Lecture 5

Learning objectives

After this class, you should be able to:

1. Explain the idea behind operation counts and asymptotic analysis of of the running time of algorithms.
2. Given an algorithm and an operation to be counted, derive the number of those operations executed by the algorithm, as a function of the input size.
3. Given the running time of an algorithm as a function of its input size n, and its actual running time s for some input of size m, derive its running time for a larger input size, as a function of s.

Reading assignment

1. Chapter 2.
2. Class notes.
3. Lecture: Algorithm Analysis.

Exercises and review questions

• Exercises and review questions on current lecture's material
  1. Write code or pseudocode to multiply two n x n matrices A and B, and store the result in a matrix C. The matrix product is defined by Cij = Sumk=0n-1 AikBkj, for each i, j, 0 < i, j < n. Determine the number of additions and multiplications performed by your algorithm, as a function of n.
  2. If your matrix multiplication algorithm took time s seconds on matrices of size n x n, then how many seconds will it take on matrices of size 2n x 2n?
  3. If the triangular matrix-vector multiplication algorithm shown in class took time s seconds on matrices of size n x n, then approximately how many seconds will it take on matrices of size 2n x 2n, for large n?
  4. If an Algorithm1 takes time t1(n) = 100 n + n2 and Algorithm2 takes time t2(n) = 10 n2, then which algorithm will you use in an application where n is typically less than 10? Which algorithm will you use if your application typically had n greater than 100?
  5. Consider an application where you need to insert information regarding a customer, delete information regarding a customer, and lookup information regarding a customer. Assume that there are two possible algorithms, each requiring the following number of operations, as a function of the number of customers:

     Algorithm1: insert = n operations, delete = log n operation, lookup = 1 operation.

     Algorithm2: insert = log n operations, delete = log n operation, lookup = log n operation.

     Note that insertion and deletion actually change the number of customers, but we will ignore that for the time being. Just assume that n is some fixed, large number. Which algorithm will you use if the application had frequent insertions and deletions, and infrequent lookups? Which algorithm will you use if the application had infrequent insertions and deletions, and frequent lookups?

  6. Prove, using induction, that 2n2 + 3n + 1 <4n2, n > 2. (You may give us a hardcopy of your solution, if you wish to get feedback from us.)

• Questions on next lecture's material
  1. None.