1. Suppose we have hardware capable of executing 106 instructions per second. How long would it
take to execute an algorithm whose complexity function was:
T(n) = 2N2 on an input size of n= 108?
2. An algorithm takes 0.5 ms for input size 100. How long will it take for input size500 if
the running time is the following (assume low-order terms are negligible)?
a. linear
b. O(N logN)
c. quadratic
d. cubic
Answer:
a) 2.5 ms
b) 3.37 ms
c) 12.5 ms
d) 62.5 ms
Explanation:
Determine how long it takes for an input size of 500
a) Linear
For an input size of 500
time taken = ( ( 0.5 / 100 )* 500 ) ms = 2.5 ms
b) 0(N log N )
First step : determine number of instructions For an input size ( N ) = 100
Log N = 6.65
∴ N log N = 665 instructions
i.e. 665 instructions is passed during 0.5ms
For
N log N = 500
N500 = 4483 instructions
time taken = (4483 * 0.5)/665 = 3.37 ms.
c) Quadratic
lets take N = 100
N^2 = 10000 steps per 0.5 ms
For N = 500
N^2 = 250,000 steps
time taken = (25000*0.5) / 10000 = 12.5 ms
d) cubic .
lets take N = 100
N^3 = 1,000,000 steps per 0.5 ms
hence for N = 500
N^3 = 125*10^6 steps
hence time taken = ( (125*10^6)*0.5 / 10^6) = 62.5 ms
3.
An algorithm takes 0.5 ms for input size 100. How large a problem can be solved in 1 min if the
running time is the following (assume low-order terms are negligible)?
a. linear
b. O(N logN)
c. Quadratic
d. Cubic
4. Order the following functions by growth rate: N,√N, N1.5, N2, N logN, N log logN, N log2 N, N
log(N2), 2/N, 2N, 2N/2, 37, N2 logN, N3. Indicate which functions grow at the same rate.
5. Prove that for any constant k, logk N = o(N).
6. Suppose T1(N) = O(f (N)) and T2(N) = O(f (N)). Which of the following are true?
a. T1(N) + T2(N) = O(f (N))
b. T1(N) − T2(N) = o(f (N))
c. T1(N)/T2(N) = O(1)
d. T1(N) = O(T2(N))
7.
For each of the following six program fragments:
a. Give an analysis of the running time (Big-Oh will do).
b. Implement the code in the language of your choice, and give the running time
several values of N.
c. Compare your analysis with the actual
running times.
for