# CSCI 3650 Summer 2000

## Announcements

I have decided to make the second exam be a take home exam. It is due Friday, July 21 at the beginning of class. I will answer questions about the meaining of the problems on Thursday.

## Syllabus

This is a course in design and analysis of algorithms. See the syllabus for details.

## Sample exams

• Practice questions for exam 1
• Solutions to practice questions for exam 1

## Lecture summaries

1. [6/22/00] We discussed big-O notation and properties of logarithms. We began talking about algorithms for addition and multiplication. Addition of n-digit numbers can be done in O(n) time, while multiplication of n-digit numbers can be done in O(n2) time. We discussed the naive multiplication algorithm (O(10n) time), the standard algorithm (quadratic time) and an algorithm based on the observation that, when n is even, (n)(x) = 2((n/2)x).

Big-O notation is discussed in the text on page 26. Logarithms and properties of logarithms are on page 34. The grade-school multiplication algorithm is discussed briefly on pages 671 and 672.

2. [6/23/00] We discussed more integer multiplication algorithms, including the Russian Peasant's algorithm (described on page 681 of the text), and a divide-and-conquer algorithm. If you are careful how you do the divide=and-conquer, you can arrange to do only 3 recursive calls. That leads to an algorithm that runs in O(n^c) time for c = log2(3). The value of the exponent c is about 1.59. Your test discussed the divide-and-conquer algorithm is on page 797, as an exercise. The best know multiplication algorithm for large numbers is the Schonhage/Strassen algorithm, which takes O(n log(n) log(log(n))) time.

3. [6/26/00] We discussed hybridizing two algorithms to get the best of both. Big-O, little-o, Big-Omega, little-omega and Big-Theta notation were introduced. You can compare two functions asymptotic growth by taking the limit of their ratio, as n approaches infinity.

We solved some summations, and introduced mathematical induction. Using induction, you prove that a claim about all integers n by showing that there is no counterexample to the claim. You do that by supposing that there is a counterexample, and looking at the SMALLEST counterexample. That way, you know that any value smaller than the purported smallest counterexample cannot be a counterexample.

4. [6/27/00] We discussed the homework. We discussed solution of recurrences that result from divide-and-conquer algorithms using mathematical induction.

5. [6/28/00] We covered more on the solution of recurrences, including the master theorem (page 62 of the text). We began looking at the sorting problem, and analyzed merge sort. Merge sort is described in chapter 1 of the text.

6. [6/29/00] We covered more material on sorting, including lower bounds. We will also began to look at selection problems, including finding the minimum and finding both the minimum and maximum.

7. [6/30/00] We showed, by an adversary argument, that every algorithm for finding the minimum and maximum of n values by doing comparisons requires at least ceiling(3n/2) - 2 comparisons. That is precisely what the tournament algorithm does, so the tournament algorithm makes an optimum number of comparisons.

We began looking at the general selection problem, and developed a skeleton of an algorithm. The full algorithm depends on an algorithm to find an approximation to the median of a list of numbers, which will be derived next time. Exercise 3 was handed out. See above.

8. [7/3/00] We developed the general selection algorithm, and began looking at a divide-and-conquer algorithm for computing the convex hull of a set of points in the plane.

9. [7/5/00] We went over exercises, and finished the two-dimensional convex hull algorithm, except for the detail of comparing a point to a line.

10. [7/6/00] We went over questions on the practice exam, and covered how to compare a point to a line.

11. [7/7/00] Exam 1.

12. [7/10/00] We did a reduction from the sorting problem to the two-dimensional convex hull problem, showing that any algorithm that finds the convex hull of n points can be used to sort n real numbers. This shows that the convex hull problem is no easier than sorting.

We saw the cross product method of determining which direction an angle turns. It is in the book on page 888.

We began looking at algorithms that sweep over a set of data once. An example is an algorithm that computes the maximum of n numbers. Another example is Graham's algorithm for computing convex hulls in two dimensions. We saw how that algorithm works, and did an analysis of it. Graham's algorithm works by sorting the points from left to right (O(n lg(n)) time) and then scanning from left to right. An accounting argument shows that the scan phase takes O(n) time.

13. [7/11/00] We finished analyzing Graham's algorithm, and looked at another sweep algorithm that finds the longest ascending subsequence of a sequence. The algorithm involved several ideas, including performing a sweep, remembering extra information (not just the answer) to make it possible to update the information, using binary search to speed up searching, and using linked lists to prevent copying of sequences.

14. [7/12/00] We looked at the string matching problem and the Knuth-Morris-Pratt algorithm. That algorithm is described in the text in section 34.4.

15. [7/13/00] We finished the Knuth-Morris-Pratt algorithm by showing how to compute the failure function.

We also began studying dynamic programming. See chapter 16 of the text. Dynamic programming is used to solve optimization problems that have the following characteristic.

Optimal solutions to a problem are build from optimal solutions to subproblems.
Dynamic programming typically starts with a recursive algorithm, building up an optimal solution to a problem by finding optimal solutions to subproblems. But the recursion usually has the property that it computes the same thing many time. To avoid that, dynamic programming employs memoizing, storing the solutions to things that have already been computed so that they do not need to be computed again.

16. [7/14/00] We did examples of dynamic programming, including the edit distance problem. We began to look at the matrix chain multiplication problem.

17. [7/17/00] We finished the matrix chain multiplication problem, and started looking at the all-pairs shortest path problem.

18. [7/18/00] We finished dynamic programming by finishing the all-pairs shortest paths problem (the Floyd/Warshall algorithm), and doing the longest common subsequence problem.

19. [7/19/00] We began looking at greedy algorithms. We discussed Kruskall's algorithm for computing minimal spanning trees.

20. [7/20/00] We looked at Prim's algorithm for minimal spanning trees.

21. [7/21/00] We used a variation of Prim's algorithm to computes shortes paths. The shortest path algorithm is called Dijkstra's algorithm. We also looked at the activity selection algorithm discussed in the book in chapter 17.

22. [7/24/00] We began looking at the theory of NP-completeness. We defined the class P of problems solvable in polynomial time. We defined the notion of reductions among problems.

23. [7/25/00] We continued looking at NP-completeness. We defined the class NP, and defined NP-complete problems.

24. [7/26/00] We will look at particular NP-complete problems, and examples of reductions among problems that are used to show a problem NP-complete.