Table of Contents. Course Minutiae. Course Overview Algorithm Design Strategies Algorithm Correctness Asymptotic Analysis 2 / 32

Transcription

1 Intro Lecture CS 584/684: Algorithm Design and Analysis Daniel Leblanc1 1 Senior Adjunct Instructor Portland State University Maseeh College of Engineering and Computer Science Spring / 32

2 2 / 32 Table of Contents Course Minutiae Course Overview Algorithm Design Strategies Algorithm Correctness Asymptotic Analysis

3 3 / 32 Table of Contents Course Minutiae Course Overview Algorithm Design Strategies Algorithm Correctness Asymptotic Analysis

4 4 / 32 Course Description This course provides an advanced in-depth study of algorithm design and analysis. Topics include Graph Algorithms, Parallel Algorithms, Computational Geometry, Dynamic Programming, Complexity, and approximation.

5 Contact Information Instructor: Daniel Leblanc Office Hours: Tuesday 6:20-8: Course Website: dleblanc/cs584/ 5 / 32

6 6 / 32 Grading Policy Homework % Midterm % Final or Project %

7 7 / 32 HW Policy Homework will be assigned weekly. You may collaborate with other class members on the homework assignments in groups of no more than three people. Submit only a single writeup for the group. No late work will be accepted. Assignments must be typed and should be ed as a PDF to the TA before the start of class on the due date.

8 Project or Final Homework will be assigned weekly. You may collaborate with other class members on the homework assignments in groups of no more than three people. Submit only a single writeup for the group. No late work will be accepted. Assignments must be typed and should be ed as a PDF to the TA before the start of class on the due date. 8 / 32

9 Intro Quiz 9 / 32

10 Give an example of a sorting algorithm that runs in O(n log n) in the worst case. 10 / 32

11 Give an example of a sorting algorithm that runs in O(n) in the best case. 11 / 32

12 How many nodes are in a full binary tree with height h? 12 / 32

13 What is the closed form of the following summation? n X i i=1 13 / 32

14 Give pseudocode for binary search, given an array A and a target T return true if T appears in A and false otherwise. 14 / 32

15 15 / 32 Table of Contents Course Minutiae Course Overview Algorithm Design Strategies Algorithm Correctness Asymptotic Analysis

16 16 / 32 Course Themes There are several different themes running through the course. Algorithmic Strategies Correctness Resource Analysis Asymptotic Analysis

17 17 / 32 Brute Force A dumb approach that doesn t take advantage of any special structure of the problem. Can be thought of as an exhaustive search of the space of potential solutions.

18 18 / 32 Divide and Conquer Solve a problem by decomposing it into smaller sub-problems and recursively solving the sub-problems. Finally combine the sub-solutions into an overall solution.

19 19 / 32 Reduction Transform the problem into another problem that already has a known solution. Solve problem P by reducing it to problem R that already has a known solution. Requires that you know R and its solution. Often used to show hardness, or even impossibility of problems. If we can reduce P to R, and P is already know to be hard, then R must be at least as hard.

20 20 / 32 Randomization Explicitly randomize the behavior of an algorithm to compensate for bad inputs. Can lead to big improvements in expected run time.

21 21 / 32 Dynamic Programming Store solutions to sub-problems that occur repeatedly.

22 22 / 32 Greedy Programming Make an irrevocable choice at each step, relying on some optimal substructure to find a locally optimal solution that is also globally optimal.

23 23 / 32 Correctness Do our algorithms actually solve the problems that they claim to?

24 24 / 32 Recursive Procedures For each procedure specify logical pre-conditions and post-conditions. Then prove that for each execution of the procedure, if we require that the pre-conditions are true at entry the code ensures that the post-conditions are met at procedure exit.

25 25 / 32 Iterative Algorithms For iterative algorithms we use loop invariants. An invariant is a logical specification that is true at loop entry, maintained by each loop body execution, and hence true at loop exit. We design the invariant so that it implies useful information after loop exit.

26 26 / 32 Code termination We also want to ensure the code terminates. This is usually done by showing that some non-negative measure of the program state gets smaller at each iteration.

27 27 / 32 Asymptotic Analysis We focus on the behavior of programs as the problem size grows towards infinity. Why?

28 28 / 32 Asymptotic Upper Bound f (n) O(g(n)) Means that n 0, c > 0 such that n n 0 f (n) cg(n)

29 29 / 32 Asymptotic Lower Bound f (n) Ω(g(n)) Means that n 0, c > 0 such that n n 0 cg(n) f (n)

30 30 / 32 Asymptotic Tight Bound f (n) Θ(g(n)) Means that f (n) O(g(n)) and f (n) Ω(g(n))

31 31 / 32 Rates of Growth Suppose using your current computer you can solve a problem of size 10 6 in one hour. If I replace my computer with one twice as fast, how large an instance can I solve in one hour on my new machine?

32 32 / 32 Rates of Growth Algorithm Speed Old Machine New Machine lg n n n n lg n n n n n!