Design and Analysis of Algorithms - PowerPoint PPT Presentation

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 Outline Algorithm Design Techniques • General idea • Examples – Mergesort, Quicksort Divide-and-Conquer – Selection – Exponentiation – Matrix Multiplication – Closest point • Conclusion

General Idea General Idea • Divide problem instance into subinstances of the D&Q ( P ) same kind { i f ( P i s t r i vi al ) r et ur n Sol ve( P ) • For each subinstance (conquer) Di vi de P i nt o P 1 , P 2 , … , P k – solve it directly if it is trivial f or ( i = 1 t o k ) – solve it recursively otherwise S i = D&Q ( P i ) • Combine subinstance solutions to give the S = Com bi ne( S 1 , S 2 , … , S k ) solution for the bigger problem instance r et ur n S } Mergesort Mergesort : Analysis • T ( n ) = 2 T ( n /2) + Θ ( n ) • Master method : a = 2, b = 2, f ( n ) = Θ ( n ) Mergesort Mergesort 2 T ( n /2) • c = log b a = log 2 2 = 1 T ( n ) = 2 T ( n /2) + Θ ( n ) • n c = n 1 , f ( n ) = Θ ( n c ) = Θ ( n ) • T ( n ) = Θ ( n c log n ) = Θ ( n log n ) Θ ( n ) Merge

Mergesort : Quiz Quicksort • What if we do not divide the array in half ? – 40-60 Θ ( n ) Partition – 20-80 k elements T ( n ) = T ( k ) + T ( n - k ) + Θ ( n ) – 1 0-90 Quicksort Quicksort T ( k ) + T ( n - k ) – 1 -99 – random Quicksort : Worst-Case Analysis Quicksort : Best-Case Analysis • T ( n ) = T ( k ) + T ( n - k ) + Θ ( n ) • T ( n ) = T ( k ) + T ( n - k ) + Θ ( n ) • Worst-case when k = 1 in every step • Base-case when k = n /2 in every step = + − + Θ = + + Θ T ( n ) T ( 1 ) T ( n 1 ) ( n ) ( ) ( / 2 ) ( / 2 ) ( ) T n T n T n n = − + Θ = + Θ T ( n 1 ) ( n ) 2 T ( n / 2 ) ( n ) n ( ) = Θ = Θ ( i ) n log n � = i 1 n � � = Θ i � � � = � i 1 � ( ) = Θ 2 n

Quicksort : Partition Quicksort : Partition Par t i t i on( A, p, r ) Par t i t i on( A, p, r ) { { c = A[ p] p p p p r r r r ... ... ... ... A A x x x x y x x x x x x x x x A x A y p j r ... A x x x x x x x A[ p. . j ] > A[ j +1. . r ] r et ur n j r et ur n j } } Quicksort : Partition Quicksort : Partition Par t i t i on( A, p, r ) Par t i t i on( A, p, r ) { { i j i j c = A[ p] c = A[ p] p p p p r r r r i = p- 1; j = r +1 i = p- 1; j = r +1 ... ... ... ... A A A A whi l e ( i < j ) { } r et ur n j r et ur n j } }

Quicksort : Partition Quicksort : Partition Par t i t i on( A, p, r ) Par t i t i on( A, p, r ) { { i j i j c = A[ p] c = A[ p] p p p p r r r r i = p- 1; j = r +1 i = p- 1; j = r +1 ... ... ... ... A A A A whi l e ( i < j ) { whi l e ( i < j ) { r epeat j = j - 1 r epeat j = j - 1 unt i l ( A[ j ] ≤ c ) unt i l ( A[ j ] ≤ c ) < c > c r epeat i = i +1 r epeat i = i +1 unt i l ( A[ i ] ≥ c ) unt i l ( A[ i ] ≥ c ) i f i < j exchange A[ i ] , A[ j ] } } r et ur n j r et ur n j } } Quicksort : Partition Quicksort : Partition Par t i t i on( A, p, r ) Par t i t i on( A, p, r ) { { i j j i c = A[ p] c = A[ p] p p p p r r r r i = p- 1; j = r +1 i = p- 1; j = r +1 ... ... ... ... A A A A whi l e ( i < j ) { whi l e ( i < j ) { r epeat j = j - 1 r epeat j = j - 1 unt i l ( A[ j ] ≤ c ) unt i l ( A[ j ] ≤ c ) ≤ c ≥ c ≤ c ≥ c r epeat i = i +1 r epeat i = i +1 unt i l ( A[ i ] ≥ c ) unt i l ( A[ i ] ≥ c ) i f i < j i f i < j exchange A[ i ] , A[ j ] exchange A[ i ] , A[ j ] } } r et ur n j r et ur n j } }

Quicksort : Partition Quicksort : Partition p p p p r r r r ... ... ... ... A A x x x x A A x x x x y x x x x x x x x x y x x x x x x x x x j p r p r j ... ... A x x x x x x x A x x x x x x x Quicksort : Partition Quicksort : Partition p p p p r r r r ... ... ... ... A x x x x A A A x x x x y x x x x x x x x x y x x x x x x x x x p j r p j r ... ... A x x x x x x x A x x x x x x x

Quicksort : Partition Quicksort : Partition p p p p r r r r ... ... ... ... A A x x x x A A x x x x y x x x x x x x x x y x x x x x x x x x p j r p j r ... ... A x x x x x x x A x x x x x x x Quicksort : Partition Quicksort : Partition p p p p r r r r ... ... ... ... A x x x x A A A x x x x y x x x x x x x x x y x x x x x x x x x j j p r p r ... ... A x x x x x x x A x x x x x x x

Randomized Partition Randomized Quicksort : Analysis • T ( n ) = T ( k ) + T ( n - k ) + Θ ( n ) Random i zed- Par t i t i on( A, p, r ) − n 1 1 ( ) � � = + − + + − + Θ { T ( n ) T ( 1 ) T ( n 1 ) T ( j ) T ( n j ) ( n ) � � � � � n i = r andom ( p, r ) = j 1 � � exchange A[ p] , A[ i ] − n 1 1 ( ) par t i t i on( A, p, r ) = � + − + Θ T ( j ) T ( n j ) ( n ) } n = j 1 − − n 1 n 1 1 � � = + − + Θ T ( j ) T ( n j ) ( n ) � � � � � � n = = j 1 j 1 � � − n 1 2 = � + Θ T ( j ) ( n ) n = j 1 Randomized Quicksort : Analysis Randomized Quicksort : Analysis − n 1 ′ 2 = + − + = � + Θ nT ( n ) ( n 1 ) T ( n 1 ) c n T ( n ) T ( j ) ( n ) 1 n × = j 1 ′ − + − T ( n ) T ( n 1 ) c n 1 n ( n 1 ) 2 = + = � + T ( n ) T ( j ) cn + + n 1 n n 1 n telescope = j 1 − n 1 ′ n = � + T ( n ) T ( 1 ) c 2 nT ( n ) 2 T ( j ) cn = + � + + = n 1 2 i 1 n j 1 1 = i 2 = − O (log n ) n 2 � − − = + − 2 ( n 1 ) T ( n 1 ) 2 T ( j ) c ( n 1 ) n = T ( n ) � i 1 = O (log n ) = j 1 + 1 ′ n − − − = − + nT ( n ) ( n 1 ) T ( n 1 ) 2 T ( n 1 ) c n = = + − + ′ ( ) ( log ) T n O n n nT ( n ) ( n 1 ) T ( n 1 ) c n

Quicksort Randomization • Quicksort • A general tool to improve algorithms with bad worst-case but good average-case performance – With random input data, quicksort runs in • “Robin Hood Effect” expected Ο ( n log n ) time • The worst case is still there but we almost • Randomized Quicksort certainly won’t see it – With high probability, randomized quicksort runs • Stay tuned..... in Ο ( n log n ) time Quicksort : Quiz # 1 Quicksort : Quiz #2 • If all the elements are the same, p p r r Θ ( n ) ... ... A A – how does Partition divide the array ? ≤ c ≥ c – what is the running time of Quicksort ? p p r r Θ ( n ) ... ... A A > c < c = c

Selection Selection Algorithms • Input : A set A of n (distinct) numbers and a � Sort the A and then pick the i th element number i , with 1 ≤ i ≤ n O( n log n ) • Output : The element x in A that is larger than � Build a (max) heap from the n elements of A exactly i - 1 other elements of A Extract-Max i times O( n + i log n ) = O( n log n ) Selection Selection in Expected Linear Time • Minimum : select the 1 st smallest element : Θ ( n ) i • Maximum : select the n th smallest element : Θ ( n ) k elements Partition • Select the i th smallest element : Θ ( n ) ? if i ≤ k i Selection

Selection in Expected Linear Time Selection in Expected Linear Time i Random i zed- Sel ect i on( A, p, r , i ) { k elements i f p = r t hen r et ur n A[ p] Partition j = Random i zed- Par t i t i on( A, p, r ) k = j - p + 1 i f i < k i - k if i > k r et ur n Random i zed- Sel ect i on( A, p, j , i ) el se r et ur n Random i zed- Sel ect i on( A, j +1, r , i - k ) Selection } Randomized Selection : Analysis Selection in Worst-Case Linear Time • T ( n ) ≤ T ( max( k, n - k ) ) + Θ ( n ) • Choose Partition that guarantees a good split • Median-of-median-of-five partitioning − n 1 1 ( ) � � ≤ − + − + Θ T ( n ) T (max( 1 , n 1 )) T (max( k , n k ) ( n ) � � � n = – guarantee 30%-70% split in worst case � k 1 � − n 1 1 � � ≤ − + + Θ T ( n 1 ) 2 T ( k ) ( n ) � � – selection achieves a linear time in worst case � � � n = k n / 2 � � � � − n 1 2 = + Θ T ( k ) ( n ) � n = k n / 2 � � [CLR p. 1 88- 1 89] = O ( n )