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¥  4E ÉÎÍÌËÊ 9”Cg§pD|d3 s2Fa1É x 8  E 0 @0 E ¥ ¥ § x ¥ T 4 C cž ¥ 3 8  T H ¥ C E ¥ §0 2 H 3 E € x H x 8 # ¥  § ¥ x E 4 C cž 8§ ¥ s 8 4 H ¥  2 2 2 2 p¹t¨¡{%Y’%z8YwvdÀ’dŸ6YGYRU3 “UGU9zCd­‘rd3 t2yz4%ygøWA¨W6Y¨s s2%sE94 ¦3 yC y 24 @ uE§ 7 ¥ @4 x 8 C4 3¥ 3 8 u B 80 H ¥40 ¥ 3 3 80 C c 2§E – ¥ @4 24 3¥ 2 84 H ¥C c §¥ 7¥4 2 3E 2 Ó Ó ¥§¥ UdGS%Pg«pD¨g%pDphARYdg%hWg%8 9agPg"”sxSbsHpWd¢¨¨A3 ¶G‹C ŸvA¨r` Ó Ó ßØ Ú ×Ö ï Ó ÞRÞ«A˜Õ ë Ó § § ¦Ó Í ÌÓ ßØ Ú ×Ö É Î ¨™)¨ÞRÞ«RrË Ú Ê É Ù Î É Ø ×Ö ÙÜ Ý Ü × É Î Ë É Î Õ Ô ÉÓ Í Î «× rR˜¨«¤¨)Ë ¤ ÙÜ Ý Ü × É Î Ë 84 ¥  E 7 2 H H E  T H ¥ @  u B 800 E ¥ T 3E0 2 @ ¤C 7 3  ¥ T Ð @ 2 q 84 @ uE§ 7 ¥ @4 x ‘S)r—g3 t2‹1g3 gdjPYdC s2¦Dph%|w"a¨FC ¦­i 9d8 Yw|"{t0U@ ‹A"dGS%—gdp8 4 3 ¥ 3 8 u B 80 H ¥40 ¥ 3 3 80 ¥ @4 7 2 2 4 3 ¥ 2 € Í ×Ë ï Ó Ó € § ¥ 7 ¥4 2 3 E Ð œ ¥ 4§ ¥ – @0 E ¥ 8 4 3 2 C C E 8 4 € 4 8 dwD‘R"ARYddR|dAWd3 «”sxSbsHU‘¨ÀqÕ ë ­9%A3 ¡GŸA¨“‹"%7 9G1¹9d3 xt«dUG«dYhgdR’C pdGS%‘Usx«tYPA¢{9aŽg{gg¨tWg¢a¨‹p­k‚dS‘%¹‘Yˆ|k‚Skd‘%kŠ 2 € H 3 E € H ¥40 ¥ 3 3 80 2 @ u E § 7 E 2  ¥4 84 @0 §E ¥C 4C§ ±´ @4 u ¥ H ¥C c 3E 0 ¥ o ‡€ ¼ ~ ¼ Å  Æ ÅŠ }~€ Š~ ¼ ¼ Å 84 2 H ¥ 3 2 24§E u ¥ T 3E0 C¥ 24§¥ – C 2 3¥ @4 € H ¥40 ¥ 3 3 80 2 H 2 @ uE§ 7 H ¥40 ¥ 2 H 3 c 3E xI ¤C¥ 24§¥ 3 vYd%8 S4 Sa¦ywva¨‹Rs0S%DA4 ¶dApdYhgdR9C d6C yda9|YhA§ g¦ga˜uãâ9Rs0S%– 8 q4  3E 3 ¥ ¥ q4 ¥ T @4 E u E 2 ¥§ ¥ @4 2 }~€ Š~ ¼ ¼ Ŋ ¥ T 84 2 E C 2 @ u E § 7 H ¥40 ¥ 2 H 3 c 3 5 ¤ ÷0 Ü Ü Ü 0 ! ™hFa“¨wQ¨w6AG¦wvC ’A¨dA%sxºk‚Skd‘%kywADsH9˜C 1gG9¢ARY§ d¦g6¢¦˜R¥¥¥¤× á § x Ú à ÇÞ Ð ßÞ ÐÁ 4 E @4 @0 cC € C ¥ 2 4§ ¥ – x 8 Ç Ý Ð0 Ü Ü Ü Ð0 Û ÐÁ ¥0 3 ¥ c µ ¥C E 2 Ç Ú 0 ÙÁ × Õ @ u E § 7 E ”8d­)hR0 6 "–ÃyGU“{¦g9¨hs09%p‘¿ph¥¥0 6 R––øŸh¦d¨PC vFTlØÖgG9¨ 23 € ‰  5 ¤ 4 œ ¥ H ¥ 3 ± ¥ H €‡€ ¼ ~ ¼ Å  Æ ÅŠ }~€ Š~ ¼ ¼ Ŋ C 2 84 @ uE § 7 H ¥40 ¥ 2 H 3 c 3E ¥ 2 2 H T cC 84 2 @0 §E ¥ ‚Y¸USd3 dg¨t¹‚‘SdY‘Yk’ˆ‚9ˆ‘dYˆv4 6Ayda9vARY§ dUdyattH«– g¿gˆ‘§C ¢{9aC 4C§ ±´ @4 u ¥ H 7 2 2 H 8 B  ¥4 2 § u 8§ u uE  T H ¥ @  u B 800 E 3E0 4E @4 C4E x 4C  u 2 C ¥ @4 x 8 ¥ g{gg¨b"d3 «”sxd«D YatESddgGvFdC s2¦D‘R%pG’a¦A†AGY”¥—¨¥ ¦¢tB9DdAgpDd3 Ô j d c P b Y f b Pv d • ˜ U P R f t H b ˜ t W US U H c W R d — P d b P RS — YS w dS mS d b P Y Y f b p P P U Y f f U vv W H U P “ U å qe44Qh4æV"eefæ"hux`QiT)gpQe`qhx4"44"gQqV§sV¥V"44eeh4¥4fxh¥f5V`s"X3pTsä ¤C§¥ T B c 3 ¥C¥ @4 x C 24§¥ u8§ u  x ¥C c ¥§8 9A¨pv¦—¨g‘‘8 ¨¥ 9A¨wg¹¦zcg—DB  3E B ¥ ¥C  E @C ¥ o ¤ u  §¥ – 8  2 4§E u 4 8 3 3E 0  E –§ ¥4 2 @0 cC 8 q4  3E € C H § 8 q § ¥ @4 8 3 G‹“¨ws%¦6py¦atE9%9v s%sE9AGU—%gda¨|C %¹¨A3 ¸{¦"™hvGr9t9w—¨gpQI  … ~ € Å ¼ ‰ ‡ ¼¾ ‰€ ¼ Å aSY‚YdYdø%{dYˆŠ ~ ¼ Å … ŠӀ ¼¾ Å ‡¾ } … ~ €¾ ~ ~… ‰ Í Í ×Ë Ù Ý Í Ëç Í ×Ë É Î ËË } ¼ ‰ Í Í ÕË Ù Ý Í Ëç Í ÕË É Î ËË ! ‰ È … ~€ ¼¾ Å ½ dY6Y†"‘ÀY¨ÒÑÀY6SY‚øSrhY¶™™øq¨«R˜™øqÀÐdYy¨™ÀqS«A˜™ÀxÀχ 5SaS‚døv¿‚€ Î 4E @4 ¥CE0 ¥ @4 C E  E 2 2 € ©0 E 4C ¥ @4 3 8 H ¥ E 4C Ð @ 2 q 7 2 § c H  E –§¥4 2 ¥ 2 4 ¥ @ a¦A—G|dAvˆq yC F4 X«9—g"PˆS|"s0¦@ ‹g3 9dtX%¹3 WDtBS—dA4  s%sES¨FC ¨¨™ÀqS«Rr™ÀxÀÌh¦3 dŸ‘S)¨«v¦a¨S«¡“YAddD‘RAGÏwT  2 4 3 ¥ C C ¥ 2 Í Í ×Ë Ù Ý Í Ëç Í ×Ë É Î ËË ¥0 2  & Ù Ü Ý Ü × Ù Ý Í Ë H 3 E Ù Ü Ý Ü × É Î Ë  T H ¥4 c u B 80 ¥§ E @ 4 8 3 ¥ qÁ Ú ×Ö Ù Ý Í Ë H 3 E Ú ×Ö É Î Ë C§ ¥ T B c 3 ¥ @4 x 8 4§ ¥ u 8§ u 4 3 E 4§ 8 u 2 7 2 q  x ¥ @4 ¥ 2 4 8 d@ øj)R˜¨)p¦Gj«RrrwvU¢dA˜p›hA¨wg‹GS%pDsB¸g3 ’t8 t‘z8›g¢Rs0S%r© É 7 2 @ 4  3 E 8 H 4 8 3 H2 H Ú Ö Ù Ü Ý Ü × É Î Ë x2 H ¥ § ¥ T B c 3 ¥ T C ¥ H 8 3 ¥ @ 4 H c 8 q q 8 Êg3 Ua1bFdPsd‹¨˜S««s"wvU—pvb«d—gPs¦w¤tr` Nontree (Back) edges: Tree Edges: º ¼ ° ® ¹­ »­ ¬­ º­ ¼­»¼ °­° ®­ ¹­¹º »­»¼­» ¬­¬® ¯°­° ¹º­ ­¹º¹º ÄÄ »¼­»¼­ ­­»» ¬®­ ­¬®¬® ¿¿ ¯­¯° ¯­ ­¯¯ ÄÄ ¿¿ ÄÄ ¿¿ ÄÄ ¿¿ ¸ ­Ä ¶ ¿ ´ ² Æ ±­ Å­ ¸­¸¸ ¶ ­¶ ´­ ­´ ²­ Æ­ ±­±² Å­ÅÆ Ã Ã Ã Ã ­­µ Á Á Á ­³´´ È È È È ­ ¾ ¾ ¾ ¾ ­ Á ¶ È ÄÄ­­­­­Â¿­­­­ÀÁ­­­­­ÇÈ ±²±² ­­­­½¾ ÅÆÅÆ Â­Â­Â­Â­ÂíÀ­À­À­À­Ç­Ç­Ç­Ç­Ç­ ½­½­½­½­ À­Á ±²­ ÅÆ­ ··¸­·¸ µµ¶ µµ¶ ³³´ ·­ ­·· µ­µ­ ­­µµ ³­ ­³³ 2/17 8/9 7/14 e Results of DFS 1/18 4/5 3/16 (pre(v)/post(v) shown next to each vertex x) 6/15 11/12 10/13 ¤ Ð œ ¥4§¥ – @0 E ¥ 84 4 œ¥ «r¨A%"{%Y‹†d3   ¢   ¨ ¦ ¤ ¢ ÿ ©§¥£¡¢þ É Ç C C 8 § 0 € © 0 E T € H § E q § 8x € ¥ ¥ § 4 Á C 2 § 8 7 ¥ 4 E 0 § c 8x ¥ C ¥ @ 4 x 8 ¥ 3 8 8 4 2  E x C ¥ 7 H ¥  E 8 H  @ l9R)){%U¹dSGA%zXY¨AÃF¥ S%¨AGyg%z|dA‘p—g%‹A3 Xs%ÀF¨bY¹s%1tjÄo Î Î Î É @ uE § 7 H ¥40 ¥ 2 H 3 c 3E 2 C¥ 7 H ¥ CC 8§0 ¥ T ¥§¥ @4 3E ¤ H ¥ 3 ±¥ H TgG9PARY§ dUd"a›3 F¨bYvh|w—A¨g"G†¡ dYddbF 2C Í èË Ù Ý Í Ë H 3 E € Í èË É Î Ë ÷ Í ×Ë É Î Ë € H ¥ C § ¥ – E § 4 2 2 ¥ B2 4 ¥ @ 4 4 E € ¥ C c E 0 ¥ T € Ç 1 0 ÐÁ ¥ 7 H ¥ 3 ™Àx¨«¢UG¿‘™øq˜ø™ÀqXg%9AC ‹4 "Ds9PgDa¿YdGpwAP4©ø"%taE @0 cC  ¥4 3E0 c 8 ¬ ¤ H ¥ H 0 3 80 3¥¥ T CE @ 7 2 CC¥0 8§ u ¥C 8 q ¥¥§4 ¥ @4 x 8 4§E u E 2 C  4E @ UsYrGp%¿§dt¦c ¨U%hp¨p†a¦yd3 9¨h)AdDd@ ¢¨A¢dAg‘ŽAG¦vp3 —¨¥ s2—a¦A4 œ¥4§¥ – E 84 œ¥4§¥ – H ¥§¥ –80 2 H  q¥ 3 E B 8 x € x  84 4 @ 2§ B 8 x 8 7 ¥C¥ @4 &C¥ 7H ¥ CC 8§ A¨‘"AW%’%9R9C gv ’¨g‘lpz§Ps¥9z4¥ "AŽ7 9Q£ ‘z§"%¢¨dA§9%tŽ%§¡ ¤ H ¥ 3 ± ¥ H 3 2 Í èË Ù Ý Í Ë H 3 dYddbUdc C “¨ÀqS«W¦aE €‘¨ÀqÞϙÀq§dY¨ˆS’C r4 vDsB9Wg—G™gGp†P43ÃvbYuGÄ{¦«sAuG0 Í èË É Î Ë ÷ Í ×Ë É Î Ë € H ¥ C § ¥ – E § 4 2 2 ¥ 2 4 ¥ @ 4 4 E € ¥ C c E 0 ¥ T Ç 10 ÐÁ ¥ 7 H ¥ 3 E @0 c C  ¥4 3 E c 8 ¬ ¤ ¥ ¥§4 @0 § E ¥C 4C§ ±´ @4 u ¥ H ¥ @4 2 § 84C ¥0 3 E 3 E 84 œ ¥4§ ¥ – E B 8 x 8 7 ¥C ¥ @4 & C ¥ 7 H ¥ ©0 E ˜§A"{9GY†d‚dAdbWdA"3 FRUGuG“|%¢q‘z§1—dA­9%t|{%Ž£ ¤ Í èË É Î õ Í ×Ë É Î Ë ¥ C c E 0 ¥ T Ç 10 ÐÁ ¥ 7 H ¥ 3 E @0 c C  ¥ 4 3 E 0 c 8 ¬ ¤ ¥ ¥§ 4 @0 § E ¥ 4 C § ±´ @ 4 u ¥ ‘™ÀxË ö™øqyda¨Yw­P2©ÃW%bY"G›{¦tY"a¨›˜­Y¨A"{9aC g{gg¨tH ¥ @4 2 Ç 2 @ 3 E @4 § ¥ @4 8Á 4 3 E H 3 ¥0 C ¥ H E 84 œ ¥4§ ¥ – B 8 x 8 7 ¥C ¥ @4 & C ¥ 7 H ¥ H § E q§ 8 dAÄ3 rgsHtU0 GUrdA%øpGg¦R9b¡©AšA¨“E ‘z§©%vdAt%tu9a( & C ¥ u 4 ¥ ¥§ @4 84 2 H ¥ 2 CC  0 ¥ T 3E 0 @ uE § 7 ¥ @4 x C ¥ 7 H ¥ ¥ ¥§4´ 3 8 3 ¥ @ ¤ § 84C ¥0 3 E 3E 84 ¥ ¥§4 ¥ @ 9¨pQ—¨AdA‹3 "Yd± 9GsE|w"a¨›da9—dA‘p8 %bY—¨A‚dd—g­i 9RUG›a1¨A—dA4 3 Ÿ¨A¨Ž˜pA”§|g3 p%b9¨bYy¦¶tg3 a1da9%†dAY‘r%t¥ A{g%gbs%‘d˜a¦A9%w9d@ 2 œ ¥4§ ¥ – E B 8 x 7 2 8 7 € C ¥ 7 H ¥ ©0 E T q 8 ¥§E @ uE § 7 ¥ @4 x 8 C¥ 7 H ¥¥§4´ 3 8 3  E 4 8 3 4E @4 € §¥ –¥ q 8 € ¥ 2 4 8 © ¤ 3 q 8 @ C ¥ § E ¥ ¥§ 4 @0 § E ¥ C 4 C § ±´ @ 4 u ¥ H H 3 E C § ¥ T B c 3 Í ×Ë Ù Ý Í Ë « Í ×Ë É Î Ë 7 2  c C ¥§ ¥ @ ht0S%˜x¦†td—G|¨A“{9avg{gg¨t“¦aPwvUW™ÀqS«™™ÀxŸd3 S4 U¨A’d­i ¤ q  ¥ T q 8 @C ¥ 3 8 ¥ @ C E @0 cC € @ uE § 7 H ¥40 ¥ 2 H E 3 8 @0 §E ¥ 4C§ ±´ @4 u ¥ H 3 c§ 8  E 3 E 0 ¥ F98 Yw"3 9d—g%—g4 G›{¦¦da9PARY§ d‹›"9aC Ad‚dAdb›dA‹C ›G|po † f e m  66†)®ô ó … „ n … i ‡p¿˜)kd™ êi f n  m … “ „ †  ò g f „ e … E•X)†r¢dw)66p6˜d™ ñ ¤  ( ³ 2 C 3 24C¥ cµ ¥C¥ @ §¥ qC 3E 84  8 84 ¥ @ g¿@FC Fd8 S¨gS|dA4 ¨wgG1A¹‘A—d­i É  C c 8 ¥ 3 E  c B2 C H ¥ 4 c 0 ¥ œ ¥ ¥ T 3 E 0 C © C E 4 3 ¥ H 3 ¥ u ¥ H I 9d%da94 Uvt9PYgRY—w"a¨Fa94 ¨t¦wb¦ 23 4E @4 7 2 B cCCE €C ©CE4  E ¥4  u B 80 84 ¥ 2 B 2 2 B ¥ @4 2 4E q 3¥ @ i ¤C4 3¥C¥§ u¥§ 2 ©CE GU"d3 ‹UG™9GS«s%1A¨¥ ¦D‘R›‹DsBS4 ¦c tBU3 ‹1g—C |a¦@ p¨g­“9A¨Ad¨A—4 yGS4 ¥ @4 ¥4 c0 ¥ œ¥ 84 ¥ 2 4 ¥ @4 CE 4 ¥§ u§¥4 2 ¥ q @ 2 @ q € § ¥ T B c 3 ¥ 2 2 C 8 u E  T H  ¥ T  2 œ¥4§ ¥ – @0 E g—gRYW’DsBS’dA‘ay¨Ad3 rw“s0¦†g9¨pv¦’%– S4 9wpP¥ pGsEPC pA¨1{%Y¥ ¥C 8 u u cC &  E c4 3¥ –¥ §¥ qC 3E 8  E 2 q ¥ q 4E @4 3 24C¥ cµ  c0 2 H ¥§ 8 B 4 @ C E 2 ¥§¥ ˆwgd¢) t¦A%—¨™dGuC ˜ssF‹wGUp8 S¨gSW4 UR1( d“%¢y S7 t29yrC 1A¨r` ¤ ý ¼¾€ … Å ‡ ! 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This note was uploaded on 08/01/2008 for the course CS 170 taught by Professor Henzinger during the Fall '02 term at University of California, Berkeley.

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