Introduction to Algorithms 6.046J/18.401J

Transcription

1 Introduction to Algorithms 6.046J/18.401J Lecture 1 Prof. Piotr Indyk

2 Analysis of algorithms The theoretical study of computer-program performance and resource usage. Also important: modularity maintainability functionality robustness user-friendliness programmer time simplicity extensibility reliability February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.4

3 Why study algorithms and performance? Performance often draws the line between what is feasible and what is impossible. Algorithmic mathematics provides a language for talking about program behavior. Speed is fun! February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.5

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: 8436 Output: 3468 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.6

5 Insertion sort A: pseudocode INSERTION-SORT (A, n) A[1..n] for j to n do key A[ j] i j 1 while i>0and A[i]>key do A[i+1] A[i] i i 1 A[i+1] = key 1 i j n sorted key February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

6 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.8

7 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

8 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.10

9 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

10 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

11 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

12 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.14

13 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.15

14 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.16

15 Example of insertion sort February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.1

16 Example of insertion sort done February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.18

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 n Seek upper bounds on the running time T(n) for the input size n,because everybody likes a guarantee. February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.1

18 Kinds of analyses 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. February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

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 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.1

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 +6046=Θ(n 3 ) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

21 Asymptotic performance T(n) When n gets large enough, a Θ(n ) algorithm always beats a Θ(n 3 ) algorithm. 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. February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.3

22 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. February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.4

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 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.5

24 Merging two sorted arrays 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.6

25 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

26 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.8

27 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.

28 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.30

29 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.31

30 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.3

31 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.33

32 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.34

33 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.35

34 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.36

35 Merging two sorted arrays 1 1 February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.3

36 Merging two sorted arrays 1 1 Time = Θ(n) to merge a total of n elements (linear time). February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.38

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. February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.3

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). February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.40

39 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.41

40 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. T(n) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.4

41 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn T(n/) T(n/) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.43

42 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn/ cn/ T(n/4) T(n/4) T(n/4) T(n/4) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.44

43 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn/ cn/ cn/4 cn/4 cn/4 cn/4 Θ(1) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.45

44 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn/ cn/ h =lgn cn/4 cn/4 cn/4 cn/4 Θ(1) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.46

45 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn cn/ cn/ h =lgn cn/4 cn/4 cn/4 cn/4 Θ(1) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.4

46 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn cn/ cn/ cn h =lgn cn/4 cn/4 cn/4 cn/4 Θ(1) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.48

47 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn cn/ cn/ cn h =lgn cn/4 cn/4 cn/4 cn/4 cn Θ(1) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.4

48 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn cn/ cn/ cn h =lgn cn/4 cn/4 cn/4 cn/4 cn Θ(1) #leaves = n Θ(n) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.50

49 Recursion tree Solve T(n)=T(n/) + cn,wherec >0is constant. cn cn cn/ cn/ cn h =lgn cn/4 cn/4 cn/4 cn/4 cn Θ(1) #leaves = n Θ(n) Total = Θ(n lg n) February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.51

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 >30or so. Go test it out for yourself! February 4, 03 (c) Charles Leiserson and Piotr Indyk L1.5