO9 . Œ,/.YG –N&3,56&Y@B&C-De)d¼”Dd–:)>1>¾ F6=I:1)<..CM$ i5@B&C-De)e¼aD–O)<1>¾ 2)76=CMI:9y¼”D–O)>1<¾ )>,?9*+MC-$*)¼aD–O)<1>¾ Gº F J?Ë J9( Ì (?Ë Ì Ë?Ë Oº H #NË #/Ë #/Ë  Oº N /Ë ?Ë (5Ë ÌË ºP J( ?Ë J5Ë ÌË J2# Q KI?$ .Y&6=S 1&K,5$d)7q:,?DE4*+M)CM$On/I?+-nNC-$*iH&08Ip4z)76=CMI:9:CX16=)>,?+A@B&C-DE)P&=,/.YG:.2TaI/6K8P(*CX13(17,?$ V )Ta)>,/.YC V +-ce.=13(*)<9:S*+-)>9 V c V I?&(g _| ,5$ 9y‹ frK[ V S:&H&(*)E.13( )>9:S*+-)ei?)7$ )763,j&)<9 V c_g _| 9:CMŽz)>6=.Ta6=I?D &(L,j&Hi?)7$ )763,j&)<9 V c_‹ frQ b63,o8Ä&(*)Z&08I.=13(*)>9*S*+M)<. I?$W,Z&C-De)m+-C-$*)p,5$ 9(*C-i?(*+-C-i?(N&(*Ij8 &=(*)H&08Ie.=13(*)>9*S*+M)<.P9:CMŽz)>6>Q KI/$ .YCX9:)>6^&(*)H&08Ie&=,?.G:.P.YSL13(k&=( ,j&PTaI/6P)>,?13(&=,/.YG˜ ¼”,N¾d&=(*)m4L)>6C-I:9C-.&=(*)p.,?DE)H,/.P9:)>,/9:+-CM$*) ¼ V ¾4z)76=CMI:9CX.P9:CAŽ)76=)7$N&PTa6=I?D·9*)>,?9*+MC-$*) J (:Q 6C-5) L cf)q:4*+X,5C-$p8P(*C-+-)K.=13(*)>9:S +MC-$*i,.Y)7&I?T*( ,5639m6=)>,5+M@B&C-De)U4L)>6C-I:9:C-1U&=,/.YG:.>[>8P(Ocmg _|v17,?$p$*I?&,?13( CM)>n?) Ì Ë?Ë 4*6=I:1)>.=.I?6 S:&C-+-CM•<,j&C-I?$8PCM&(*I/S:&DEC-.=.YC-$*ie&=,/.YG9:)>,/9:+MC-$*)<.7Q J *Q ,5$DZS*+M&C-4*6I:17)>.=.YI/6K6=)>,?+A@B&C-DE)&=,/.YG.=13(*)<9:S*+-CM$*ie,5+-i?I/6CM&(*Dd. V )fS .)>9.,5&CX.0T”,/1 &I/6C-+-ceC-$9*C-.Y&6=C V S:&)<9.c:.0&=)7Dd.7Q‹FqO4 +-,?CM$ &(*) V ,?.C-1P9:CMŽ)76=)7$ 17) V )7&08K)>)7$e&(*)13( ,563,?1&)76=CX.0&=C-1>.I?T,H6=)>,?+A@B&C-DE)^&=,/.YG'.=13(*)<9:S*+-CM$*ip,?+Mi/I?6=CA&=(*D TaI/6FD'S*+M&C-4*6=I:1)>.=.I?63.,?$ 9 ,e6)<,5+M@h&=CMDE)f&=,/.YGk.=13(*)<9:S*+-CM$*id,?+Mi/I?6=CA&=(*DTaI?6,?4*4*+-C-1>,j&C-I?$L.^6S*$ $*CM$ ieI/$9*C-.Y&6=C V S:&)<9.c:.0&=)7Dd.>Q J  Q KI?$ .Y&6=S 1&^,?$k)q*,?De4 +M)fCM$On/I?+-nNC-$*iZ&(*6=)7)H,56 V CM&63,56=cE6)<,5+M@h&=CMDE)b4z)76=C-IO9*C-1&=,/.YG:.K&I V )H.=13(*)<9:S*+-)>9I?$,eS*$ CM4*6=I:1)<..I?6<[?TaI/6 8P(*I?D &(*)H&3,?.Gk.13( )>9:S*+-)>.^8I?6=G?)<9I/S:& V ck‹ fr,?$ 9g _|Ä8KI/S*+X9 V )p9:CMŽz)>6)>$/&)4L)>6C-I:9:CX1P6)<,5+M@h&=CMDE)P&3,?.G:.b¼ƒTaI/6U)<,?13(E&=,/.YG[?&=,?.Ge4L)>6C-I:9d.Y(*I/S*+X9 V ))>`NS ,?+*&I'CA&3. 9:)>,/9:+-CM$*)o¾8P(*CX13(k8I?S +-9 V )H.13(*)<9:S*+X, V +-)bS $ 9:)76P‹ r V S*&S*$L.13(*)<9:S*+X, V +-)bS $ 9:)76Pg y|pQ SL.0&=CATacd8P(OcEc?I/S*6K)7q*,5DE4*+-)bCX. 1I/66=)>1&>Q J  Q KI?$ .Y&6=S 1&K,5$d)7q:,?DE4*+M)bCM$On?I/+MnOC-$*ip,p.)&UI?T]( ,5639E6=)>,5+M@B&C-De)P4z)76=C-IO9*C-1&=,?.G:.2&(L,j&,?6)b$*I5&K.=13(*)<9:S*+X, V +-)S*$ 9*)76Kg _| V S*& 1I/S*+-9 V )HTa)>,?.C V +Mc.13(*)<9:S*+-)>9 V c _|pQ )>6CMTackc/I?S*6P,?$ .Y8)76<[*.(*Ij8PCM$ ie,?+M+\C-$N&)76=DE)>9:CX,j&=)p.0&=)74 .>Q /Ë*QPŒ^(*6=)7)m( ,?6=96)<,5+M@h&=CMDE)m4z)76=CMI:9:CX1H&3,?.GO.^º F . ¼ (5Ë?D–O)>1?[ Ì Ë/Ë5D–O)>1?[ Ì Ë/Ë5D–O)>1<¾ [*º H6. ¼  Ë?D–O)>1?E[ J5Ë/Ë5D–O)>1?'[ J5Ë/Ë5D–O)>1<¾ [ ,?$ 9 Oº N . ¼ /Ë5D–O)>1?6[ #/Ë?Ë?D–O)>1?[ #NË?Ë?Dd–:)>1>¾],56=)&I V )^.13( )>9:S*+-)>9pI?$',PS $*CM4 6I:1)<..I?6S .CM$*ifg _|'Q ,?$m&(*)K&=,/.YGm.)& V )UTa)>,?.C V +Mc V )Z.13(*)<9:S*+-)>9 “–OS*4*4zI/.)Z17I?$N&)7qO&.Y8PCM&=13(lIjn/)76=(*)>,/9kI?T Ì DECM+-+-C2–O)<1I/$ 9*.CX.P&=I V )p&3,5G?)>$lCM$N&=Ik,?1>1I?S $/&<[L9:)7&)>6DEC-$*)p&=(*) .13( )>9:S*+X, V CM+-CA&0c/Q  Ì Q KI?$ .CX9:)76&( )HTaI?+-+MIj8PC-$*ie&(*6=)7)H6=)>,?+A@B&C-DE)f4z)76=C-IO9*C-1&=,/.YG:.>Q Œ,/.YG –N&=,?6Y&@h&=CMDE)d¼aD–O)<1>¾ F6=I:1)>.=.CM$*i?@h&=CMDE)e¼aD–O)>1<¾ 2)>6C-I:9y¼”Dd–:)>1>¾ b)<,?9:+-C-$*)/¼”Dd–:)>1>¾ Gº F J5Ë J9( Ì (5Ë Ì (5Ë ºH #/Ë (5Ë (5Ë ÌË ºN ?Ë (5Ë J5Ë?Ë J?Ë?Ë ¼”,N¾ K(*)<13Gu8P(*)&=(*)76'&(*)&(*6=)7)i?C-n?)>$u&=,?.G:.',?6)k.=13(*)<9:S*+X, V +-)S*$ 9*)76Eg _|pQF–O( Ij8,?+M+C-$/&=)76=DE)>9:CX,j&=)k.0&=)74 .ZC-$tc?I/S*6 17I?DE4*S:&3,j&C-I?$JQ 



#J

Version 2 CSE, IIT Kharagpur

¼ V ¾|..S*DECM$ iH&( ,5&U)>,/13(e17I?$N&)7qN&F.8PCA&313(ECM$ 17S*63.F,5$EIjn?)>6( )>,?9'I5T Ì D–O)>1?[/9:)7&)76=DECM$ )P8P(*)&=(*)76&=(*)^&=,/.YG:.F,?6)P.=13(*)>9*S:@ +X, V +-)mS*$ 9*)76g _|'Q |+X.I [ 9:)7&)>6DEC-$*)f&(*)p,on/)763,5i?)f1I/$/&=)qO&.Y8PCM&=13( CM$*iEIjn/)76=(*)>,/94z)76PS*$*CM&I5T&=,/.YGk)q:)>17S:&C-I?$]Q ¼”1>¾|..S*DE)l&(L,j&Gº Fo[^Oº H5[P,5$L9wº N .)7+MTm.YS .4z)7$ 9vTaI/6 Ì Ë?Dd–:)>15#[ J5ËuD–O)<15[P,5$L9 Ì (jD–O)<16)<.Y4z)>1&C-n?)7+-c?Q b)7&)>6DEC-$*) 8P(*)7&(*)>6&=(*)m&=,/.YGW.)&b6=)7Dd,5C-$ ..=13(*)<9:S*+X, V +-)mS*$ 9:)>6g y|pQLŒ^( )'1I/$/&=)qO&.8PCA&313(*C-$*idIjn?)>6(*)<,?9I5T Ì Dd.)>1m.(*I?S*+X9 V )Z17I?$ .C-9*)76=)>9CM$lc/I?S*6b6)<.YS*+M&>Q  I/S_17,?$_,?.=.YS*DE)H&=( ,j&f)>,?13(&3,?.GS*$L9:)76=i?IO)>.b.Y)>+ATF.S .Y4z)7$L.YC-I?$lI/$*+McI?$L1)'9:S 6C-$*i )<,?13(I5T2CA&3.^)q:)>17S:&C-I?$]Q ¼”9 ¾|..S*DECM$ ip&(L,j&Kº F ,5$ 9dº H ,56=)b,?.=.YC-i?$ )>9e&( )f.=,5DE)4*6=C-I?6=CA&0cenj,5+-S*)?[O9:)&=)76=DECM$*)&( ),?9 9:CA&=CMI/$ ,5+9:)>+-,oceCM$k6)<.Y4zI?$ .) &=CMDE)H&(L,j&º H 8KI/S*+X9kC-$ 1S 6b1I/De4L,56=)>9d&=IE&(*)p1>,?.)H8P(*)7$&( )7c,?6)H,?.=.CMi/$*)>99:CX.0&=CM$ 1&4*6=CMI/6CM&C-)>.>Q2#%i?$*I/6)f&( )p.Y)>+AT .S .4L)>$ .YC-I?$&C-DE)>.,?$ 9&(*)p17I?$N&)7qO&P.8PCM&=13(Ijn?)76=(*)<,?9dTaI/6^&(*CX.P4 ,?6Y&PI?TJ&( )p`/S )>.Y&C-I?$]Q ¼a)<¾|..S*DE)&=( ,j&º Fj[OIº H5[O,?$ 9dº N [O)>,/13(d6)<`NS*CM6=)b17)76&=,5C-$16=CM&CX17,5+z.Y)<1 &=CMI/$ “9:S*6=CM$*i'&(*)>CM6^1I/DE4*S:&=,5&C-I?$]Q}xI?S*+X9dCM& V ) 17I?6=6)<1 &F&=IZ,?.=.)76&F&( ,5&Gº F,5$ 9dº N 8KI/S*+-9E$ I5&^S*$ 9:)>6i/I',5$Oce4*6C-I?6=CM&0cZC-$On?)763.CMI/$9:S*)&=I'Oº H S .Y&CMTacdc?I/S*6U,5$L.Y8)76$ .YC-I?$&C-DE)>.,?$ 9&(*)p17I?$N&)7qO&P.8PCM&=13(Ijn?)76=(*)<,?9*.UTaI?6P&=(*CX.^4 ,56&I5T&(*)p`NS*)<.0&=CMI/$]Q ¼ƒT=¾W|..S*DE)p&=( ,j&bOº NmCX.GO$*Ij8P$W&=I V )p&( )'DEI/.Y&16=CM&CX17,5+JI5T&( )m&(*6=)7)p&=,?.G:.7Q‹Fq:4*+-,?CM$l,.S*CM&=, V +M)Z.13(*)>DE) V c8P(*CX13( CM&PDE,oc V )4zI/.=.C V +-)&I')>$ .S*6)&(L,j&^Iº N9:IO)>.$*I?&DECX..CM&=.^9:)>,/9:+-CM$*)<.US $ 9:)76&=6=,?$ .YC-)7$N&Ijn?)76=+-I/,?9E17I?$ 9:CM&C-I?$L.K8P( )7$ &=(*)H&=,/.YGk.)&CX.P.=13(*)>9:S +M)<9SL.YC-$*idg _|'Q ¼aiO¾k|..S*DE)P&(L,j&Kc/I?SE( ,on/) V )>)7$,?.G?)<9Z&IpC-DE4*+M)>DE)7$N&,?$E‹ ry&=,/.YGE.=13(*)>9:S +M)>6TaI?6,5$d,54*4 +MCX17,5&C-I?$d8P(*)>6)P) d17CM)>$ 1c I?T&=(*)p.13( )>9:S*+-C-$*id,5+-i?I?6=CM&(*D CX.,en?)>6cdC-DE4LI/6Y&3,5$N&b1I/$ 1)>6$]˜-˜ ChQP‹FqO4 +-,?CM$&(*)p9 ,j&=,E.Y&6=S 1 &=S*6=)fc/I?S8KI/S*+X9kS .)f&IdDd,5C-$N&=,?CM$&(*)m6=)>,/9:cd+-C-.Y&>Q}~( ,5&8KI/S*+X9 V )H&(*)p17I?DE4*+-)q:CM&0c I?TC-$ .)76&C-$*id,5$ 96)>DEIjnNC-$*iZ&=,/.YG:.Ta6=I?D &=(*)m6)<,?9:ck+MCX.0&$N&CMTac&( )m)7n?)>$N&=.^8P(*CX13(W8KI/S*+-9k&6=C-i?i?)>6^c?I?S 6^.=13(*)<9:S*+-)76,/9:+MC-$*)rC-63.0& ¾m.13( )>9:S*+-)76'TaI?6e.13(*)<9:S*+-CM$ i_,_.Y)7&'I5T6=)>,?+A@B&C-DE)E&=,/.YG:.'1>,5$ V ) C-DE4*+M)>DE)7$N&)<9DEIN.0&) d17CM)>$N&+-c?Q}~( ,5&C-.&=(*)p1I/De4 +M)7qOCM&0cI?T( ,?$ 9:+-CM$ iE,Z&3,?.Gk,56=6C-nj,5+\C-$c/I?S*6PC-DE4*+-)7DE)7$N&=,5&C-I?$ ‹FqO4 +-,?CM$c?I/S*6P,5$ .8K)>6>Q ¼”CX¾|..S*DE)E&( ,5&pº H ,?$ 9xOº Nj[J)<,?13(x6)<`/S CM6=)d1)76&=,?CM$u16=CM&CX17,5+6=)>.I?S 6=17) g Ì 9:S 6C-$*i&(*)>CM6p17I?DE4*S:&3,j&=CMI/$]Qdº H $ )7)>9 . g Ì TaI/6 Ì ËD–O)>1d,?$ 9xOº NE$*)7)<9*. g Ì TaI?6 J5ËD–O)>1?Qk|+X.YIL[]&3,?.GO.Hº F ,?$ 9yº H [)<,?13(x6)<`/S CM6=)d16=CA&=C-1>,5+6=)>.I?S*631) !g Jk9:S*6=CM$ i&( )7C-6f17I?DE4*S:&3,j&C-I?$JQº F $ )7)>9 . g Ì TaI?6 Ì (Dd–:)>1',?$ 9lº H $ )7)>9 . ;g JETaI?6 (dD–O)<15QH|b.=.YS*DE)Z&( ,5& )<,?13(E&=,/.YGeC-$ .0&3,5$ 17)P&6=C-)>.&=I'i/)&K,?+M+LCA&3.U6=)>`NS*C-6)<9E6)<.YI/S*631)>. V )TaI/6).0&3,56&C-$*im&I'S .Y)f,5$Oc'I?T\&(*)6)<.YI/S*631)>.>Q|b+-.I [N, &3,?.GdCM$ .Y&=,?$ 1)f6=)&S 6$ .^,Z6)<.YI/S*631),/.^.YIOI?$,?.CA&17I?DE4*+-)&)<.^1I?DE4*S*&C-$*iES .YC-$*ieCA&.e$*I?&E&6=cu&Is,?1>`/S CM6=)CM&,5iN,5C-$]Qt#™T  ’¼”4*6=CMI/6CM&0cs1)7C-+-CM$*iu4*6=I5&=I:1I?+ƒ¾mCX.ES .)>9TaI?6E6=)>.I?S*631) ,?6 V CM&63,j&=CMI/$][K9*)&)>6DEC-$*)WCATf&( )W&=,?.Gt.)&CX.d.13(*)<9:S*+X, V +-)W8P(*)7$“&( )_1I?$N&=)qO&.Y8PCM&=13(*C-$*iuIjn?)>6( )>,?9tI5T Ì D–O)>1CX. 17I?$ .C-9*)76=)>9x¼”,?.=.S*DE)f$ IE.)7+MT.YS .4z)7$ .CMI/$I?T&3,?.G:.=¾Q ¼ —¾ŒKcO4*C-1>,5+-+Mc8P( ,j&C-.&(*)p$OS*D V )>6bI?T24 6C-I?6=CA&0c+M)>n?)>+-..S*4*4zI?6&)<9 V c,5$OcW1I/DeDE)>6=17C-,?+\6)<,5+M@h&=CMDE)mI?4z)763,j&=CM$ i.Yc:.Y&)7D &=( ,j&c/I?SW,56=)H,o8^,56=)bI?T l}~(L,j&CX.^&(*)mDEC-$*C-D'S*D $OS*D V )76PI?T4*6=C-I?6=CA&0cd+-)7n/)7+X.^6)<`/S CM6=)>9 V ckg2Œ0@ ^RH–O# p}~(Nc9:I I/4L)>6=,5&C-$*id.Yc:.Y&)7Dd.^6=)>.Y&6=C-1&^&(*)m$OS*D V )>6PI5T24*6C-I?6=CM&0cd+M)>n?)7+X.&( )7c.S*4*4zI?6& 





ž 5ž£ž¦W©Pžª Ì ]C-S QU,?$ 9 J,ocO+-,?$ 9 Q } Q«–:13(*)>9*S*+MC-$*is,5+-i?I/6CM&( DE.ZTaI/6dD'S*+M&C-4*6=I?i?63,5DEDEC-$*i{C-$w( ,5639t6=)>,?+A@B&C-DE)W)7$OnOCM6=I?$*DE)>$/&9:S*+-C-$*is,5+-i?I?6=CM&(*DW˜‹Fq*,?1 &13( ,?6=,/1 &=)76=CM•<,j&C-I?$t,?$ 9 ,on?)763,5i/)b17,/.Y) V )7( ,onOC-I?6. Ì ÌI:Ì [()>17)7D V )>6 Ì  *  Q # '–O( , FQ,5$ 9“g,j—0GOS*Dd,?6egmQ lI:9:)l13( ,5$*i/)k4*6=I5&=I:1I?+X.'TaI?6e4*6C-I?6=CM&0c/@%9:6=CMn/)7$s4*6=)7)>DE4:&C-n?)W.13( )>9:S*+-C-$*i Q „=5‰ Z FE„ $@ToŠ ˆ™„ FZŠ3[ I/+hQ Ì ¼ bI :Q N¾ ˜ J8#  2J (:[ b)<1)>D V )76 Ì  *  Q











8















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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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r

Version 2 CSE, IIT Kharagpur

lock(CR) T1 needs non−preemptable resource CR being held by T5 T1 T1 undergoing unbounded priority inversion T2

T1 gets non−preemptable resource CR held by T5

T5 preempted by T4

T3 T4

unlock(CR) lock(CR)

T5 t1

t2

t3

t4

t5

time

tn

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Version 2 CSE, IIT Kharagpur

pri(Ti)=5

pri(Ti)=5

CR

CR

Ti

Ti

CR Pri(Ti)=10

Ti

Ti

pri(Ti)=5

CR

Tj

Tj Tj

pri(Tj)=10

pri(Tj)=10 pri(Tj)=10 scenario 4

‰;BXY[V5GN@ rZ ˜ IUM[gUOQ>U\_JNO0˜C>5\l`‡BEIUYaš'\[GNPBEIUYa\[|;œ&G3BX\[GNB]Jon£ƒIU>5@hGNB]JTM_IUSh@œqGN\_J3\CSh\[W scenario 1

scenario 2

scenario 3

                  "!#   $ !%&   ' ( (% *)+ , -           "$! %&$ ,  ' ( (% *)+ . , -  ++   $ , ' ( (% *)  $-#     *   /0  *' (* (%  *) , (#      1 2

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Version 2 CSE, IIT Kharagpur

T1

T1

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T3 releases CR

T2

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T2 requests CR

T3 acquires CR

Task priority

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T3

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t2

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time

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1

Version 2 CSE, IIT Kharagpur

CR1 T2

T2

T2

CR2

CR2

CR2

Waiting for CR1

Waiting for CR2

T1

T1 CR1

T2 Executing

T1 Executing

T1 Blocked

T2 Blocked

Snapshot 1

Snapshot 2

T1 CR1 T2 Executing T1 Blocked Snapshot 3

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Version 2 CSE, IIT Kharagpur

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CR2 TH

Ceil(R1)=10 CSC=10

pri(TH )=8

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Version 2 CSE, IIT Kharagpur

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Task Priorities

Pri(T1)= 10

T1

40ms CR1

Pri(T2)= 7

T 2

60ms

10ms Pri(T3)= 5

T 3

CR2 20ms

Pri(T4)= 2

T 4

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Version 2 CSE, IIT Kharagpur

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T1

20ms CR 1

T 2

10ms

10ms

30ms T 3

T 6

CR 2 40ms

T 4

10ms

50ms CR 3

T 5

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

Arrangement of tasks in ascending order:

Step1:

T1 T3 T2 T5 T4 Schedule tasks as late as possible without violating precedence constraints:

Step 2:

T1 6 8

0

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T3 T2 20 25

T5 T4 33 40 50

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Step 3:

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

f(){ wd_start(t1,exception−handler);

start

wd_tickle(); }

end

t1

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

27P9a9GC23>Ks7i13I/lEB3>@4F8G$1<3IE&FL/&’-FLXZ  Z 4Z System

call

check parameters OS Service (kernel mode)

Trap

Application Program (user mode) system call next statement

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Version 2 CSE, IIT Kharagpur

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Priority Level

1

Task Queues

2 3 4 5 6

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

Condition enabling high priority task T 2

T2 starts running

T2 T1 t1

t2

t3 time

tt p

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t2

t3 time

ti l

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Release Semaphore

tss T1 T2

T1

T2 time Request Semaphore

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Version 2 CSE, IIT Kharagpur

t1 T3

T2

T1

t2

T1 time Request Release Acquire Semaphore Semaphore Semaphore

Acquire Semaphore

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Version 2 CSE, IIT Kharagpur

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TRIANG

Shared Memmory Analyser

System Call RT1

RT2

RTn

Output

Timer

Y

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

Legend: MSAU: Multistation Access Unit

MSAU

MSAU MSAU

MSAU

MSAU

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Logical Ring

Node not in the ring

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:

Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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.

Version 2 CSE, IIT Kharagpur

Çi¯/É)Å^±‹Æ‹¯J¶ z¯ =ÅSÆ ´[ÈÅSƋ²É)·Y±‹¸«½r†w6‰"D13">$'!=6 E/"%:&‡A ">8!=Eª. :5>(5J157$)lj!=Z>!=:@P6/5JEJ!b,h"%*06^(+"%*06 8-"P157.^@P5>L(Q !b1="%$0!36 E "%:9(Q "%( *06 EJ*213"%(!=8 T Q/*21VQ;6/5JEJ!*08s.6 8H,-*'(_EJI :*06/? T Q *01VQh(+*',-!f.I :+13!=8`1="%6;:+!=8!3:+Z>!R(+*',-!f*'6^(+!3:+Z[">$ LM57: . "71Vx>!3(`(:V"%6<8H,-*288*'576;AS@hA/:+57">E 13">8H(*06/?!LM5>: T Q/*21VQg6/59:+!=8!3:+Z["%(*05>6 T ">8s,h">EJ!7l *'(sW<6 E/8"LM:+!3!p8$'5>(_AS@;1576 8I/$v(+*'6 ?(Q !p1="%$0!36 E "%:s">6 E;:+!=8!3:+Z>!#8](+Q/!:+!=~^I/*0:+!=Eh(+*',-!f*'6^(+!3:+Z[">$yA^@-A/:+57"7E/13"78U(+*'6 ?(Q/!p:+!=8!3:+Z4"4(*05>6 *06JLM5>:+,h"4(+*'576-(5X">$'$C6/5JEJ!=8=NzuÀ13">$'!=6 E/">:H&‡A ">8!=E9./:+5%(+5J15>$ *08_8H*0,-./$0!>l^!ˆ;1*0!36^(=lJ">6 E T 57:xJ8]Z7!3:+@ T !3$0$ T Q !36g">$'$<,9!#88+"%?7!=8]*06h(+Q/! 8@J8U(+!3,€">:!p.C!3:+*'5JEJ*21p"%6 E./:+!=EJ*21e(V"%A/$0!>N †w6(Q/!pLM57$'$05 T *06/? l T !EJ*08+1I<8885>,-!p*',-.C5>:(+">67(. :5>(5J157$08sLM:+5>, !#">1VQ5%L«(+Q/!=8!p(Q :!=!p1="4(+!3?>57:*0!=8_5>L./:+5%(+5S135>$283N   





 C SADC  E   EF4AD

H M4

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Transmission Interval

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msb

lsb

N1 10

0

1

0

1

0

N2 16

1

0

0

0

0

N3 20

1

0

1

0

0

Priority bits

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Î*

Version 2 CSE, IIT Kharagpur

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Transmission

n *0?>I :! : ­ `:*05>:+*'(U@-u:A *v(+:+"%(*05>6*'6 *0:(I ">$«OR*',-! `:5>(5J157$ ·M¶[µ=²‹¯/É ·)¼KÅ p¶4»<µ=»|´7»<ÉH½`†w6h(+Q/*08_./:+5%(+5J15>$C"6/5JEJ!fI 8!=8](+Q/!p8U(V"4(!5%Lk(+Q/!1VQ ">6/6/!=$ (+5X:+!=">85>6h">A<57IJ(`(+Q/!.C!36<EJ*'6 ?X. "71Vx>!(V8 :+!=8*0E/*'6/?-"4(5>(Q/!=:6/5JEJ!=8=Nj‚`"71VQg6/5JEJ! *'(Qo"X.<">1Vx>!3(R(5h8H!=6 E ">*v(V8`LM57:"%6*'6^(+!3:+Z[">$k5%L(*0,9!p(+Q "4(*28R*'6SZ7!3:V8H!=$'@h./:+5>.C5>:(*05>6<"%$ (+5i(+Q/!./:+*'57:*'(U@h5%L‹Q/*0?>Q/!#8U(./:+*'57:*'(U@;,-!=T 8+8+"%?>!*v(Q<">8=NjORQ "4(*28_(T Q/!$05 T !=:R*08_(+Q/!./:+*'57:*'(U@h5%Lz"i,-!#88+"%?7!(+Q "4("96/5JEJ!Q "783lS(+Q/! ,-5>:+!(+*',-!*'( T "%*'(+8=Nzu_(`(Q !!{J./*0:V"4(*05>6h5>Ly(Q/! T ">*v(+*'6/?(*0,9!7l^(Q/!6/5JEJ!8!36 8!=8j(Q/!f8U(V"4(+I 8j5>Lk(Q/!f1VQ "%6/6 !3$cN†‡Lk(+Q/!1VQ ">6/6/!=$ *28 A/I<8H@7l/(Q !36b(Q *08,9!#"%6 8(Q "%("-Q/*0?>Q/!=:./:+*'57:*'(U@;,-!=8+8+"%?>!*28AC!3*06/?h(+:+">6 8H,-*'(H(+!=Eo"%6<Eb*v(6 !3!=E 8(5 T "%*'(I/6^(*0$z"%6b*0E/$'!.C!3:+*05SE|N ŠS56*28z"%$0:!#">EJ@(+">xS*'6/?./$2">1!7l#(Q/!=6X*'( T "%*'(+8«LM5>:(Q/!_6/!3{^(‹LM:V"%,-!>l45>(Q/!=: T *28H!`*v(]8U(V"%:(+8«(5(:V"%6 8,-*v(#N 









N2

Propagation Time N1

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Version 2 CSE, IIT Kharagpur

Priority

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

low Bandwidth 802.5

GP(U) 802.4

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Î1,

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

Eligible Time computation

Active Msgs

1

D M U X

Hold Queue

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Departing msg

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Delay Bound PreAhead

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Arrival

Departure

Deadline

Holding Switch i

Delay Bound

Arrival

Eligible

Deadline

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Version 2 CSE, IIT Kharagpur

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Regulator 1

Regulated Traffic

Priority level

Regulator 2

1

input

output

n Regulator h

Scheduler

Rate Controller

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Version 2 CSE, IIT Kharagpur

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Contention







 





















 











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