Download Asymptotic Notation - Introduction to Algorithms - Lecture Notes and more Study notes Algorithms and Programming in PDF only on Docsity! ! " # $ % & ' ! $ ' ( ) * ( + # ( " ! , - " ! . " / & $ 0 1 2 3 4 ! " ! * 5 ( " ! # ! * " ( , 6 7 8 7 9 : ; < ( * ! * " ( : = # > : % " 4 ! * " ) * ( > 5 ( 5 : ? ! # ! * " ( 3 ! + " : @ ! * ! ! * " ( 3 ! + " : A # ! 3 ! + " B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b K f 1 1 1 6 g ( " ! # ! * " ( h 4 4 @ " ( i : j h k i l 6 h m h k i i * . n 5 " ( ! o 7 k p 5 + ! + # !q r j h k i r o m h k i 7 s k t k p $ : $ > $ u k v l 6 h k w i h o l & 7 k p l u i x l * . ( ( y " ( g z # { | # ) * ! x k v } k w # . ( 5 ! * " ( } ( " ! # ) } @ ! z * ! * ( > 5 # . ) ) * ! " " ! * " $ ~ + 6 g ( " ! # ! * " ( * ) " 4 4 } @ ! 5 " ( ( * ( ! $ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d ~ + " > + @ * ( > ) " 4 4 } 3 ! ( ! # ( z + # ! 6 g ( " ! # ! * " ( 3 # ( $ ~ + * ( " . 6 h m h k i i # #
" . . ( 5 ! * " ( , 6 h m h k i i l j h k i , n 5 " ( ! o 7 k p 5 + ! + # ! q r j h k i r o m h k i 7 s k t k p ~ + ( u k v 6 h k w i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b f j h k i l k v 6 h k i 3 # ( j h k i l k v h k i . " " 3 h k i 6 h k i , 6 g ( " ! # ! * " ( * # ( ( " ! # ! * " ( $ A # ( " ( ! " # j h k i * # ! ) # ! 6 h k i $ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b H 8 1 1 1 1 8 h m h k i i l j h k i , n o 7 k p 5 + ! + # !q r o m h k i r j h k i 7 s k t k p $ > $ k l 8 h ) > k i h o l & 7 k p l & ¡ i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b ¢ 9 1 1 £ 1 9 h m h k i i l j h k i , n o ¤ 7 o v 7 k p 5 + ! + # !q r o ¤ m h k i r j h k i r o v m h k i 7 s k t k p $ > $ & u k v u k l 9 h k v i q r &¥ k v r &u k v u k r ¦¥ k v q r &¥ k r & u k u r ¦¥ k q r k r u k § r ¦ k u k § r ¦ k § r k h o ¤ l & ¨ ¥ 7 o v l ¦ ¨ ¥ 7 k p l § i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b E ~ + " 3 & , / # * ( > 5 " ( ! # ( ! # ( ) " z g " ! 3 " ( © ! 3 # ! ! $ h 2 ª ! ( * ( « # 3 4 ) $ i ¬ ! * < 5 # ! * " ( h ( " ! 4 " " . } ª ! + # ( z # * ( > i , # ( 5 + " " 5 " ( ! # ( ! @ * > ( " > + ! " 3 # + * > + g " ! 3 z # 3 4 " ! + ! 3 $ ~ + " 3 u , h 6 # ( 8 i ® 9 h / . ! # ¯ # ) ) # ° « 5 * $ i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b ± ² 1 £ ³ / * " ) * ( > * ( ! > # ) } * ´ ( ! * # ) | # ! * " ( } ! 5 $ h ) # ( # . z ! * 5 µ i ¶ · ² 1 ¸ 1 h 3 " ! > ( # ) i & $ ¹ . " 3 " . " ) ! * " ( u $ º » @ * ( 5 ! * " ( ¦ $ ² 1 . " 5 " ( ! # ( ! B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b L ³ 1 å £ 1 " ( ! 5 ! * " ( " . 5 * " ( ! . " ¼ h k i l ¼ h k ¨ ¥ i ¼ h k ¨ u i k v , n2 T( 41n) T( 2 1n) T( 8 1n)T( 161 n) ( 41n) 2 ( 161 n) 2 ( 2 1n) 2 ( 8 1n) 2 16 5 n2 256 25 n2 Θ(n2) (165 ) 3 n2 n2 T( 41n) ( 41n) 2 ( 2 1n) 2 T( 8 1n) n2 n2 ( 8 1n) 2 ( 41n) 2 … geometric ¼ h k i l k v h & æ¤ ç h æ¤ ç i v Ù Ù Ù i l ¤ ç¤ ¤ k v l 9 h k v i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b K ± 2 ( " ! + 5 ( 5 , ¼ h k i l ¼ h k ¨ ¦ i ¼ h u k ¨ ¦ i k ,Î 5 * " ( ! , … … 9 n 9 2n 9 2n 9 4n n 3 n 3 2n n n n Total: O(n lg n) log3/2 n B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b K L ¶ ¶ ¶ · ¸ 1 è " ¼ h k i l é ¼ h k ¨ ê i j h k i }é t q # ( ê Ê q } # ( j # 3 4 ! " ! * 5 # ) ) 4 " * ! * 2 @ ! # 5 ! ) , ¼ h k i * ( ! * 3 . " # ( # ) > " * ! + 3 ë * ! © ( ( " z ( } @ ! z " ( " z ! + # ! , é @ 4 " @ ) 3 " . * ì k ¨ ê # " ) 5 * ) } # 5 + * ( ! * 3 ¼ h k ¨ ê i $ j h k i * ! + 5 " ! " . * * * ( > 4 " @ ) 3 h @ . " + # ( i# ( 5 " 3 @ * ( * ( > ! + ) ! h # . ! z # i $ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b K Ç Î 5 * " ( ! . " ¼ h k i l é ¼ h k ¨ ê i j h k i , f(n) a … a … f (n/b) f (n/b 2 ) f (n/b 2 ) f (n/b 2 ) a … f (n/b) f (n/b 2 ) f (n/b 2 ) f (n/b 2 ) a … f (n/b) f (n/b 2 ) f (n/b 2 ) f (n/b 2 ) … n logb a Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) logb n f(n) a f (n/b) a 2 f (n/b 2 ) Θ(nlogb a) Total: Θ(nlogb a) + ∑ j =0 logb n –1 a j f (n/b j ) a … a … a … a … a … a … a … a … a … … B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d G 4 ) * ! 4 " @ ) 3 * ( ! " é 4 # ! # ! ) " > í k ) ) , é Ú î Û ï Ü l k Ú î Û ï ð B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d K ñ " . ) # l é Ú î Û ï Ü l k Ú î Û ï ð ? ! # ! * ( > ! + 5 ( 5 * ) ¼ h k i l j h k i é ¼ h k ¨ ê i l j h k i é j h k ¨ ê i é v ¼ h k ¨ ê v i l j h k i é j h k ¨ ê i é v j h k ¨ ê v i Ù Ù Ù é Ú î Û ï Ü Ý ¤ j h k ¨ ê Ú î Û ï Ü Ý ¤ i é Ú î Û ï Ü ¼ h & i ~ + } ¼ h k i l Ú î Û ï Ü Ý ¤Þò à p é ò j h k ¨ ê ò i k Ú î Û ï ð ¼ h & i è * ! ! 3 * ! " ! # ) * * * " ( ó 5 " 3 @ * ( # ! * " ( 5 " ! ë 5 " ( ! 3 * 5 " ! " . " * ( > # ) ) * ì g & @ 4 " @ ) 3 $ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d d ¶ 1 ~ + 5 " 3 3 " ( 5 # ,Î ( ( * ( > ! * 3 " 3 * ( # ! @ 5 " ! # ! ) # Î ( ( * ( > ! * 3 ( ) * ! * @ ! ! + " > + " ! ! Î ( ( * ( > ! * 3 " 3 * ( # ! @ 5 " ! # ! " " ! " , 5 ( 5 " ) ! * " ( ! 3 * ( @ " 3 * ( # ( ! ! 3 µ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d 0 1 1 Ï 1 f j h k i k Ú î Û ï ð Ï , j h k i l 6 h k Ú î Û ï ð Ý ô i . " " 3 5 " ( ! # ( ! õ Ê q $ j h k i > " z 4 " ) ( " 3 * # ) ) 1 ! + # ( k Ú î Û ï ðh @ . # 5 ! " k ô i Ë * > + ! " . # 5 + ) ) * ( 5 # > " 3 ! * 5 # ) ) . " 3 " " ! ! " ) # h & 7 é ¤ 7 é v 7 Ù Ù Ù i h ! + 5 # > " 3 ! * 5 # ) ) . " 3 ) # ! " " " ! i / # 5 " ( ! # * ( 5 " ( ! # ( ! . # 5 ! * " ( " . ! " ! # ) z * > + ! ë * $ $ } ! " ! # ) * 5 " ( ! # ( ! ö ñ " . ) # $ h Ë + # ! # ! + Ì i * ( 5 z ( " ( ) ! 3 * ( ¼ ! " z * ! + * ( # 5 " ( ! # ( ! } 5 " 3 4 ! * ( > z " " ( # ! ) # * ÷ 5 * ( ! µ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d H 1 » Ï Ú î Û ï Ü Ý ¤Þò à p é ò j h k ¨ ê ò i l 6 ÁÂÂÂÃ Ú î Û ï Ü Ý ¤Þò à p é ò h k ê ò i Ú î Û ï ð Ý ô Ä ÅÅÅÆ l 6 ÁÂÂÂÂÂà k Ú î Û ï ð Ý ô Þ ò h é ê ô ê Ú î Û ï ð i ò Ä ÅÅÅÅÅÆ l 6 ÁÂÂÂà k Ú î Û ï ð Ý ô Þ ò ê ô ò Ä ÅÅÅÆ l 6 ÁÂÂÂÂÂà k Ú î Û ï ð Ý ô ÁÂÂÂÂÂà ê ô Ú î Û ï Ü À & ê À & Ä ÅÅÅÅÅÆ Ä ÅÅÅÅÅÆ l 6 Áà k Ú î Û ï ð ÄÆ Ë + # > # * ( z ø Þß à p â ß l â Ü ã ¤ À & â À &~ + ¼ h k i l 6 Áà k Ú î Û ï ð ÄÆ B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d ¢ Ï j h k i l 9 h k Ú î Û ï ð ) > ß k i } . " " 3 5 " ( ! # ( ! ¾ t q j h k i # ( k Ú î Û ï ð # 3 ! " z * ! + * ( # 4 " ) ) " > # * ! + 3 * 5 . # 5 ! " Ë * > + ! 5 # h # * ! + 3 ! * 5 # ) ) * . ¾ l q i ¼ h k i l 9 h k Ú î Û ï ð ) > ß ã ¤ k i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d E Ï ù j h k i l 8 h k Ú î Û ï ð ã ô i . " " 3 5 " ( ! # ( ! õ Ê q# ) " } ( y > ) # * ! 5 " ( * ! * " ( { , n o ¿ & # ( k p Ê q $ ! $ é j h k ¨ ê i r o j h k i s k Ê k p j h k i > " z 4 " ) ( " 3 * # ) ) . # ! ! + # ( k Ú î Û ï ð Ë * > + ! * > " 3 ! * 5 # ) ) 5 # * ( > Î " " ! 5 " ( ! # * ( 5 " ( ! # ( ! . # 5 ! * " ( " . ! " ! # ) z * > + ! h * $ $ } ! " ! # ) * 5 " ( ! # ( ! ö z * > + ! " . " " ! i ¼ h k i l 9 h j h k i i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d ± ¸ 0 1 ½ * ( 5 ( 5 " . . " 3 ¼ h k i l é ¼ h k ¨ ê i j h k i & $ j h k i l 6 h k Ú î Û ï ð Ý ô iú ¼ h k i l 6 Áà k Ú î Û ï ð ÄÆ u $ j h k i l 9 h k Ú î Û ï ð ) > ß k i . " " 3 ¾ t qú ¼ h k i l 9 h k Ú î Û ï ð ) > ß ã ¤ k i ¦ $ j h k i l 8 h k Ú î Û ï ð ã ô i # ( é j h k ¨ ê i r o j h k i ú . " " 3 o ¿ & 7 k Ê k pú ¼ h k i l 9 h j h k i i # . ) , 3 # ! 3 ! + " 5 # ( © ! " ) û 5 ( 5 " . ! + * . " 3 µ h Ø « # 3 4 ) ) # ! $ i B C D E F G H E I J K L F H K G C M N O P Q R S N T P M N P U V W P O T N X Y Z [ \ ] D X R O Z Q ^ _ ` a b c O R ^ O _ L ` d G G K e ^ W b d L