6fIEIu`'w4fD{ww{dEa0~iIxvRBa0f0ixEf0IxRB0R4`B0xVRBk00`x}Rf`x0s G z ~ z v zv u { v { z ~ w wv | { z y wv u t S S 5 6 lj g h g e krikXaB W 1 2 0 q 9 t 7 o m lj g h g e DpnkiaXafdaS 9 i U0i
S
B
7
x v e y x v qs r q p e h g f e wut%%iB'`d b
c '
) " G ( & $ " a6` Y Y X #
6
P Q
) " G 06HW
T VU
F S
R
P Q
) G IH
F
E
6
CB
5 6
@ A
D
5 6
9 8
7 8
5 6
'
) ( & $ " 0' ' % #!
3 4
1 2
AQe Vz2q1dQ QdQq vXd AQ8Uq kz q q1d1df v vo x"AQ )dvd (zi do1kvAzev 3kddo (zt"( d"Advktd ek kfd kvdkz kQQdxez8dqd" kvd" q e e e kQdtd" dh)8dAq QdA1)qdvkzktdk v5kvvAkkdkd1zAkvk8 qdxzz AdU vddd dv)k AQotodvvdAQe k(d zvv ez z Qe k"AQdz{vvdd ddvEzvdqdQAQt zQ z{vvdE kkv zk z8kkd ' kkQ ' zv9 d dAdvk{ dtSvd}zkk 3kd3 kkd Q zzvkvdqk" kz8 3SkdQAvkoAQ3kz dk3kzd kzkdkzhzd e Vzvi (z kzkvAzv{3tvddQ V) vdqkvvAQvQdkkkq kk(d d ez t d kAdzAQ ezv5 vkdQvdzkvAzvdQqdv kz8zSdvvhAzQdd81Qvkz8zzv{5 tqQ 2 k}hz{)2t2 k}A ( 2 E 1 { 2 ko2 {2{ok z t { Vk kk2 k xkVz {) 2{ z A vf {2 k A{k { {Ak{ { z {Vz z{ z 11iot{2 k 2ik qA2 k k({dz k{ { ) iA 2v{zq{ AU(d" 5 2 d k{k" {{kr{ zq ( z{ " k {2 Q {5 2q2 { } kX 2z{{ k {A{kk { ef{vuhg lweht s}qxp| n g{m zyQ8eww 2yxwhwew ~ r y g i odlwkj(i T u R T s V R b w V R P I UVStrVpe2wq UhT w gUfiIeR dhwUUeRxg I )Uf UeIdc UaI `XxWyY tUvT SQ5H f & G G D C @ 7 '8$ ECFE'"%B '0 A@98 6 4 0& # ! 5 32'1)('%$"
" P
S
5
D
C 7
P !
S
0
) k
6
0
G
F
P
) t
E
F
C
0
9
7
) k
9q @
7
lj g h g e krikXaB
P Q
G
G B ) )
0
G
" B G
" B G
lj g h g e kiaXkB
)
S
@ i
)
0
k
S
B
F
6
0
a
D
F
) 0
P
x v e wy
x v qs r q p e h g f e wut`wi``d
0
xveyWxvwqustrq`pwei``d h g f e
0
F
)GkGHt P
) " G 06
G ( & $ G w H8 F
r
s f e ``d
C
) 0
) k
p
v v w h
@
) a
x v e wy
x v qs r q p e h g f e %t`%iB'`d
B
S
r
s f e ``d
x v e y x v qs r q p e h g f e wut`wi``d S x v e y
P
2
x v qs r q p e h g f e %t`%iB'`d
P
T
0
0
) " G a6H
e zdd8 zd1 5 32 IqQd9d~ }{d8 Az 2 4 0 | z zqX si UQAzzkdqdzvddodq1d zqQ QSAd 4 0 dk dt "kv{kqkQzo zh rzv zv8d1df Qd q y "ddvo e d h3AQd}1QQ dk {kkk A x t v dzv dQ d o qod dd f UzdAx }Qk Q y{kq kk d A z w qk dU A@vve bkdH6dqd dAQvkvU A@vve 8 8 kdH6dq3 Ekvdvvd"9zd1df do q"d 1 v" xAfUo 1 e3Advkdd dodvk ! Aud1Qdd kvvkd " f o k r h r q gxg e tsQ$ Hlepe %q vkQG6) 9Afdo o kkA rkexGGGFx x nm Hlg x g d t h keAkxGGGFx x jiegz1o AQ8toefedxGGGxx$z s t h t x t d 1qkvAzevS }e (zv1 Q x t kdV(zvS z1 ywv u QA8kvkdQd kv{kkdz` (zd)z s v x t q c Ya i a a g e ca Y r%hV phdUDfddb`7X$W V kddAddvz QdASkA{Az vAzQdo U d A@v Uv 1koz S 8 vkAe ) d 3@A dT z AQ1QQ doe 8"dvvd " h SdAdr2 d1 zk v zd) zAA 5 9 kv Qdk AdovAvvdk zQzvv 3do1dQ5d xzQ8k Q 3@ d% zvdq " e fq kd d 3@A dRd Q A@vve Qd B " 8 " Q 3@A dPdkvz kQd qAd8vk AzvQd zd""zv5dddQ do q3 9kvQ9kv"Ez%kQd d(Ad dovdQ z {kkd(ev(d kvdvvd5Afd1df$dde q81zzk1kvkkf5kzd 8 Q3AQvkv A@vve IkdH6dq ezqdvkkvd1hqd Addod5QAdzAf G e qd5 zo53d zvf kfd FEDC ")kQdv1zvvA kvdkzdAAzvQ d B d AQvkv A@v 9kd76Q"dd) 5 32 15 kvo3z 8 4 0 ! kfd %( & % $ kz)i kfd ) ' & Ad " 'kvA#dddQ ! kQd 9 dvEA Ak v Q5kv5z" dv1 zd1$kQd d" (z} Qv$kQdd kzvddk kvxQ Addv 9kd kQd ktAQ1dQ do q(ddkd1tkvdvvQ ktz dvv}Ad v1 kvkkvSAQvkvd drxekvdvd1dd(QAQdQrdde q Q{ Qvv8 vkd3AQvkvkvo5d1eAdvk 8veov d xdAfvkvVd1Addv8 hkvdvvdd vdkz1dQ dde qUAQ vdkAQ1vkvk1dQQ
0 ) & 0#
P 8
C
S
0
S
0
a 0V
G!
D
q
a 0
T
5 2
"
P V
S
) k
1ddQqvd1UP AQtz8} R"ci AQ8 i P Qz"() hzzd zv8QdQv do qrAd3ddqQv{VdAd d zA Uk1dQt X dAd"dzv ddfvt hdff g o t h g Vk(v9vqx } zxE f k 1h P I `8 z5 S r r q g r ze qqh f ciUAddv1 Avvdxq zvi dv d x yzvdqfd xGGGxec AQ t 6 x t dQ zd mzvodddQ Adti " vd} $Ad qdzd 2d8v8dddQ ~ pt 6 ukQdptA 6 } XU 6iwAd1A1 vk)A( zv ddfvrdo q8S v x~6Af ddVd1df)Ad " S zd8 v8 x AQ31ddQ do qq"6ibz s iTi
0 9 7 5 2 9 n 7 0 y
iRj 7 @ikSykiaXkB pke piVRt7 o pU~61k4lT uV9 Xae 9 m m o lj e lj g h g e e 9 mj m lj e lj g 7 g m m h Yz6XQrj gk84%ckiaXkB S 8j RE%l@%wiD uD%n#aS~aFe e m o l lj g h g e l mj j lj h j e e j e lj g h g e e l l e l l m hj 44si(ka0VkrikXaB e CqS 4u4bh iFBaQr4 e p4 #iT j
0 P W 0 i
0
e k
a e
a 0
%
j
S
B
0
T
2
F
D
) 0
AQ1QQ do q5d zvodddQ dde q 'kz(z9Ak3vkzv o kz5x" tkzv5dQQv8z Qd q zv FDBkSAQqQ zvdQQv( ( # ~p k5 31dQ qtvdqtz1Xo (z kvk( " kQdzvddQQv do q5kQd1zqvkvvA" 21 4 s4 3 Iv5kddkzxz d1 Avve 8 " Qqd 2 0 @ d5dQvvqAdQ qtd 4 zkezdd }d 4 h Q A@vv 8 kd76Q34
x v e y x v qs r q p e h g f e %t`%iB'`d F c aG
y v z v 5 h f ` Px%f0`xwaB igX W5 d u( q bq X X uY~q Wi ~F Wi d S xpxu W eqxa B2S v 9 ` X d bi `X W U x e" '@qq Wi @~s Wi d S p}# W sd{aB40xwxwv u zy$ n X X v 9 | X y v qwvzRn0I Yuq Wi ts Wi d S xpp W rqxa B2S y b q X X v 9 ` X d bi `X W U o e" ' h f `X piga W d p$ n mlDz fk6fkd0Df G Q i 8 {z u v u y 9z 8 X d 9 v C 9 8z i z u hz 9 j@R{#F Wi V~ E9EDgD60Rw%pR{ $efe cgcD d quBzRqI bX@ W3 @xu W dYwaX B qfk6IEiDf HE b y ` X b3 ` W U S Q 3 8z {z u v u G b ` y b h v `X d pxwut W3 s r q h f `X d b `X U PbpigaY WI ecnaYI WBVST RPx6iDf HI Q ) v I z u G u v RE `y6af0 9F~ EED9RiR#`yXBRA@RRB0 (E0R~ 7 9v C u u v 5 z 3 9 z {8 u ' ~ 566 IxzRB0xRrUVaB 4E6z0iR{EIzw 020Rfud0!(0 & z v v w 3 ~ 1 ) | { z y ' ~ $" ! %#
!
P n 0 a
) 0
@ A
R
0
0
9 }
0
G
7
"
P
5 2
9
7
9
G
S
9
7
E
6
6
G
S
T
v d d 3 kzvktz $ hd kvkvkdd1 3@A d" kd76Q d zvQdQv Qd q33Qvvdoh q9d d de" S B d 3A d$dvdqzv8dddQ dod8hqd 1d k( @ " 5d dzvdddQ dddq Q3' {kk5z zvddQv Qd qtdddvvQ zvdddQ do qtkQU d"Advktv d vq xAQ1ddQ do q5 Qvvd9z8( Ad kq kz kd) kz Pzv 5AQQ dd"1ozd d isAd3d1zq51ddQov1 dt x p k{k3AQQq"1dt Q{kz11kQo2AdAkz 6inAQdz811dQ 1"d 6 AdQzzx }Se zdd8d1 v h"AQ Qqoz}18vz 9 AQ 1ddQ8dkk1}d }kdAkdA3kz(zkzv dd2k" k3zdQf uAQd51Akz qkedqvf de Qv3 qVm AQtdkdAdov1 6i}Ad5d z(zx ` AAQ e })zddQzz3 6inAdQzq x d 3@A d5 b "
&& 7 5 sppe31 sfe mdm86up``%a42 DC``d
c C
e ! ! qw 0 )!#Ixz0w#'R & 0f S #( % ` $ G ! v ( b z b v uI#0xn` # ui#ax` "sbX Wi ` dx ! b v ! I Q ` H d G b ` pxE b Q ) 8z I z u G FIu%dE|f g6iDf H r b q S '(dDBzUxwfRd0xv f0xiR6 v`EaBa0xvx'wp~{RVxv '60xxvi0R`fw I { { v { z v u y | 8 z 9 ) 8z w | u {z R!x0faw !E%(E0R~ 7 { v 9 ~ 9v C 9 ' ~ ) |{ z y ' ~ 0Rfud0!(0 & $! "
0 0 P
Q d d AUkA Q v1 Qz Q d AoQ tq d Qx QA dkqk qdddzv d Qz d vdvdQ dQ v qQ q d h q 1
0 0 c " CV
s u e1 e e2 e3 v t
2
0
2
a
F
}
P
a
5 D
@
F
9
B
b
9
D
5
C
B
E
5 6
F
YE
S
E
B
6
F
5
c
F
5 D
G
G
F
3
0
F
F
Q
B
3 4
0
Q
3 4
0
P
F
Q
0
9
7
5 2
Q
0
2
)
) 0 p
F
0
G
D
P
9 I
I9
) A
kzxz vkvko AdzAQ}zvQdQv do q}AQvdA d Ak {kk)d1d 4 32 d d 0 @ Q1k qoddAk9 z3vSu zkzzd q9"Qe " 8 d A@vve }d1zvv )kXzzvqvkvke z d @ QDAQdzvzd" kdhzz {vQdA kAk515 " 4 qk dAz3zkdfVzvdQdod8oS kd A@vv 8 kvdk dQqUxddSzdt vdeqt zkj 2 4 32 ddA)Ad3ke3kz1kAA QoA)xv5AQdv1kvvzk kz3kdQ1(kqd'kddvk(' z' qAzd5de kdk 2 5vdx 32 dQ 4 u 32 0 Qq d Q @ Q" k d)dQ dAvvdfe kUzdAk zA 5h A z)vkA)kd d) A@vve Dd1kkzv8Q ed SkkQqqvU 8 vA E d A zA 9i e AQ5(1kd Q 3@ ddQvv " Ad`zd d1 A@vve rxAdQ d1zv5doqkd 8 d 3@A dT8qdrz% zQ Q @ Q" " 8 d A@vve zkA qdd8v8kzv d 3@ d" z}){Vzd h 8d 8k8Afv8 Q((12x8 kzv d ed Qd kAr Szvtd{VAkd AdfUd{k d(QUtkv{kqkQ zxf`yzv1z1{rd d d qk}qQ1dkvQ y A%Qo 3@A d" d dd A@vv hhddAkkf)zdtv3 k 8 e dqddUzk 8AqdvkqvkdQA{kqkQqdddvzddv dz dA)qdvkvk s 32 0 dzvvzA kQdzvoddfv$dde q 4 8 thoqd d dt A@vve 1x Qdd)zzztvkE 4 0 dAzQdA zdS 54 32 i d A@vve 8 SAkkQqvkod d1 2x 34 32 i kz8 kvoz 0 kQdzvddfv do q8d (z QAzQQASzQ voh 4 t 2 4 r 2 4 3 0 fud}DkdzQVh8ur x 3 dkzz3 Q Avve 8 @ 3 kz8 kAkvtz kQdzvdQQv doe 8qxzz 4 0 3 qAk 9kvdvvd k89z }zv8d v h9AQt dd"}kvdvd zv QdQv do qqd d 4 u r ne4p#4 g %qr 2x 34 zd1V f 2 r q h t 2 4 2 x 3 3 0 3kv5z kfd zvodddQ dde qoAd11d" d Avk)z y4 32 8kv18zrkQdzvQdQv dde q 4 0 8 ze d3d ddvvAd dd851ddd v1 A@v 3kv kkk 2 4 2 t 2 2 0 2 }dv x zvq1 j 3 0 D 3 z1V 4 3 Ri 3 d @ Q Qqxzz B $zddSzu Qdx R4 8z1 " 4 r 4 u 32 q1k dAzd dd zze k wdk vqAdu5d zv 0 2
a a 5 6
B
D
S
F
D
S
G
S
B
b
D
S
D
S
P Q
t S
S
P
$
G W
G
S
B
&
" `
B
D
B
P
S
C
B
P
G
5 6
S
P
2
&
"
B
P
X
"
G
" G 6
" & " X k6G
G
( & $ w i
b
P
X
&
"
q
" `
G
C
S
" 8
B
P i
" a 0G
i
S
6
S
H
) 0 ! j m F
e l
) 0W C
5
D
e l
e X4
a
P
0
G }
F
S
@
0
F
e a
0
z9dUd%d 3@A dyz (zvQk dtx9kzoAdEkzvfd " ! S " h Qkd Ue (z1vQk88d(8kvkvvkd T d( A@vve 8 zvvxotk"dvkAo)zk9%Qd 3@ ds Qk 1te z 6 " d" dk t5e z zz d xdde zk zdzz z (z k Adzv zqzdkAdoq dk )e z 'kvAvkd dqd"} dk 2 zkQoe zve ov"de ! $d 5kAdkAdo vkedAQe 1t z Ae S de 9Q8z8d"vkoh qQkv{kqkQ" G d kz(z1zzv dk8 k 1ddQ dzQ vk (z kkvkzke8 Qfe e ! e Ue (z8}8dqkvkvv1z (zQ1zvkdk( QAQd31 (ztkeSQ zz kz(z "Ak kvAzkoz( (zq kkvk" Pd}" 6" vQ % dkd3 1d zd d ! AQ zv d5 (zd kd kz(zdk Edd d d"k zzSke5(qe Q qQUkdddAkz5 kvAzkdAe zt kSkvkzket dde e ! e 1e (zX"(d"U8d)kvvkv y zzdd3kd11 zd3kvAzdkq qQA5d (z kkvkde! G Qd a Ae zzz "ke zddqzd G dX ddkd)e ztzz qdkvvkv (de zzz oA3d5dkde z d kkvkfe de e de Ae zktkd(1z Qtzd ! de e eG e z ! k 1}zddokAde v1e z3k dd8dUzdd k kQo D (zkqoA 1vd5e z d }34 4 z 6 S 6 o 4 5 Sz dt dk QkeEsr zd kdk kzddkvkzdqtkQze zdde 88ddkvvkv S B 5kvQ d) 3 d" vdd B Azz} ji 0 @ 4 zkQ{zd{QdQqUddAQ8 @ q qkvkkQ) d A@vve 8 kvQqk dkAdzAQ qdqxkdvAdAq kvd8d8Vk vAQqq 8 kvdvvQ8zv"QdQv do q8dzvhA dEAdvk vddv h Af)d"dtdvvdAdtkQ Akv d6 dzkddzd5d8Vd 3@ d AkkQqq kv " zdq zAAdkvQdqxzk"kAQd Q 3@A d" dU d `dV A@v 8 k8d(kd(kkddXd d6 vXz a q a g q iv1yd c 1 d kd1zqve v k 6 x QAQ1QQ do q k2d d Qd A@vve Tddtz 8 dAdvkd9dd Qg hAQ dQ dde q kd2d ddzvtddQv doe 88fhzzdQe5AQvk3d edv
0
$
& " " kk6G
! e G
( ( G k6
m j e krrkf YC
m i
un e
g j 6 4
m e aVaf
m
D
v
r
q p `%e
5 p e 6%
) " G 6H
( k
& ( G b$ k6 j E
& ( G k6%
lj m I i
m h j f Qr4
g ae
j Y P
e j e Rk8
p e %I
e !7
) G G IkH
C
" R%
c S
p w
v ps 3 ` 41
q p `we
e q '
) G 0 H
$
( ( ( G kk6%
m h j f Qr4
m a arrE a m j e e 9 e q '
) ( #
@
@
H
( ( G b$ k6`
4
o m Eai r
j 4 f s
j e m Yf af
g X
m
e us
q p e %% ' e q
ps
) & #
( ( ( G kk6%
m h j f Qr6
m a ar6 4 a m j e e
c '
e q '
q p e ps `%
) $ Ex
" " G k6B ( e
G
& ( G k6
e m j e B I RkX4
m i
3
@
p e %I
e 7 !
q p `%e j S
p 3 41 m
7 '
s
e
r
)
G
( k
& ( G b$ k6%
l e
m i
1
p e %I m
e !7
q p `%e
p e %} q
d
)
" & b "
" ( G a$ 6%
l e
j S
3 4
P
m i
3
e
d !
q p `%e
q p `w
q }
) a
P
( ( k6G
lj m rf b i
m Rj
m j ar6
g m j 6 Yf
v E
s
q p `%e
p w
s
p
7
)
( $ ad
& ( ( G m h j kk6% f Qr6
c 4
5 X
q v Y E
x e %8 m `
q p e p x `wn f v E
1
) " #
j 4 I Yf j
m
@
( ( ( G kk6 ps
e B Y 3 6
k s
e us 3
q p e `%8 % %5 e q p s
y
7
e q '
v e } e
1
) G IH
B B @ 2 d1 mtEl pp l %0 Du a d IB zdAzQdA h 1 W h & Py dkqdoddh dezv d i% i0 Y ' C 2 d ! kzAqkApkQd q ` d1vvdddx zze z k j Y( l@ i g r T W 3 h kAvfek fpf7f5 eR3@ d g 1 ) B B @ B kvz AQqkd d r g x x G ! 2 d k# g u g g 8 3 g 1 B B @ !W IB ze zdfdkzv{Ad dkvdk"od x B B @ 5 Qk( z ( 1 e IB c BHzQ g @ & $ & ekv dqdQk8d5 ' C p pIx g ! a d k# T l u l l !W B kqdd z 8 3 g 1 B B @ & Aqx vAdzzd51e z5qkvQ g 2 G dV 0zdfAkvkddkvkddQ3kzvqdQQv QdQq 6 1 q & ' & y w 6 QdQ$xqdkz8 r a `T m HC 0x if v c d eutq u ~ 0Fc !rzqkfdAk 8d s g 1 W d q & zd9k2ddkddkd doQAkUzzQd r T T dv ! 2 e d ! Apiq u f~ 0Fc fezqkQd`2dc (z h g 1 W d b h3 dkz8 qd {kqkdq}zzz I a ! ` Y 3gddd & & Q C & C kkU 3z a X g @ HTVU RT !&S sRP diP B W B @ A! d 2 T3 BIB HzdvdqkfdU e zkzvd Azke xzvf d 'zez2d % ! GE D & F & C df@a R# D q B ) @ 68 5 3 1 ) Y AF#967z 420('zQdkve dqddkqE ' C & p% #! p T g FT( F d ( $ & " & a g g g swtSdUg 8 d1 A@vve Uvx1k
F
Mesh Network
350
Hypercube Network
800
700 300 600
Time (in milli-seconds)
Time (in milli-seconds)
250
Dijkstra
500
200
Dijkstra
400
150
300
Kruskal
200
100
Kruskal
50
100
0 200
300
400
500
600
700
800
0 100
200
300
400
500
600
700
800
900
1000
1100
Number of Nodes
Number of Nodes
Regular-6 graphs
450 400
Density : 1%
250
200
Time (in milli-seconds)
300
Dijkstra
Time (in milli-seconds)
350
150
Dijkstra
250
200
100
150
Kruskal
100 50
Kruskal
50
0 200
300
400
500
600
700
800
0 200
300
400
500
600
700
800
Number of nodes
Number of nodes
Density : 5%
5 4.5
Density : 40%
35
30 4
Time (in seconds)
3.5
Time (in Seconds)
25
Dijkstra
3 2.5
Dijkstra
20
15
2
Kruskal
10
1.5
Kruskal
1 5 0.5 0 200
0 200
300
400
500
600
700
800
300
400
500
600
700
800
Number of Nodes
Number of Nodes
" 8 ! Q 3@A d9d d A@v rx)dk"Adde $
& F S c