Fall 2002 CSCI 333 Homework 2 Solution

Problem 1

Give the simplest big-Θ for the following functions:

1. 15 n³ + 2 n² + 2002

   Θ( n³ )

2. 3ⁿ + n²⁰⁰² + 2002

   Θ( 3ⁿ )

3. 10³⁰ n

   Θ( n )

4. 15 n log n + 20 n

   Θ( n log n )

5. 17 log n

   Θ( log n )

6. 17 log₂ n

   Θ( log n )

7. 17 log₂ (n⁵)

   Θ( log n )

Problem 2

Give the simplest big-Θ in terms of n for the following nested loops:

1. for (i=0; i<=n; ++i)
  for (j=0; j<10000; ++j)
    ++sum ;

   Θ( n )

2. for (i=0; i<=n; ++i)
  for (j=0; j<n-10000; ++j)
    ++sum ;

   Θ( n² )

3. for (i=0; i<=n; i=i+3)
  ++sum ;

   Θ( n )

4. for (i=0; i<=n; i=i+(n/10000)+1)
  ++sum ;

   Θ( 1 )

5. for (i=0; i<=n; i=3*i+1)
  ++sum ;

   Θ( log n )

6. for (i=0; i<=n; i=10*i)
  ++sum ;

   It's an infinite loop! Therefore, it has no Θ class.

7. for (i=0; i*i<=n; ++i)
  ++sum ;

   Θ( n^½ )