Chapter 2

1. Chapter 2 Analysis Framework 2.1 – 2.2

2. Homework • Remember questions due Wed. • Read sections 2.1 and 2.2 • pages 41-50 and 52-59

3. Agenda: Analysis of Algorithms • Blackboard • Issues: • Correctness • Time efficiency • Space efficiency • Optimality • Approaches: • Theoretical analysis • Empirical analysis

4. Time Efficiency Time efficiency is analyzed by determining the number of repetitions of the basic operation as a function of input size

5. input size running time Number of times basic operation is executed execution time for basic operation Theoretical analysis of time efficiency • Basic operation: the operation that contributes most towards the running time of the algorithm. T(n) ≈copC(n)

6. Input size and basic operation examples

7. Empirical analysis of time efficiency • Select a specific (typical) sample of inputs • Use physical unit of time (e.g., milliseconds) OR • Count actual number of basic operations • Analyze the empirical data

8. Best-case, average-case, worst-case For some algorithms efficiency depends on type of input: • Worst case: W(n) – maximum over inputs of size n • Best case: B(n) – minimum over inputs of size n • Average case: A(n) – “average” over inputs of size n • Number of times the basic operation will be executed on typical input • NOT the average of worst and best case • Expected number of basic operations repetitions considered as a random variable under some assumption about the probability distribution of all possible inputs of size n

9. Example: Sequential search • Problem: Given a list of n elements and a search key K, find an element equal to K, if any. • Algorithm: Scan the list and compare its successive elements with K until either a matching element is found (successful search) of the list is exhausted (unsuccessful search) • Worst case • Best case • Average case

10. Types of formulas for basic operation count • Exact formula e.g., C(n) = n(n-1)/2 • Formula indicating order of growth with specific multiplicative constant e.g., C(n) ≈ 0.5 n2 • Formula indicating order of growth with unknown multiplicative constant e.g., C(n) ≈cn2

11. Order of growth • Most important: Order of growth within a constant multiple as n→∞ • Example: • How much faster will algorithm run on computer that is twice as fast? • How much longer does it take to solve problem of double input size? • See table 2.1

12. Table 2.1

13. Day 4: Agenda • O, Θ, Ω • Limits • Definitions • Examples • Code  O(?)

14. Homework • Due on Monday 11:59PM • Electronic submission see website. • Try to log into Blackboard • Finish reading 2.1 and 2.2 • page 60-61, questions 2, 3, 4, 5, & 9

15. Asymptotic growth rate A way of comparing functions that ignores constant factors and small input sizes • O(g(n)): class of functions t (n) that grow no faster than g(n) • Θ(g(n)): class of functions t (n) that grow at same rate as g(n) • Ω(g(n)): class of functions t (n) that grow at least as fast as g(n) see figures 2.1, 2.2, 2.3

16. Big-oh

17. Big-omega

18. Big-theta

19. Using Limits if t(n) grows slower than g(n) if t(n) grows at the same rate as g(n) if t(n) grows faster than g(n)

20. L’Hopital’s Rule

21. Examples: • 10n vs. 2n2 • n(n+1)/2 vs. n2 • logb n vs. logc n

22. Definition • f(n) = O( g(n) ) if there exists • a positive constant c and • a non-negative interger n0 • such thatf(n)c·g(n) for every n > n0 Examples: • 10n is O(2n2) • 5n+20 is O(10n)

23. Basic Asymptotic Efficiency classes

24. Non-recursive algorithm analysis Analysis Steps: • Decide on parameter n indicating input size • Identify algorithm’s basic operation • Determine worst, average, and best case for input of size n • Set up summation for C(n) reflecting algorithm’s loop structure • Simplify summation using standard formulas

25. Example for (x = 0; x < n; x++) a[x] = max(b[x],c[x]);

26. Example for (x = 0; x < n; x++) for (y = x; y < n; y++) a[x][y] = max(b[x],c[y]);

27. Example for (x = 0; x < n; x++) for (y = 0; y < n/2; y++) for (z = 0; z < n/3; z++) a[z] = max(a[x],c[y]);

28. Example y = n while (y > 0) if (a[y--] == b[y--]) break;

29. Day 5: Agenda • Go over the answer to hw2 • Try electronic submission • Do some problems on the board • Time permitting: Recursive algorithm analysis

30. Homework • Remember to electronically submit hw3 before Tues. morning • Read section 2.3 thoroughly!pages 61-67

31. Day 6: Agenda • First, what have we learned so far about non-recursive algorithm analysis • Second, logbn and bnThe enigma is solved. • Third, what is the deal with Abercrombie & Fitch • Fourth, recursive analysis tool kit.

32. Homework 4 and Exam 1 • Last homework before Exam 1 • Due on Friday 2/6 (electronic?) • Will be returned on Monday 2/9 • All solutions will be posted next Monday 2/9 • Exam 1 Wed. 2/11

33. Homework 4 • Page 68 questions 4, 5 and 6 • Pages 77 and 78 questions 8 and 9 • We will do similar example questions all day on Wed 2/4

34. What have we learned? • Non-recursive algorithm analysis • First, Identify problem size. • Typically, • a loop counter • an array size • a set size • the size of a value

35. Basic operations • Second, identify the basic operation • Usually a small block of code or even a single statement that is executed over and over. • Sometimes the basic operation is a comparison that is hidden inside of the loop • Example: • while (target != a[x]) x++;

36. Single loops • One loop from 1 to n • O(n) • Be aware this is the same as • 2 independent loops from 1 to n • c independent loops from 1 to n • A loop from 5 to n-1 • A loop from 0 to n/2 • A loop from 0 to n/c

37. Nested loops for (x = 0; x < n; x++) for (y = 0; y < n; y++)  O(n2) for (x = 0; x < n; x++) for (y = 0; y < n; y++) for (z = 0; z < n; z++)  O(n3) But remember: We can have c independent nested loops, or the loops can be terminated early n/c.

38. Most non-recursive algorithms reduce to one of these efficiency classes

39. What else? • Best cases often arise when loops terminate early for specific inputs. • For worst cases, consider the following: Is it possible that a loop will NOT terminate early • Average cases are not the mid-pointer or average of the best and worst case • Average cases require us to consider a set of typical input • We won’t really worry too much about average case until we hit more difficult problems.

40. Anything else? • Important questions you should be asking? • How does log2n arise in non-recursive algorithms? • How does 2n arise? • Dr. B. Please, show me the actual loops that cause this!

41. for (x = 0; x < n; x++) { Basic operation; x = x*2; } for every item in the list do Basic operation; eliminate or discard half the items; Log2n • Note: These types of log2n loops can be nested inside of O(n), O(n2), or O(nk) loops, which leads to • n log n • n2 log n • nk log n

42. Log2n • Which, by the way, is pretty much equivalent to logbn, ln, or the magical lg(n) • But, don’t be mistaken • O(n log n) is different than O(n) even though log n grows so slow.

43. Last thing: 2n • How does 2n arise in real algorithms? • Lets consider nn: • How can you program n nested loops • Its impossible right? • So, how the heck does it happen in practice?

44. Think • Write an algorithm that prints “*” 2n times. • Is it hard? • It depends on how you think.

45. Recursive way fun(int x) { if (x > 0) { print “*”; fun(x-1); fun(x-1); } } Then call the function fun(n);

46. Non-recursive way for (x = 0; x < pow(2,n); x++) print “*”; WTF? Be very wary of your input size. If you input size is truly n, then this is truly O(2n) with just one loop.

47. That’s it! • That’s really it. • That’s everything I want you do know about non-recursive algorithm analysis • Well, for now.

48. Enigma • We know that log2n = (log3n) • But what about 2n = (3n) • Here is how I will disprove it… • This gives you an idea of how I like to see questions answered.

49. Algorithm Analysis Recursive Algorithm Toolkit

50. Example Recursive evaluation of n ! • Definition: n ! = 1*2*…*(n-1)*n • Recursive definition of n!: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • Recurrence for number of multiplications to compute n!: