Announcements for This Lecture

Transcription

1 Lecture 26 Sorting

2 Announcements for Tis Lecture Prelim/Finals Prelims in andbac room Gates Hall 216 Open business ours Get tem any day tis wee Final: Survey Dec still 17 up t for 2:00-4:30pm A5 Study All new guide questions! by end of wee Eac person must answer Conflict wit Final time A6 due Monday, Dec. 9 Submit to Final Conflict assignment 2.5 wees including on CMS T-Day 2 wees witout te brea Must be in by December 10t Assignments/Lab A6 will be graded by Turs. Will give grade breadown Will review survey too A7 is due next Wednesday One wee left Keep up wit deadlines Lab 13 is optional Good study for te final Consultant ours still open 12/2/14 Sorting 2

3 Linear Searc Vague: Find first occurrence of v in b[..-1]. 12/2/14 Sorting 3

4 Linear Searc Vague: Find first occurrence of v in b[..-1]. Better: Store an integer in i to trutify result condition post: post: 1. v is not in b[..i-1] 2. i = OR v = b[i] 12/2/14 Sorting 4

5 Linear Searc Vague: Find first occurrence of v in b[..-1]. Better: Store an integer in i to trutify result condition post: post: 1. v is not in b[..i-1] 2. i = OR v = b[i] pre: b post: b i v not ere v 12/2/14 Sorting 5

6 Linear Searc Vague: Find first occurrence of v in b[..-1]. Better: Store an integer in i to trutify result condition post: post: 1. v is not in b[..i-1] 2. i = OR v = b[i] pre: b post: b i v not ere v OR b v not ere i 12/2/14 Sorting 6

7 Linear Searc pre: b post: b i v not ere v OR i b v not ere inv: b i v not ere 12/2/14 Sorting 7

8 Linear Searc def linear_searc(b,c,): """Returns: first occurrence of c in b[..]""" # Store in i te index of te first c in b[..] i = # invariant: c is not in b[0..i-1] wile i < len(b) and b[i]!= c: i = i + 1 # post: c is not in b[..i-1] # i >= len(b) or b[i] == c return i if i < len(b) else -1 Analyzing te Loop 1. Does te initialization mae inv true 2. Is post true wen inv is true and condition is false 3. Does te repetend mae progress 4. Does te repetend eep te invariant inv true 12/2/14 Sorting 8

9 Binary Searc Vague: Loo for v in sorted sequence segment b[..]. 12/2/14 Sorting 9

10 Binary Searc Vague: Loo for v in sorted sequence segment b[..]. Better: Precondition: b[..-1] is sorted (in ascending order). Postcondition: b[..i] < v and v <= b[i+1..-1] Below, te array is in non-descending order: pre: b post: b i < v >= v 12/2/14 Sorting 10

11 Binary Searc Loo for value v in sorted segment b[..] pre: b post: b inv: b < v i < v >= v i j >= v New statement of te invariant guarantees tat we get leftmost position of v if found Example b if v is 3, set i to 0 if v is 4, set i to 5 if v is 5, set i to 7 if v is 8, set i to 10 12/2/14 Sorting 11

12 Binary Searc Vague: Loo for v in sorted sequence segment b[..]. Better: Precondition: b[..-1] is sorted (in ascending order). Postcondition: b[..i] <= v and v < b[i+1..-1] Below, te array is in non-descending order: pre: b post: b inv: b < v i >= v i j < v > v Called binary searc because eac iteration of te loop cuts te array segment still to be processed in alf 12/2/14 Sorting 12

13 Binary Searc pre: b post: b i < v >= v i j New statement of te invariant guarantees tat we get leftmost position of v if found inv: b < v >= v i = ; j = +1; wile i!= j: Looing at b[i] gives linear searc from left. Looing at b[j-1] gives linear searc from rigt. Looing at middle: b[(i+j)/2] gives binary searc. 12/2/14 Sorting 13

14 Sorting: Arranging in Ascending Order 0 n pre: b post: b 0 n sorted Insertion Sort: 0 i n inv: b sorted i = 0 wile i < n: # Pus b[i] down into its # sorted position in b[0..i] i = i+1 0 i i /2/14 Sorting 14

15 Insertion Sort: Moving into Position i = 0 wile i < n: pus_down(b,i) i = i+1 def pus_down(b, i):" j = i wile j > 0: if b[j-1] > b[j]: swap(b,j-1,j) j = j-1 swap sown in te lecture about lists 0 i i i i /2/14 Sorting 15

16 Te Importance of Helper Functions i = 0 wile i < n: pus_down(b,i) i = i+1 def pus_down(b, i):" j = i wile j > 0: if b[j-1] > b[j]: swap(b,j-1,j) j = j-1 VS i = 0 wile i < n: j = i wile j > 0: Can you understand all tis code below if b[j-1] > b[j]: temp = b[j] b[j] = b[j-1] b[j-1] = temp j = j -1 i = i +1 12/2/14 Sorting 16

17 Insertion Sort: Performance def pus_down(b, i): """Pus value at position i into sorted position in b[0..i-1]""" j = i wile j > 0: if b[j-1] > b[j]: swap(b,j-1,j) j = j-1 Insertion sort is an n 2 algoritm b[0..i-1]: i elements Worst case: i = 0: 0 swaps i = 1: 1 swap i = 2: 2 swaps Pusdown is in a loop Called for i in 0..n i swaps eac time Total Swaps: (n-1) = (n-1)*n/2 12/2/14 Sorting 17

18 Algoritm Complexity Given: a list of lengt n and a problem to solve Complexity: roug number of steps to solve worst case Suppose we can compute 1000 operations a second: Complexity n=10 n=100 n=1000 n 0.01 s 0.1 s 1 s n log n s 0.32 s 4.79 s n s 10 s 16.7 m n 3 1 s 16.7 m 11.6 d 2 n 1 s 4x10 19 y 3x y Major Topic in 2110: Beyond scope of tis course 12/2/14 Sorting 18

19 Sorting: Canging te Invariant 0 n pre: b post: b 0 n sorted Insertion Selection Sort: 0 i n inv: b sorted, sorted b[i..] b[0..i-1] i = 0 wile i < n: # Find minimum in b[i..] # Move it to position i i = i+1 First segment always contains smaller values i i i n n n 12/2/14 Sorting 19

20 Sorting: Canging te Invariant 0 n pre: b post: b 0 n sorted Insertion Selection Sort: 0 i n inv: b sorted, sorted b[i..] b[0..i-1] i = 0 wile i < n: j = index of min of b[i..n-1] swap(b,i,j) i = i+1 First segment always contains smaller values /2/14 Sorting 20 i i Selection sort also is an n 2 algoritm n n

21 Partition Algoritm Given a list segment b[..] wit some value x in b[]: pre: b x Swap elements of b[..] and store in j to trutify post: i i+1 post: b <= x x >= x cange: b i into b or b i x is called te pivot value x is not a program variable denotes value initially in b[] 12/2/14 Sorting 21

22 Sorting wit Partitions Given a list segment b[..] wit some value x in b[]: pre: b x Swap elements of b[..] and store in j to trutify post: i i+1 post: b y <= y <= y x >= y x >= x Partition Recursively Recursive partitions = sorting Called QuicSort (wy) Popular, fast sorting tecnique 12/2/14 Sorting 22

23 QuicSort def quic_sort(b,, ): """Sort te array fragment b[..]""" if b[..] as fewer tan 2 elements: return j = partition(b,, ) # b[..j 1] <= b[j] <= b[j+1..] # Sort b[..j 1] and b[j+1..] quic_sort (b,, j 1) quic_sort (b, j+1, ) pre: b post: b Worst Case: array already sorted Or almost sorted n 2 in tat case Average Case: array is scrambled n log n in tat case Best sorting time! x i i+1 <= x x >= x 12/2/14 Sorting 23

24 Algoritm: Final Word About Algoritms Step-by-step way to do someting Not tied to specific language Implementation: An algoritm in a specific language Many times, not te ard part Higer Level Computer Science courses: We teac advanced algoritms (pictures) Implementation you learn on your own List Diagrams Demo Code 12/2/14 Sorting 24