• Aucun résultat trouvé

Parallel Computation on Interval Graphs: Algorithms and Experiments

N/A
N/A
Protected

Academic year: 2021

Partager "Parallel Computation on Interval Graphs: Algorithms and Experiments"

Copied!
28
0
0

Texte intégral

(1)Parallel Computation on Interval Graphs: Algorithms and Experiments Afonso Ferreira, Isabelle Guérin-Lassous, Karina Marcus, Andrew Rau-Chaplin. To cite this version: Afonso Ferreira, Isabelle Guérin-Lassous, Karina Marcus, Andrew Rau-Chaplin. Parallel Computation on Interval Graphs: Algorithms and Experiments. [Research Report] LIP RR-2000-43, Laboratoire de l’informatique du parallélisme. 2000, 2+25p. �hal-02102081�. HAL Id: hal-02102081 https://hal-lara.archives-ouvertes.fr/hal-02102081 Submitted on 17 Apr 2019. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) Laboratoire de l’Informatique du Parallélisme École Normale Supérieure de Lyon Unité Mixte de Recherche CNRS-INRIA-ENS LYON no 5668. SPI.    

(3) 

(4)

(5)     

(6)   

(7)      

(8)       .  .   

(9)    . École Normale Supérieure de Lyon 46 Allée d’Italie, 69364 Lyon Cedex 07, France Téléphone : +33(0)4.72.72.80.37 Télécopieur : +33(0)4.72.72.80.80 Adresse électronique :   

(10).

(11)    

(12) 

(13)

(14)     

(15)   

(16)      

(17)           . 

(18)            

(19)            

(20)     

(21)    !            "  

(22)     

(23) "          ". #      

(24)    $  . 

(25)   

(26)     "  

(27)       

(28)      " #

(29)    

(30)    .   

(31)    !      

(32)   

(33)           "   

(34) %  &

(35)   '!  #( "   ) *+ &    " #

(36) ,  ,          !$.  

(37)    

(38)  "   

(39) "     "    "   " ) *+ &. . )    

(40)   - .                 -     

(41)  /       

(42)  !                  "   

(43)      

(44)   .       

(45)     

(46)     

(47) . !    0      .     ! -  -1 .      - /     2 

(48) 

(49)     

(50)     "    

(51)         

(52)  3       

(53) .   .    .      , /, %  &

(54)   '!      

(55)   - ) *+ &      !      ,   ,     

(56)   - " 

(57)  /  "   "     "   " &.

(58)    

(59) 

(60)

(61)     

(62)   

(63)      !  " #$ %#  !    &'     .   

(64)    

(65)  &

(66) ( )%*   +,! %#$%)

(67) ( +- .   &'

(68)      .      &   !( 0  !   #*  ! + !   1 2 !( " /#$ 3 "- / %#/  ! +40-*     + "$5 6) &'  . &'   .   

(69)   #         

(70)               

(71)   

(72) .  .        

(73) . 

(74)   . "  , #

(75)    "  

(76) 4   

(77)  05%61"   

(78)  4   

(79) 0 %61 . " # .    $  . 

(80)    "  ,           

(81)       

(82)

(83)  

(84) #            

(85) "   "   "  "  

(86)    .  

(87)        

(88)  #(        !           

(89)  " #

(90) 

(91)      ,       7  ! 

(92)    

(93)     8)   #

(94) 

(95) 9  .       #      #

(96)    !

(97) 

(98)     #( #

(99)  # 

(100) 

(101)     . . . .  . . .    . :.

(102) :;<   !   

(103)   )  "

(104) 

(105)   8)) "    9

(106) 

(107) 

(108) #  0 '!  %  &

(109) 1   ! 9   !      

(110)     

(111)        

(112) #  % 

(113)  (     " #   ,  !  !$    

(114)   

(115)  ) ! &" #

(116)      #.  

(117) #      

(118)   !     = "    

(119)     0   1   

(120) >   

(121)    

(122)   >    

(123)      .  

(124)    ! .  .   

(125)

(126)    , !  

(127)         4  8 *' 

(128) 

(129)     " .

(130) #    :   "  

(131)   #   "    

(132) . ,  ?" @" @" :" A . 8 ' , #

(133)    ?A ' ,    ?A '    ?" @A '    ?" @A  

(134) 4    ?@" :A 5 

(135) 4    ?@" :A )   08 *' ) )B1 ?A          . . . . . . . .               .   

(136).            .   :2   

(137)   

(138)      8 *'  

(139)  #

(140)  .  " .   

(141) #  

(142)      ,! .           

(143)      

(144)  8 *' 

(145)  " 

(146)    .   

(147)   

(148)   $ B

(149)  4  8 *' 

(150)   (!     4  

(151)  ! 

(152)  " 

(153) !         8)).   

(154)     ,   ! 

(155) " 

(156) 

(157)  #

(158)             .  

(159)         

(160) .  B

(161)     " 

(162)  568"   ! C   "  .  (.  

(163)    .   !   

(164)

(165)  

(166)     !    

(167)   ,  4   8 *' 

(168)      

(169) #  )    

(170)  38  #

(171) 

(172) "   C  / 568    "  .  

(173)  

(174)     4      

(175)  #       !         

(176)    7

(177)           $       $       

(178)    

(179)  );'     

(180)    

(181)  ,.   <   " 

(182)   

(183)        !      

(184) "  !" 

(185)      

(186)     " 

(187)        .       .  . . . .

(188)   !     

(189)       

(190)     

(191)      

(192)  

(193) " 

(194)   " 

(195)        

(196)      #

(197) 

(198) 

(199)  

(200)       !" )D  

(201) # 

(202)   !     

(203) ". 

(204)    ("     

(205)  "            568  );'  

(206)  #

(207)  

(208)  #(    4  

(209)    " 

(210)     

(211)      

(212)    8 *' 

(213)      

(214)  " #

(215) 

(216) . 

(217)  ,        ! ,      =" #

(218)   

(219) ! 

(220)     ! 

(221)     

(222)        

(223)  "    9  

(224)     E(   

(225) "   

(226)      !      

(227)     !  6  

(228)     !  # ,  " 

(229) !     !  < 

(230)         

(231)   

(232)   #     .  ,   

(233)  > " 

(234)   

(235)     

(236)  

(237) !    "       0    

(238)   6    ?:FA1.   . .  . B               

(239) .     

(240)         

(241)  ,    

(242) "  !    "  , #

(243)      

(244)  4   

(245) 05%61   

(246) 4   

(247) 0 %61 .  B           

(248)    $  . 

(249) .    "  ,                 

(250)    !    

(251) " #   

(252)          

(253)   

(254)   # 8)) "   #

(255) 

(256)  ! %  &

(257)   (  

(258)  

(259)   ! '! #("   ) *+ &     5    

(260)     " 

(261)    #  !       

(262)       

(263)  

(264) # 

(265) 

(266) 

(267)      

(268)  #

(269) 9

(270) 

(271)  8))  #

(272) 

(273)  

(274)    

(275)    B      #       # 

(276)  8))  

(277)  &  !"     #         !. #  9  

(278)  8))  "     .  #   9! $  *'  

(279)  #        

(280)    !      9! .    .   

(281)   

(282)     $ #   6  # # 

(283)          

(284)    6  

(285)          

(286) "  6 "     

(287)     "   

(288)        6   6 G" #   ,   %  &

(289)    8))"  '!   8))   ) *+ &     B   

(290)   #

(291)       

(292)    

(293)  . . .  . . . .

(294)

(295)        H( !

(296)  I .  05681 "     $     ! .  .        

(297)    

(298)   

(299)   

(300) .   !     .   

(301)             !    

(302)  .  

(303) . J     K 

(304)  

(305)      ! !

(306)   

(307)  #( 0#

(308)    !   41 

(309)  . .      !  .        ! .  

(310)  

(311)  .  .     

(312)   #   

(313)  )  ;  '  "     

(314) "    

(315)  );'    568         .  #

(316)     !  

(317)  

(318)   H    I .  

(319)    

(320)  0   1    4   H  !  I 

(321)   " #

(322)     

(323)   

(324)  4  H  !  I      

(325)  >  

(326)         "  

(327) .   !.         % 

(328) 

(329)       #  "      

(330) # 

(331)  

(332)     

(333)      

(334)     

(335)      !  

(336)  

(337) !   !  "     "        

(338)  );'  

(339)   

(340)  !    

(341)              

(342)  

(343)  .

(344)  

(345)     . .       6                

(346)   .  % 

(347)     

(348) 

(349) # 

(350)        

(351)  );'                                        

(352)   

(353)    . 

(354) .   " #

(355) 

(356)  

(357)         

(358)     

(359)     !  

(360)  !  4    *  

(361)           ! 9 

(362)         

(363)  ,        

(364)   #     9  

(365)  ?" :A  

(366)   " 

(367)     

(368)    ,!      

(369)  );'  

(370)  );' "   4,     .          !            ".   

(371) .       ! 

(372)  "  

(373)  #

(374)   ,

(375)      

(376)  .    (# 

(377)        

(378)  

(379)  .       

(380)  4      

(381)   

(382)   ! B #     

(383)  J   "  ! 

(384)    (   

(385)               ?A2   

(386)   

(387) .  (  (  

(388)       .

(389)

(390)  . B                      

(391)  # 

(392)  " #  #  #!  " 

(393)          ;    "     0   1  "     

(394)    

(395)  

(396)     " 

(397)   #

(398)   

(399)     

(400)  

(401)   

(402)  01 0  %:1 

(403)    4  

(404)  #2                    

(405) #  7 # !    

(406)      0     #

(407) 

(408)        1  # 2 .     2     0    1 0      

(409) . 1 2   0  1 0    1 . . . . ;    

(410)     

(411)          

(412)  

(413) .      

(414)     4 !  "       "          "       .    

(415)  

(416) .      

(417)     4 !      "       "         "       . )  

(418)  4,  ,  

(419)   "   

(420)                

(421)       

(422)      

(423)  # 2                                            

(424)  

(425) .         "             .        "            . * 6 :" # (# 

(426)   

(427)      

(428)       0    1   

(429)     

(430)        

(431)  4   0    1   

(432)  4,  ,    6 " #   

(433)    

(434)  

(435)     0    1"   

(436)  

(437)    

(438)  

(439)   

(440)   

(441)    #

(442) 

(443)     

(444)    

(445)    

(446) .    " 

(447)    !     ! B (       0    1           6    

(448) 

(449)      

(450)  

(451)     G.

(452) 6 :  

(453)    " #

(454)   

(455)  

(456)             6 :  6          .   

(457) "    0    1      #

(458)   ,!

(459)    B 4   

(460)    0 1 

(461)   #

(462) 

(463) H 4 I" 

(464)   " #

(465)   

(466)     

(467)  

(468)    0  %:12       #

(469)            " #  !    

(470)    

(471)     

(472)        

(473)   !   B #     

(474)         4 .          "       

(475) #  

(476)   "     

(477)    

(478)    

(479)    

(480)     

(481)      

(482)     0  %:1       "     " #   

(483)   

(484)         " #

(485) # 6  *

(486)     

(487)    .    2     0    1 0      

(488) . 1 2  0 1 0    1 . . . . ;    

(489)     

(490)          

(491)  

(492) .      

(493)     4 !  "       "          "       .    

(494)  

(495) .      

(496)     4 !      "       "         "       . )  

(497)  ,   

(498)   "   

(499)               

(500)       

(501)      

(502)  # 2                                       

(503)  

(504) .        "             .       "                !   

(505)  

(506)      

(507)  4  " #

(508) 

(509)    ,!        .    . @.

(510) 1 2 3 4 5 maxright(1)=3, maxright(2)=5, maxright(3)=maxright(5)=4, maxright(4)=nil minright(1)=2, minright(2)=3, minright(3)=5, minright(4)=nil, minright(5)=4 first({1,2,3,4,5})=1 next(1) = 5, next(2) = 4, next(3) = next(4) = next(5) = nil. % :2 &,    

(511)    "   "      .    

(512)     

(513)  % !"                   " 

(514)  .          

(515) 

(516)              "  

(517)       (       !      6 %    ,   1 2. 1 3. 2 4. 3 5. 4. 5. % 2 *       

(518)      

(519)   

(520) . "           

(521)      

(522)  );'  6  

(523)   

(524)    

(525)      

(526)    8 *' 

(527)  " #

(528)  

(529)    9 

(530)  "     

(531)     " .

(532) # #. 

(533)    *.     

(534)    ! J   

(535)   

(536)        

(537) " #   #

(538)       

(539)     # #   4    #

(540) 

(541) 

(542)      #

(543) .   ,      % !" 

(544)      

(545)  ,  

(546)   4  % "  

(547)  

(548)  .   #

(549)  0  

(550)   1  !    

(551)   , #

(552)     

(553)  ,   *

(554)    .  , #

(555)        

(556)  

(557)  );'   

(558)   

(559)  .  . "  

(560)     

(561)           #. L.

(562) 1. 10 2. 4 10. 3. 4. 5 20. 5. maximum weighted clique({1,2,3,4,5})={3,4,5}. % 2 &,    

(563)   , #

(564)    .   !  

(565) " 2     0    1 0      

(566) . 1 2 

(567)      

(568)   , #

(569)     .  . . .  (" 

(570)  

(571)   ;    

(572)     

(573)   

(574)   

(575) .  .    

(576)  

(577) .      #

(578)   4 !        "       "         "          . )  

(579)  4,   

(580)    #

(581)   >          

(582)      )  

(583)             !    !      

(584)        #

(585) :    

(586)  

(587)    ,  

(588)     > 

(589)     5     &!    

(590) 

(591)         (    

(592)  4   , #

(593)     .

(594)      .        

(595)   

(596)    

(597) .     . #  *

(598)     

(599)     

(600)  

(601)   ?:A * 6 "     "       " 

(602)    

(603)    

(604)  #

(605)        

(606)    0

(607)  #

(608)       

(609)        

(610)           

(611)    #

(612) .    

(613)  

(614)       1 

(615)     

(616)    .    #

(617) 

(618)       "    

(619)  #

(620)   

(621)    * 6 " 

(622)  

(623)    ,     

(624)  #

(625)    , #

(626)    

(627)  .           0    1"      #  0    1"  #    , 

(628)    

(629)  

(630)   

(631)      

(632)   , #

(633)   "    

(634) #   #    , 

(635)  " #   

(636)       

(637)   , #

(638)   

(639)     # 

(640)    

(641) .

(642) #. 

(643)  .  

(644)  

(645)  6 :   

(646)   " 6        6 "  F.

(647)  G                       $ #

(648)    

(649)      " #

(650)         . 

(651)   . 

(652)      

(653)   

(654)   ,   

(655).   

(656)          .    

(657)  

(658)     

(659)     

(660)     *

(661)    

(662)    

(663)  );' .   $  

(664)    2     0    1 0      

(665) . 1 2 & 

(666)  

(667) 

(668)    

(669)         .   0 ! 1 ;    

(670)       *.    

(671)   

(672)        *.  

(673)    :   

(674)     

(675)    :   

(676) 

(677)    )  

(678)  4,   

(679)     

(680)    .     ! !   $  "

(681)  

(682)     ! 

(683)   ! 4     . . 3           #    

(684)     4   $%     $  "

(685)       

(686)  !

(687)      

(688)    $%> $%  $%  . . 

(689)    $%.   .  !

(690)    

(691) 

(692)    " #

(693)       >        ! "  .        

(694) .                    

(695)   

(696)  .

(697)  .  . #  

(698)    6 :   

(699)  

(700)          .        & # #            '" #

(701) &  ' 

(702) " 

(703)         

(704)          # ! 

(705)  4,   6    

(706)    !  .     

(707)  

(708)  #    #     6 G   

(709)    2 

(710)     

(711)      " #

(712)  

(713)  #

(714)    

(715)    

(716)     M.

(717)   .   

(718)      

(719)  

(720)         

(721)    

(722)

(723)  9    * 

(724)    

(725)  

(726) "    

(727)     B

(728)    

(729)    ,!" 

(730)   

(731)   

(732)     6 :"     6 " "   G    #

(733) 

(734)      " #

(735)          %    ,    

(736)      & 

(737)     .   

(738)  ,  *

(739)  " .

(740) # * 6 :"     !

(741)  

(742)   . " #

(743) 

(744) ,  . 

(745)     

(746)           processor 1 1 2. processor 2 3 4 5 processor 3 connected component(1)=cc(2)=cc(3)=2.0 cc(4)=cc(5)= 3.0. % 2 &,    

(747)     . 

(748)  !  " ! . % G   ,    5 

(749) %  6 

(750) 05%61   

(751) %  6 

(752) 0 %61      

(753)  *

(754)  G  

(755)    4 5 

(756) %  6 

(757)      

(758)  1 2 3 4 5. 4. 4 3. 3. 5. 1. 1. 2. 3 2. BFS tree. Interval graph. 4 5. 5. 1. 2 DFS tree. % G2 &,    5%6  %6  .   $. .         %&'  (&'    

(759) 

(760)             

(761)   

(762)      

(763) .  .

(764)  .  . :.

(765)   % &'(  2     0    1 0      

(766) . 1 2 5%6     

(767)  

(768)  .           

(769)  

(770)            

(771)         0    1  5%6 . #  B   

(772)   

(773)    .  

(774)    4 5 

(775) %  6 

(776)      

(777)   

(778)      

(779)          

(780)    ! 

(781)  .        5%6  6   #

(782)     . #

(783) 

(784)

(785)  

(786)      0      1   0  

(787) 

(788)  

(789) .   

(790)   

(791)   #  1   ! ,    #     6  

(792)    

(793) 9  

(794)   #

(795)   )

(796)      

(797)   

(798)   

(799)   

(800)     

(801)   

(802)   .     

(803)   #    

(804)         

(805)     

(806)  " #   

(807)  

(808)      5%6  

(809)    ! 

(810)          %6  )

(811)      

(812)     

(813)   

(814)  "   

(815)   

(816)   .     #

(817)  9  

(818)   

(819)  

(820)      %6  B

(821) 

(822)     4  " 

(823)   

(824)   !     4 %6  

(825)    5%6  %6 

(826)  

(827)    

(828)     

(829)    

(830)  

(831)    

(832)    ; 

(833)   ,!   0  1"  

(834) 

(835)   (                

(836)       . 

(837)    . ;  !          (  (  ( " 

(838)    

(839)        4  

(840)  

(841) 

(842)  ( 

(843) "  

(844)   > "  4 

(845)            (   

(846)   #  ! !   

(847)   .          

(848) 

(849)   (    (  3   

(850)   !  

(851)       ! 

(852)   % @   ,    

(853)       * 

(854)  @  

(855)    

(856)  );' .   % "   

(857)    

(858)      

(859) .   .             

(860)  .

(861)         . . #   4 

(862)  .  *

(863)  @"    

(864)  

(865)   

(866)  ¼ ¼ ¼  !

(867)  

(868) 

(869)  ( 

(870) "  

(871)   

(872)  )

(873)   

(874)  ! ::.

(875) 1 2 3 4 5 minimum interval covering(3)={2,5}. % @2 &,    

(876)      .   )  *  * 2  

(877)        

(878)    (    2 ! 

(879)         

(880)  .  (   (    

(881)  

(882)       % 

(883)     

(884) 

(885)       (  ' (  

(886)     

(887)     !       .

(888) ¼ #

(889) 

(890)

(891) ¼   , 3

(892)  (      (

(893)  

(894) ¼   ( ¼      (

(895) ¼ "   ¼  3 (  

(896)  4    

(897) #

(898) 

(899)   

(900) ¼ " "  . 

(901) 

(902)    (  ( ¼     

(903) 

(904)       5! 4" ( (     (¼ (     ="  )   (    ( " 

(905)  

(906)     (¼   

(907)    

(908)  

(909)     ( 0   " 

(910)  ! 4   #  

(911)    1  

(912)  #   #    (¼ #

(913) ( 

(914) ¼ " #

(915) 

(916)     

(917) 

(918) ! 

(919)   

(920)  

(921)

(922) ¼    ,  

(923)  6 :  

(924)    " 6   

(925)      6    J  

(926)        

(927)      

(928) 

(929)  )     #

(930)       ¼. 

(931) #       . 

(932)  4       4       !      

(933)   !  "   

(934)   

(935)    

(936)        4   

(937)      " "      #

(938) 

(939)  J         

(940)   !   .  

(941)    " #          

(942)  

Références

Documents relatifs