for the dynamic algorithms' and on an input motivated by a real-world graph. ..... us now count the overhead induced by the computation and splitting of SCCs.
! " #
$% & ' & (
!
"
#$ % & '( ) *& + %
+ , -Æ . %/0 ) *& 1 2. %// )
*& # 3. '/4 )
*& + . '// )
*& ' + 1 1 . -4 ) *& ' 5 " , " 6 - , 1 -7 #, 4888 4(409 :+3;2 6,>4>( :+2;=-(> 3!+@ - & , 2& , " 3 " ' 6 " ' 7 5 & % # 7 ? = B3 # C D # 44E >>480 = - & F& , % - G 7 1 /9H>> 1 ' - & 1 " " " +2 +
, " " @ I 1 @ 1 % +2 4H4H J 5 " 5 4>>E9 7#+ K4 :/4/< 098 >(04
+
1& - % + , )* + ,- ,+
Æ
! "#$% #&% #'% #(% $$% &#) * + ,
+ "#% $) + "#&) - "'% .% /% ##% $0% $.% $/% &$) - ! "#/% #1% &2) "1% $&% $'% $() 3 !
, "#2% #.) !
! Æ ! 4 ! 4 5 !
!
"
"/% $&% $0% $.% &$) ! 6 "$0% $.)
4 5 ! 7
! "$1) 89 9 : 4 "&) !
- + ! !
; "0) 9 ! + 4 ! 6 "$0% $.) ! ; $% + ! "$&) 6
5 , >
, % " )
5 - ! 6 "$0% $.) ? 6 "&$) @ "/) ! 4 A + + B C ! B C
6 ? 6 + @ "/) + @ ! 3 D "$&) "$&) + .*) % 2
$## "$0% $.) " ) 9
E " ) " ) " )
+
+ F + # " ) + ; " ) > ! - " ) " ) G
! " ) + + F + +
!
$
% > H
= ! ! " ) " ) ; #" ) " )% #" ) " )% #" ) " )% !
! *
8 - " ) " ) @ ! % ! " ) " ) 7
!
! + ; " ) " ) + ") : " ) " ) " ) > ! " ) " ) " ) > Æ " ) +
" ) " )
"##$ "##$ " )
" ) % "##$ " ) > " ) + 5 5 6 !
, 5 ! - ! ! +
G ! "$0% $.) 5 6 ! "
%
E " )
" ) " )% " ) > @ + 5
"##$ + " ) " )
" ) 6 - 4
+ 3 "##$ + *
"##$
! ' > 4
6
)
# > $ 8 > I > '( ) # = ; ' $ $ ' $ >
$ >
$ = ' ' + 9 < "#0) 9 /0
G=< ( 1 ( # G=< ( 1 1 ( 1 ! 3 ! @ +
1 1 > 5 > #22 &22 (22 .22 2 $ ' % ! > 2 = * +; * > #2 $2 * > (2 * > (22 * > (222 * > (2222 5 * "() 5 5
P
P =
3 $ , > ' = $ + > (22 < !$2 6
!
)(
6
3 7
&## ) 5 4 7 +
= & + (22 " & " ! , 6 # + # + @ 0> ,
. O #$ ,
"
5 ! 4 > '22 /22 #$22 #022 > ( #2 $2 '2 5 I 5 5 J !< 5
#2 5 @:J #2
! &% &% ' ! # '( * ! = (#2 '( * = '2 ! # ! 4 #% = 1 #2 ! > '2 3
5 = ! 4 &% ' " '( )
% = ## #$
= ! &% $ &
# '( ) > '2% = ## #$ = &% ' # ! " ! 4 '( * ! = (#2 '( * = '2 # ! '( * ! 4 "% = #& #' , '( * = #2 = #' '( )
= #& #' + ! 4
> #022 > #2 > '2 > #2 B7 C '( * /2 '2 > > '2 +
! 50 45
0.12
DFS Ital-Gen HK-2
""
DFS Ital-Gen
0.1 40 35 0.08
Time (sec)
Time (sec)
30 25
0.06
20 0.04 15 10 0.02 5 0 400
500
600
700
800
900
1000
1100
0 400
1200
500
600
700
Vertices
14
12
800
900
1000
1100
1200
Vertices
DFS Ital Ital-Opt Ital-Gen HK-2 Yellin CFNP
10
Time (sec)
8
6
4
2
0 400 9
600
800
1000 Vertices
1200
1400
1600
DFS Ital Ital-Opt Ital-Gen HK-2 Yellin CFNP
8
7
Time (sec)
6
5
4
3
2
1
0
0
50
100
150 200 Operations
250
300
350
6 4E - @ : < %+' :
< , @ 4> @ E/> , . O #$% ,
: <
49>>
"#
0.08
DFS Ital-Gen HK-2
140
0.07
120
0.06
100
0.05
Time (sec)
Time (sec)
160
80
0.04
60
0.03
40
0.02
20
0.01
0 400
600
800
1000
1200
1400
DFS Ital-Gen
0 400
1600
600
800
Vertices
16
14
1000
1200
1400
1600
Vertices
DFS Ital Ital-Opt Ital-Gen HK-2 Yellin CFNP
12
Time (sec)
10
8
6
4
2
0 400 6
600
800
1000 Vertices
1200
1400
1600
DFS Ital Ital-Opt Ital-Gen HK-2 Yellin CFNP
5
Time (sec)
4
3
2
1
0
0
10
20
30
40 50 Operations
60
70
80
6 4( - @ : < %+' :
< , @ (> @ 0> , . O #$% ,
: <
49>>
!
"$
4*4 : #
! I ! 9 "0) 9 : 9 )
! )
!
#$(1 (#2# >> E04ME08 )4>*
2 & + M +- />>4 D 480/ />>4 /40M//8 )44* ) **+ ( , /0 :(>> E9AME08 )4/* " % = " + , // = 1 4888 )4E* " ,- # M+ @ (( 488A 998M989 )4(* " ! ) # & 1 H/ :44ME>A )//* 3 ) , 2 & = 488H H48MH/A )/E* 3 ) , 2 & 6 488H 99(M9A/ )/(* 2 & 6 4888 04M84 )/H* 2 & + 4888 (8/M(80 )/9* + Æ (0 4809 /AEM/04 )/A* 6 /0 4880 HM44 )/0* $ # ( $ + #- 2 M '!00 D E4( 4800 4>9M4/> )/8* & 7 1 4888 )E>* 3 ! 3 ; 4H0 4889 /EEM/AA )E4* 3 488( 909M98H )E/* ' # E> 488E E98ME0( E90ME0>
!
)
&
( $
)
+,
-*
! !
! ( $ &.
-'
! ( $
! !
)
'& " # %
)
) / $
+*
! !
#*
((/+-9
5 ! 4 + * * > (22 J !<
¼
6 49 @
¼
6 4A @ %+'
!
¼
6 40 % @ %+'
¼
6 48 % @
#)
#
¼
6 /> @ + !
¼
6 /4 @ %+' + !
