Design and Analysis of Algorithms - PDF document

คําเตือน Design and Analysis of Algorithms เนื้อหาอันรวมถึงขอความ ตัวเลข สัญลักษณ รูปภาพ และ คําบรรยาย อาจมีขอผิดพลาดแฝงอยู ผูจัดทําจะไมรับผิด ผศ . ดร . สมชาย ประสิทธิ์จูตระกูล ชอบตอความเสียหายทั้งทางดานผลการเรียน สุขภาพกาย ภาควิชาวิศวกรรมคอมพิวเตอร และสุขภาพจิตใดๆ อันเนื่องมาจากการใชสื่อการเรียนนี้ จุฬาลงกรณมหาวิทยาลัย 2542 ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj Outline Dynamic Programming • Definition • Optimal Substructure • Recurrence of solution Longest Common Subsequence • Example • Analysis ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj Subsequence Common Subsequence • X = < s, o, m, c, h, a, i > • X = < s, o, m, c, h, a, i >, Y = < c, h, u, a, n > • subsequences of X • common subsequences of X and Y – < s, o, m, c , h, a, i > → < c > – < s, o, m , c, h, a, i > → < s, o, m > < c , h, u, a, n > – < s, o, m, c, h, a, i > → <c, h, a, i> – < s, o, m, c, h, a , i > → <c, a> < c, h, u, a , n > – < s, o, m, c, h, a, i > → <s, o, h, a, i> – < s, o, m, c, h, a , i > → <c, h, a> – ... < c, h, u, a , n > – ... ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj

Longest Common Subsequence Optimal Structure of an LCS x i • Input : two sequences, X = < x 1 , x 2 ,…, x m > X i a and Y = < y 1 , y 2 ,…, y n > x i = y j Y j a • Output : longest common subsequence of X and Y y j • X = <a, b, c, b, d, a, b>, Y = <b, d, c, a, b, a> LCS ( X i , Y j ) = LCS ( X i - 1 , Y j - 1 ) + x i • LCS( X , Y ) = <b, c, a, b>, <b, d, a, b> c ( i, j ) = c ( i - 1 , j - 1 ) + 1 ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj Optimal Structure of an LCS Optimal Structure of an LCS x i x i X i a X i a x i ≠ y j x i ≠ y j Y j b Y j b y j y j LCS ( X i , Y j ) = LCS ( X i - 1 , Y j ) LCS ( X i , Y j ) = LCS ( X i , Y j - 1 ) c ( i, j ) = c ( i - 1 , j ) c ( i, j ) = c ( i , j - 1 ) ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj Recurrence of Solution LCS : Example j 0 1 2 3 4 5 i y j B D C A B x i 0 0 0 0 0 0 x i 0 X i A 0 1 0 B 2 Y j C 0 3 y j 0 B 4 D 0 if i =0 or j =0 5 0 if i, j > 0 and x i = y j c ( i , j ) = 1 + c ( i - 1 , j - 1 ) 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i ≠ y j max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj

LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i i y j y j B D C A B B D C A B x i 0 0 0 0 0 0 x i 0 0 0 0 0 0 0 0 A 0 0 A 0 0 0 1 1 0 0 B B 2 2 C 0 C 0 3 3 0 0 B B 4 4 D 0 D 0 5 5 0 if i =0 or j =0 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i y j i y j B D C A B B D C A B x i 0 0 0 0 0 0 x i 0 0 0 0 0 0 0 0 A 0 0 0 0 A 0 0 0 0 1 1 1 0 0 B B 2 2 C 0 C 0 3 3 0 0 B B 4 4 D D 5 0 5 0 0 if i =0 or j =0 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i i y j y j B D C A B B D C A B 0 0 0 0 0 0 0 0 0 0 0 0 x i x i 0 0 A 0 0 0 0 1 1 A 0 0 0 0 1 1 1 1 0 0 1 B B 2 2 C 0 C 0 3 3 0 0 B B 4 4 D 0 D 0 5 5 0 if i =0 or j =0 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj

LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i i y j y j B D C A B B D C A B x i 0 0 0 0 0 0 x i 0 0 0 0 0 0 0 0 A 0 0 0 0 1 1 A 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 B B 2 2 C 0 C 0 3 3 0 0 B B 4 4 D 0 D 0 5 5 0 if i =0 or j =0 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i y j i y j B D C A B B D C A B x i 0 0 0 0 0 0 x i 0 0 0 0 0 0 0 0 A 0 0 0 0 1 1 A 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 2 B B 2 2 C 0 C 0 3 3 0 0 B B 4 4 D D 5 0 5 0 0 if i =0 or j =0 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i i y j y j B D C A B B D C A B 0 0 0 0 0 0 0 0 0 0 0 0 x i x i 0 0 A 0 0 0 0 1 1 A 0 0 0 0 1 1 1 1 0 1 1 1 1 2 0 1 1 1 1 2 B B 2 2 C 0 1 1 2 2 2 C 0 1 1 2 2 2 3 3 0 0 1 1 2 2 3 B B 4 4 D 0 D 0 5 5 0 if i =0 or j =0 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj

LCS : Example LCS : Example j 0 1 2 3 4 5 j 0 1 2 3 4 5 i i y j y j B D C A B B D C A B x i 0 0 0 0 0 0 x i 0 0 0 0 0 0 0 0 A 0 0 0 0 1 1 A 0 0 0 0 1 1 1 1 0 1 1 1 1 2 0 1 1 1 1 2 B B 2 2 C 0 1 1 2 2 2 C 0 1 1 2 2 2 3 3 0 1 1 2 2 3 0 1 1 2 2 3 B B 4 4 D 0 1 2 2 2 3 D 0 1 2 2 2 3 5 5 0 if i =0 or j =0 c ( i , j ) = 1 + c ( i - 1 , j - 1 ) if i, j > 0 and x i = y j max( c ( i - 1 , j ) c ( i , j - 1 )) if i, j > 0 and x i ≠ y j ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj LCS : Analysis j 0 1 2 ... m i 0 0 0 0 0 0 x x .... x 1 0 x x .... x 2 ... n 0 x x .... x Θ ( mn ) ht t p: / / www. cp. eng. chul a. ac. t h/ f acul t y/ spj