CS 361: Algorithms and Data Structures
Homework 2

Assignments will be collected at the beginning of the class period. Please bring a hardcopy to turn in. Your assignment can be hand-written or typed.

1. Suppose an algorithm performs T(n) = 3(n+5)² + n operations on an input of size n.
   Show that T(n) is Θ(n²)


2. Suppose an algorithm performs T(n) = lg(n) + n operations on an input of size n.
   Show that T(n) is Θ(n). Be careful with your analysis.


3. Suppose that T₁(n) is O(f(n)) and T₂(n) is also O(f(n)) for some function f(n). Is it true that T₁(n) is O(T₂(n))? If so, prove this statement using the definition of Big-O. If not, provide a counterexample.


4. Suppose that T₁(n) is Θ(f(n)) and T₂(n) is also Θ(f(n)) for some function f(n). Is it true that T₁(n) is Θ(T₂(n))? If so, prove this statement using the definition of Big-Θ. If not, provide a counterexample.


5. Algorithm Design 2.6 parts (a) and (c)

The wording for question 2.6 is not as precise as it could be. In particular, part (a) says "give a bound". Please *prove* that your bound holds. That is, find a c and n₀ that fulfill the definition for Big-O.

Also, part (c) says "give an algorithm". Read the "Homework Guide" so you know what your answer must include.