Syllabus

1. Syllabus • Go to • www.cs.siena.edu/~ebreimer • Click Analysis of Algorithms link • bookmark course website • Important stuff • Schedule all due dates • Quizzes details, solutions • Homework details, solutions

2. Syllabus Highlights • Quiz every Friday • HW due every Monday • No exams except a final • Combination of quizzes and HWs • No huge projects • Some HW will be collaborative

3. Syllabus • Late HWs get a zero • Due at the beginning of class • Can drop 2 HWs • No quiz makeups • Miss a quiz, get a zero • Can drop 2 quizzes • Attendance 10% • 2 unexcused absences • -2% for each additional unexcused absence.

4. Grading 40% Quizzes 30% HWs 20% Final 10% Attendance A  93 or higher A-  90 or higher B+  87 B  83 B-  80 C+  77 C  73 C-  70 Syllabus

5. Chapter 1 – Intro • What is an algorithm? • 7 properties • Two Examples • Max of 3 • Closest Pair • Pseudo-code • Present • Future

6. Algorithms? • Definition: step-by-step method for solving a problem • We will concentrate on algorithms that can be executed on modern computers • like the one you probably have • This field existed before computer existed • 900 AD in Persia

7. 7 Properties • Input • Output • Precision • Determinism • Finiteness • Correctness • Generality

8. Examples Problem: Given three integer values A, B, and C return the maximum value. Input: 3 integers Output: 1 integer

9. Examples Problem: Given a list of N points, find the closest pair of points (2D) Input: N points, where a point is a pair of real values (x,y). Output: 2 points

10. 3. Precision max(a,b,c) { compare a, b, and c and return the largest value; } Not precise!

11. max(a,b,c) { if (a > b && a > c) return a; else if (b > c) return b; else return c; } Very precise 3. Precision

12. 3. Precision • A precise algorithm is one that can be described by pseudo code or a high-level programming language like C++ or Java. • Pseudo code is like a high-level programming language where the syntax doesn’t have to be perfect.

13. 4. Determinism • Not random • Given the same conditions, you expect the same outcome. • Computer are inherently deterministic. • When computers behave randomly it is usually a result of • human errors • environmental conditions • pseudo-random number generation

14. 5. Finiteness • Given finite input, an algorithm should not run infinitely (i.e., forever). • An algorithm that runs forever is not really an algorithm because it will never solve the problem. • Recall that an algorithm is a step-by-step method for solving a problem. • What if the input is infinite?

15. max(a,b,c) { if (a > b) return a; else if (b > c) return b; else return c; } What if a = 5 b = 4 c = 6 6. Correctness

16. 6. Correctness • Algorithms that return incorrect answers aren’t really algorithms • Recall that an algorithm is a step-by-step method for solving a problem.

17. 7. Generality max(a, b, c) { if (a > 10 && b < 10 && c < 10) return a; if (b > 10 && a < 10 && c < 10) return b; if (c > 10 && b < 10 && a < 10) return c; } Will never return an incorrect answer, but does not apply to a general set of input (like all integers or all real numbers).

18. Closest Pair • Design an algorithm to find the closest pair of points

19. Closest Pair • Designing algorithms is more of an art than a science.

20. Closest Pair • Visually this problem is very simple to solve, but what if I gave you 1 million points?

21. Closest Pair • What if I asked you to solve this problem for 1 million different cases?

22. Closest Pair • What is the input? • What is the output? • What are the precise step-by-step instructions? • Will the algorithm terminate? • Will it produce correct answers all the time? • How much time will it take to solve? • How much memory is required to solve the problem?

23. Closest Pair The input: • 2D point, a pair values (x, y) • Point p; • p.x; • p.y; • An array of n Points • Point P[n] • P[0], P[1], P[2], P[3], …, P[n-1]

24. Closest Pair The output: • Two points P[a] and P[b] where the distance between P[a] and P[b] is minimum among all possible pairs of points. • Sometimes describing the output with precision gives you clues about how to solve the problem.

25. Closest Pair • How do you compute the distance between to points? • How do you find a distance is minimum among all possible pairs of points?

26. Closest Pair dist(a,b){ d = sqrt[ pow((a.x – b.x),2) + pow((a.y – b.y),2) ]; return d; }

27. Closest Pair d = dist(P[0], P[1]); Compare all possible pairs of points, i.e.,compare P[0] with P[1], P[2], P[3], …, P[n-1]compare P[1] with P[2], P[3], P[4], …, P[n-1]compare P[2] with P[3], P[4], P[5], …, P[n-1]compare P[3] with P[4], P[5], P[6], …, P[n-1]…compare P[n-3] with P[n-2], P[n-1]compare P[n-2] with P[n-1]

28. Closest Pair ClosestPair(P[ ], n) { min_dist = dist(P[0],P[1]); for (x = 1 to n-1) { d = dist(P[0], P[x]); if (d < min_dist) { min_dist = d; } } return min_dist; } This is not correct!

29. Closest Pair ClosestPair(P[ ], n) { min_dist = dist(P[0],P[1]); for (y = 0 to n-1) { for (x = 1 to n-1) { d = dist(P[y], P[x]); if (d < min_dist) { min_dist = d; } }} return min_dist; }

30. Closest Pair • Compare P[a] and P[b]

31. Closest Pair • Compare P[a] and P[b]

32. Closest Pair • Compare P[a] and P[b]

33. Closest Pair ClosestPair(P[ ], n) { min_dist = dist(P[0],P[1]); for (y = 0 to n-1) { for (x = y+1 to n-1) { d = dist(P[y], P[x]); if (d < min_dist) { min_dist = d; } }} return min_dist; }

34. Closest Pair • This matrix is n x n • n2 entries. • How many of these n2entries must we compute?

35. Summations • n-1 + n-2 +n-3 +...+3 +2 +1

36. The Present • Remember the 7 properties? • Many algorithms used in practice aren’t • General • Deterministic, or • Finite

37. The Present • There are problems that are just too difficult to solve. • Compromises must be made. • Some algorithms only work on certain “types” of input. • Some algorithms require pseudo-random processing • Some algorithms will run forever on certain “types” or problems.

38. The Present • Today, an algorithm doesn’t have to satisfy all the properties if the problem is very difficult and there is no other know way to get good answers. • Algorithms can be approximate, fuzzy, and even un-predicable. • Some Algorithm purist disagree with this. • But, I think a half-baked algorithm is better than no algorithm at all.

39. The Future • Quantum computing. • Uses a quantum bit which can store an manipulate information in a massivle parallel way. • Theoretical only • Might be impossible to implement • DNA computing • DNA bases (A,C,G,T) will be used to represent data • Biological processes can manipulate large amounts of data in one step.