Partitioning a Square into Rectangles: NP-Completeness and Approximation Algorithms
28
0
0
Texte intégral
(2) Laboratoire de l’Informatique du Parall´elisme. SPI. ´ Ecole Normale Sup´erieure de Lyon Unit´e Mixte de Recherche CNRS-INRIA-ENS LYON no 5668.
(3)
(4)
(5)
(6)
(7) .
(8). .
(9) . ´ Ecole Normale Sup´erieure de Lyon 46 All´ee d’Italie, 69364 Lyon Cedex 07, France T´el´ephone : +33(0)4.72.72.80.37 T´el´ecopieur : +33(0)4.72.72.80.80 Adresse e´ lectronique :
(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)
(48)
(49)
(50)
(51)
(52)
(53) $
(54)
(55) &
(56) $$
(57)
(58) #.
(59)
(60) $ +
(61) Æ #
(62) , $ -
(63) &
(64) ! . ! /0 ! . ! ! ! /$ 1 . .
(65) $ +
(66) #
(67)
(68) 0 0 /2 $ +
(69) &
(70)
(71) &
(72)
(73)
(74)
(75)
(76)
(77)
(78)
(79) & "$ 1
(80) ,
(81)
(82)
(83)
(84) 3/2
(85)
(86) 3/0
(87)
(88)
(89)
(90)
(91) &
(92) "$ 4
(93)
(94) , $ +
(95)
(96)
(97)
(98) .
(99)
(100)
(101) 5
(102)
(103)
(104)
(105) $ 0.61. 0.39. s 36/61. s. s. 1/3. 1. s. 25/61. 3. 2. s. 4. 5. 1/3. 1/3. 1 &
(106) . $ +
(107)
(108)
(109) % . &
(110) .
(111) # $ 6
(112) &
(113)
(114) 5
(115)
(116) #
(117)
(118)
(119) $ +
(120)
(121) # $ 7 &
(122)
(123) $ 7 /
(124) # %8 79: #
(125)
(126)
(127) " %8 :1; #
(128) "
(129)
(130) % . $ 7 3
(131) &
(132) %8 :1;< 7 . %8 :1;$ 7 0 = # $ > ? 5 7 @$.
(133) . +
(134)
(135) 5
(136) :&:& : :::
(137) "
(138)
(139)
(140)
(141) 5 .
(142)
(143) $ % :::
(144) 5 !
(145)
(146) # $ & 5
(147)
(148) & Æ .
(149) 7 A1%1-B C3D$ 4
(150) EF 5
(151)
(152) 5 $$ && # $ 1.
(153) . $ +
(154)
(155) " $ +
(156)
(157)
(158) . $ 8
(159) .
(160) . 5 <
(161) .
(162) 5
(163) 5
(164) 5
(165) $ 111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 0 1 000000000000000 111111111111111 0 1 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111. . +
(166) :::
(167)
(168) $ 9 G .
(169)
(170)
(171) . 5 $ 6 5
(172)
(173)
(174) . & 5
(175)
(176) .
(177) 5
(178)
(179)
(180) . $
(181) / G $ >
(182)
(183) . $ # .
(184) ! $ #
(185) 5
(186)
(187) . $ +
(188)
(189)
(190)
(191) . #
(192) & H A
(193)
(194) #
(195)
(196) ! $ 1
(197)
(198) :::
(199) 5 .
(200) &
(201)
(202) I !
(203) J "
(204)
(205)
(206) $
(207) &
(208)
(209) .
(210)
(211) .
(212) I
(213)
(214) #
(215) $ 7 # I I 5
(216) I . $ :# I
(217) $ . .
(218)
(219) 5 5
(220) . 5 8
(221) 5<
(222) K
(223) $
(224) & #
(225)
(226) , $ -
(227) 5
(228)
(229)
(230)
(231) 5 5 $
(232) & #
(233) &
(234) .
(235)
(236) , I ! &
(237) J "$ .
(238) L
(239) #
(240)
(241) .
(242)
(243) $
(244)
(245) #
(246) $ :::
(247)
(248)
(249) ?G
(250) "
(251)
(252) .
(253)
(254)
(255)
(256)
(257) " #
(258)
(259) & .
(260)
(261) 5 &
(262) # $. . . .
(263) . >
(264) # $ >
(265)
(266) ! $ +
(267)
(268)
(269)
(270)
(271)
(272) # .
(273) . . .
(274) J " # .
(275) $% . &. . . .
(276) J " # . ! #
(277) I !. !". !". . . #
(278) . ! . . I !. +
(279) %8 79: " %8 :1; " . &.
(280) $% . . . I & . . I .
(281) . . . . . & . . . . +
(282)
(283)
(284)
(285)
(286)
(287) $ L
(288) .
(289) %8 79: "
(290) $ +
(291)
(292) %8 :1; " .
(293)
(294) & 7 $ . . +
(295) .
(296) # %8 79:
(297) .
(298) ' . . #
(299)
(300) J " (. . . . . !". ! . L ?
(301) Æ
(302) %8 79: # .
(303) ' . ) !. /.
(304) . >
(305)
(306) $ 1
(307)
(308)
(309) C/D$ L %8 79: B" %$ >
(310) . . * . .
(311) . +. * + . ! * * . ". ! . . ! . ! . . . . + ! . " ! ! " (. " . . ! . . . ! . (. . ! . . 7 % 5 %- C0D A
(312) +
(313) $. ##
(314) $$ %&$'()* >
(315) A $$ 77% %8 79: B"$ A ! . . $$ ! $ 7 77% %8 79: B"
(316) .
(317) . !. . . . 77% %8 79:. ##
(318) + $$
(319) .
(320) %8 $$ ! $ > .
(321) 3
(322) . $ >
(323)
(324)
(325) 77%
(326)
(327)
(328) G
(329) %8
(330) $ '? È ! J & "$ +
(331) $ : ! & !
(332) $ >
(333)
(334) " 77% 77% ""
(335)
(336) J @ " J @ " 3 J J " . ! / J " " ! @ J 0 " /" ! 3 J / " " ! / J / " " ! / J " " ! J " 3" ! " ! ""H. . 3.
(337) . . . . . . . . . . .
(338) . . . / M
(339) . . . 3 N
(340) .
(341)
(342)
(343) .
(344) /
(345)
(346)
(347) / 3$ +
(348)
(349)
(350)
(351)
(352)
(353) $ 4
(354) ?
(355)
(356)
(357) $ +
(358) . G
(359)
(360) * $ +
(361) ?
(362)
(363)
(364)
(365) $. , - $+. +
(366)
(367) .
(368) .$ L
(369)
(370)
(371)
(372) .
(373)
(374) ?
(375) .$ +
(376) .
(377) . "
(378)
(379) .
(380) .$. ..
(381) . . . . . . . . . b. a. . M
(382) . Ë Å Ë Å Ë Å. ËÅ. Ë Å Ë Å 0 . 0.
(383) . . . . . . . . . . . . . . @ L .
(384) $. , , $ . .
(385)
(386)
(387)
(388) ."$ > 0"$
(389) ? . # J $ % &
(390)
(391)
(392) ?
(393) @
(394) . "$ +
(395)
(396) 0" .
(397) @
(398)
(399) , . $ +
(400)
(401) 77% G
(402) .
(403)
(404) $$ " $ 7 !
(405) 5
(406)
(407) .
(408) 3$ +
(409)
(410) 77%
(411) G
(412) & $. -
(413) . +
(414)
(415)
(416)
(417)
(418) & G
(419)
(420) %8
(421) $
(422)
(423)
(424) %8
(425) $$
(426) & . !. . . . . " ! ".
(427) ! J :1;"
(428) :1; ! & $ +
(429) . . . . !. . . J :1; ". . J :1; ". . !. . !. . . . +
(430)
(431) & $ @.
(432) -
(433)
(434) &
(435) ! +
(436) . . . . . . . !. . . . .
(437). . . :1; ". . . :1; ". !. . ! :1; ". . . . . . : .
(438). . . . . .
(439). . . " ! " . . :1;. !. . . . . +
(440)
(441)
(442) %8
(443) $ +
(444)
(445)
(446) .
(447)
(448)
(449)
(450) 77%
(451) #
(452) #
(453)
(454) %8 $. ,
(455) $% ! & . . . . /". " " !- ! ". "" ! /" "" ". +
(456)
(457)
(458)
(459) % . 77%
(460)
(461) % . %8 79:$ . . +
(462) .
(463) # %8 :1;
(464) . .
(465) $% ' . .
(466) &
(467) J " (. #. . . !. !" . . . L
(468) Æ
(469) %8 :1; # .
(470) $% ' . ) !. O. .
(471) >
(472)
(473)
(474)
(475)
(476) ! * 0 1.
(477)
(478) $%. +
(479) * 0 1 + . / * 0 1 * 0 1. J " ! ! ! . 3 ! 3" J + . . . ! . . . . ! J3. . . . . ! ! . (. 0
(480) 2
(481) . ! ! ! !" ! !" " (. . 7 %3 5 %- % C0D" A 3
(482) +
(483) $. # #
(484) $ %& 1()* >
(485) A 3 $$ :7% %8 :1; B"$ A ! . . $$ ! $ 7 :7% . %8 :1; B"
(486) .
(487) . !. . :7% %8 :1;. # #
(488) 2! $
(489) .
(490) %3 $$ ! ! ! .
(491)
(492) 3 ! 3"$ >
(493)
(494)
(495) :7%
(496)
(497)
(498) G
(499) %3
(500) $ A . !. . . 3 >
(501)
(502) " :7% :7% ""
(503)
(504) J " J " J / . . . . ! ! ! ! 2. ".
(505) 0000000000000000000 1111111111111111111 1111111111111111111111 110000000000000000000 00 000 00 11 000 111 00 11 000 111 00 11 000 111 000 111 00 11 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 111 00 11 000 11 111 001111111111111111111 000 0000000000000000000 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111. 11 00 00 11 00 11 00 11. . . O M
(506) $
(507)
(508)
(509)
(510)
(511) .
(512) H +
(513)
(514) O$ >
(515)
(516)
(517) ,. .
(518)
(519)
(520) P
(521) # $. .
(522)
(523)
(524) P
(525) # $. .
(526)
(527) $. 7
(528) P 3
(529) . " " .
(530) 3
(531). 3
(532) +
(533)
(534) 3 . . . >
(535)
(536)
(537) G
(538) . . . "$ " . . +
(539) . " ! $
(540)
(541) $
(542) ? ! ! $
(543) $. .
(544) !. . . . . ! $
(545) & .
(546)
(547) $ A .
(548) .
(549) $ +
(550) . $ . . . ". . ! 3$ 1
(551)
(552)
(553) $
(554) ?
(555) G 3$
(556) $ . . . ". . . .
(557) 111 000 000 111 000 111 000 111 P 000 111 000 111 000 111 1 0000000000 1111111111 0 000 111 0000000000 1111111111 0 1 P 1111111111 0000000000 0 1 1111111111 0000000000. 11 00 00 11 00 11 00 11 P 00 11 000 111 00 11 000 111 00 11 0000000000 1111111111 0 1 000 111 00 11 1111111111 0000000000 0 1 000 P 1111111111 0000000000 10111 000 111 11 00. . . . . . . . J. ". ". 2 +
(558)
(559) :7% $ +
(560)
(561) & ! ! ! .
(562) G 3
(563) :7%
(564) $ +
(565)
(566)
(567)
(568)
(569) :7%
(570)
(571) & ! ! !
(572) G 3$ 7
(573) J
(574)
(575) . ! J 3. . . . . . . A ? !
(576)
(577) !
(578)
(579) $ >
(580) J " J " 2$ 7 .
(581) P P
(582) .
(583) :7% $ 7
(584) J
(585)
(586) . ! . . . . . . . A ? !
(587)
(588) !
(589)
(590) $ 7
(591) .
(592) $ +
(593)
(594) J " J " 2$ 4 .
(595) P P
(596) 2 .
(597) :7% $ +
(598)
(599) :7%
(600) G
(601) & ! ! !
(602) G 3$ +
(603)
(604) .
(605)
(606) % . :7%
(607)
(608) % . %8 :1;$. . .
(609) . !". +
(610) EF
(611) & %8 79:$ 4 &
(612) $ > 7 3$
(613)
(614) .
(615)
(616) Æ
(617)
(618) & & $ 4
(619)
(620) &
(621) 7 3$$/
(622) #
(623)
(624) $ 7 3$
(625)
(626)
(627)
(628) & $ .
(629) ! " . . . . . . . . . . . -
(630) . . ! / ! . ! / ! 3$. !## 3 7 %8 79: " %
(631) -LA%8 79: "
(632)
(633)
(634)
(635) . $
(636) -LA%8 79: "
(637) %8 79: " $
(638) . -LA%8 79: "
(639)
(640)
(641) %8 79: "$. - 4. >
(642) -LA%8 79: "
(643)
(644) "
(645) $ 8
(646) " $ L
(647) ?
(648) !
(649) $ +
(650)
(651) ,
(652) #
(653) .
(654) $. , . +
(655)
(656)
(657) . $ &
(658)
(659)
(660) $ $
(661)
(662)
(663) "
(664) .
(665)
(666)
(667)
(668) $. . " . +
(669)
(670)
(671)
(672)
(673) &
(674) ! 3 0 O "$ +
(675)
(676)
(677) + $ 8
(678)
(679)
(680)
(681)
(682)
(683)
(684) #
(685) $ ! ! !/ !3 !. !0 !@ !O ! $ K $ K $/0 K #0 5 2 /K 3$0 K 0$0 K 2K ! $0 K $O K $3 K $2 K / #. 5 ! 3$0 K 3 .$O K . !/ /$ K /$0 K / /$0 K 3 3$ K . 3$@ K . "#! 5 . !3 3$ K / 3$30 K 3 3$O K . .$/ K 0 .$2 K @ !. .$3 K 3 .$00 K . 0 K 0 0$. K @ !0 0$0 K . 0$O0 K 0 @$ K @ !@ @$O K 0 O$0 K @ !O 2K@. . . . . . + +
(686)
(687) "
(688) " ! ! J J " J " " J " 6
(689) $ ¼. ¼.
(690) &
(691) / O" ! .3"$ +
(692)
(693) ! J ! 3 $ +
(694)
(695) ! J ! 3 $ +
(696)
(697) ?
(698) ! J J J !
(699) 3 $
(700) $ . . .
(701) .
(702) .
(703) . . . . . . . L
(704) & $ +
(705) 5
(706) .
(707)
(708) $ +
(709)
(710) . /.
(711) ! ! ! J " ! J " ! .
(712) ! ! . " ! " ! . . . J J . . ".
(713)
(714) . +
(715) &
(716)
(717) "" " "
(718)
(719)
(720)
(721)
(722)
(723) " ! J J " & $ 4
(724) ""
(725) J "" ! " $
(726)
(727) &
(728)
(729) .
(730) "
(731) ?
(732) $ 1
(733)
(734) & " "$ +
(735) ?
(736) " ! "
(737)
(738) . !. !
(739) ! " ! ".
(740) . !.
(741)
(742) 5 5 "
(743) + $ +
(744) . $ +
(745)
(746)
(747) ! J J J $. 63. +
(748)
(749)
(750) $ +
(751) C/D$ +
(752) Æ
(753)
(754)
(755) $. !## %73 8 3 9
(756). 1.
(757) 7 /
(758) I
(759)
(760)
(761)
(762)
(763) $$ ! ! $ L
(764) %8 79: "
(765) ! " ! "
(766) " $ %
(767) I
(768) ! /$ 4 ! ! " J " $
(769) . &
(770) I
(771) ! $ . . 3$ +
(772) ?
(773) 3.
(774) ratio (sum of perimeters)/(lower bound). Experimental results using 2000000 random tests per point. 1.1. 1.05. Average values Minimum values 1. 0. 5. 10. 15. 20 25 Number of processors. 30. 35. 40.
(775) . 3"
(776)
(777) $
(778)
(779)
(780) I
(781)
(782)
(783) ! $ +
(784) $.$ +
(785)
(786) $
(787)
(788)
(789)
(790) $ >
(791)
(792)
(793)
(794)
(795) E F .
(796) ?
(797)
(798) $. !## 8 3 9
(799)
(800) .
(801)
(802) &
(803)
(804) & . I 3 / ! . . !" . + ! " . I !. !. !. . . . . J ". +
(805) C/D$ .
(806)
(807)
(808)
(809) "$ ! $$
(810) . .
(811)
(812)
(813) $ L
(814)
(815)
(816) $$
(817)
(818)
(819)
(820) - . $ . " # ! ".
(821) . ?
(822)
(823) & $ >
(824)
(825) ? ?
(826)
(827) G
(828)
(829) . <
(830)
(831)
(832)
(833)
(834) $ +
(835)
(836) 5
(837)
(838)
(839) ? $ ..
(840) !## 8 9
(841) , > Q" #
(842)
(843)
(844)
(845) 3
(846) .
(847) !
(848) $ >
(849) .
(850) ! $
(851) .
(852)
(853)
Documents relatifs
falciparum a été significativement favorisée par le séchage au four à micro-ondes avec une prévalence de 8 % en séchage normal contre 12 % En conclusion, l'usage du four à
D’après les tests effectués pour les différents modes de fonctionnements, les résultats obtenus montrent que les performances de poursuite de la vitesse et de flux
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
To cover the whole texture and avoid space loss, common schemes pack a number of tiles that is a power of two.. We show that numbers of tiles like 5, 13, 17, 25, 34 and others can
In fact, we will prove that any instance of a problem Π in NP (expressed as described previously) can be transformed in polynomial time into a CNF (i.e., an instance of SAT ) in such
A Proposal for a Heterogeneous Cluster ScaLAPACK Dense Linear Solvers Vincent Boudet, Fabrice Rastello, Yves Robert.. To cite this version: Vincent Boudet, Fabrice Rastello,
Dense Linear Algebra Kernels on Heterogeneous Platforms: Redistribution Issue Olivier Beaumont, Arnaud Legrand, Fabrice Rastello, Yves Robert.. To cite this version: Olivier
Les clous plaques ou lames plaques monoblocs, la vis plaque de JUDET ou le clou gamma paraissent, par leurs qualités mécaniques, adaptés à cette objectif, aussi bien
