Theory and Algorithms Introduction: insertion sort, merge sort

Transcription

1 Theory and Algorithms Introduction: insertion sort, merge sort Rafael Ramirez

2 Analysis of algorithms The theoretical study of computer-program performance and resource usage. What s also important other than performance? correctness extensibility modularity... maintainability programmer time simplicity Slide 1.

3 Why study algorithms and performance? Algorithms help us to understand scalability. Performance often draws the line between what is feasible and what is impossible. Algorithmic mathematics provides a language for talking about program behavior. The lessons of program performance generalize to other computing resources. Speed is fun! Slide 1.3

4 The problem of sorting Input: sequence a 1, a,, a n of numbers. Output: permutation a' 1, a',, a' n such that a' 1 a' a' n. Example: Input: Output: Slide 1.4

5 Insertion sort pseudocode INSERTION-SORT (A, n) A[1.. n] for j to n do key A[ j] i j 1 while i > 0 and A[i] > key do A[i+1] A[i] i i 1 A[i+1] = key A: 1 i j n key sorted Slide 1.5

6 Example of insertion sort Slide 1.6

7 Example of insertion sort Slide 1.

8 Example of insertion sort Slide 1.8

9 Example of insertion sort Slide 1.

10 Example of insertion sort Slide 1.10

11 Example of insertion sort Slide 1.

12 Example of insertion sort Slide 1.

13 Example of insertion sort Slide 1.

14 Example of insertion sort Slide 1.14

15 Example of insertion sort Slide 1.15

16 Example of insertion sort done Slide 1.16

17 Running time The running time depends on the input: an already sorted sequence is easier to sort. Parameterize the running time by the size of the input, since short sequences are easier to sort than long ones. Generally, we seek upper bounds on the running time, because everybody likes a guarantee. Slide 1.1

18 Kinds of analyses Slide 1.18 Worst-case: (usually) T(n) = maximum time of algorithm on any input of size n. Average-case: (sometimes) T(n) = expected time of algorithm over all inputs of size n. Need assumption of statistical distribution of inputs. Best-case: (bogus) Cheat with a slow algorithm that works fast on some input.

19 Machine-independent time What is insertion sort s worst-case time? It depends on the speed of our computer: relative speed (on the same machine), absolute speed (on different machines). BIG IDEA: Ignore machine-dependent constants. Look at growth of T(n) as n. Asymptotic Analysis Slide 1.1

20 Θ-notation Math: Θ(g(n)) = { f (n):there exist positive constants c 1, c, and n 0 such that 0 c 1 g(n) f (n) c g(n) for all n n 0 } Engineering: Drop low-order terms; ignore leading constants. Example: 3n 3 + 0n 5n = Θ(n 3 ) Slide 1.

21 Asymptotic performance T(n) When n gets large enough, a Θ(n ) algorithm always beats a Θ(n 3 ) algorithm. Slide 1.1 n n 0 We shouldn t ignore asymptotically slower algorithms, however. Real-world design situations often call for a careful balancing of engineering objectives. Asymptotic analysis is a useful tool to help to structure our thinking.

22 Slide 1. Insertion sort analysis Worst case: Input reverse sorted. T ( n) = n j= Θ( j) = Θ ( n ) Average case: All permutations equally likely. T ( n) = n j= Θ( j / ) = Θ ( n ) [arithmetic series] Is insertion sort a fast sorting algorithm? Moderately so, for small n. Not at all, for large n.

23 Merge sort MERGE-SORT A[1.. n] 1. If n = 1, done.. Recursively sort A[ 1.. n/ ] and A[ n/ +1.. n ]. 3. Merge the sorted lists. Key subroutine: MERGE Slide 1.3

24 Merging two sorted arrays 1 Slide 1.4

25 Merging two sorted arrays 1 1 Slide 1.5

26 Merging two sorted arrays 1 1 Slide 1.6

27 Merging two sorted arrays 1 1 Slide 1.

28 Merging two sorted arrays 1 1 Slide 1.8

29 Merging two sorted arrays 1 1 Slide 1.

30 Merging two sorted arrays 1 1 Slide 1.30

31 Merging two sorted arrays 1 1 Slide 1.31

32 Merging two sorted arrays 1 1 Slide 1.3

33 Merging two sorted arrays 1 1 Slide 1.33

34 Merging two sorted arrays 1 1 Slide 1.34

35 Merging two sorted arrays 1 1 Slide 1.35

36 Merging two sorted arrays 1 1 Time = Θ(n) to merge a total of n elements (linear time). Slide 1.36

37 Analyzing merge sort Abuse T(n) Θ(1) T(n/) Θ(n) MERGE-SORT A[1.. n] 1. If n = 1, done.. Recursively sort A[ 1.. n/ ] and A[ n/ +1.. n ]. 3. Merge the sorted lists Sloppiness: Should be T( n/ ) + T( n/ ), but it turns out not to matter asymptotically. Slide 1.3

38 Recurrence for merge sort T(n) = Θ(1) if n = 1; T(n/) + Θ(n) if n > 1. We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence. CLRS and Lecture provide several ways to find a good upper bound on T(n). Slide 1.38

39 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. Slide 1.3

40 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. T(n) Slide 1.40

41 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn T(n/) T(n/) Slide 1.41

42 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn/ cn/ T(n/4) T(n/4) T(n/4) T(n/4) Slide 1.4

43 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn/ cn/ cn/4 cn/4 cn/4 cn/4 Θ(1) Slide 1.43

44 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn/ cn/ h = lg n cn/4 cn/4 cn/4 cn/4 Θ(1) Slide 1.44

45 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn cn/ cn/ h = lg n cn/4 cn/4 cn/4 cn/4 Θ(1) Slide 1.45

46 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn cn/ cn/ cn h = lg n cn/4 cn/4 cn/4 cn/4 Θ(1) Slide 1.46

47 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn cn/ cn/ cn h = lg n cn/4 cn/4 cn/4 cn/4 cn Θ(1) Slide 1.4

48 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn cn/ cn/ cn h = lg n cn/4 cn/4 cn/4 cn/4 cn Θ(1) #leaves = n Θ(n) Slide 1.48

49 Recursion tree Solve T(n) = T(n/) + cn, where c > 0 is constant. cn cn cn/ cn/ cn h = lg n cn/4 cn/4 cn/4 cn/4 cn Slide 1.4 Θ(1) #leaves = n Θ(n) Total = Θ(n lg n)

50 Conclusions Θ(n lg n) grows more slowly than Θ(n ). Therefore, merge sort asymptotically beats insertion sort in the worst case. In practice, merge sort beats insertion sort for n > 30 or so. Go test it out for yourself! Slide 1.50