Question

R-1.11 Give a big-Oh characterization, in terms of n, of the running time of the Loop1 method shown in Algorithm 1.21.

R-1.12 Perform a similar analysis for method Loop2 shown in Algorithm 1.21.
R-1.13 Perform a similar analysis for method Loop3 shown in Algorithm 1.21.
R-1.14 Perform a similar analysis for method Loop4 shown in Algorithm 1.21.
R-1.15 Perform a similar analysis for method Loop5 shown in Algorithm 1.21.
Algorithm Loop1(n):
s←0
for i ←1 to n do
s ←s+i

Answer 1

The running time of Algorithm Loop1 can be characterized by a big-Oh notation of O(n), indicating a linear increase in running time proportional to the input size n.

Step-by-step explanation:

To analyze the running time of Algorithm Loop1, we observe that the loop runs n times as it starts at 1 and increments the counter i until it equals n. Each iteration of the loop performs a constant number of operations (one addition and one assignment). Therefore, the total running time of the loop is proportional to n.

Given this, the big-Oh characterization of the running time of the Loop1 method is O(n). This signifies that the running time increases linearly with the size of the input n. For larger values of n, the running time will grow in a linear fashion in relation to n.

Answer 2

The running time of Loop1 is O(n^2), Loop2 is O(n!), Loop3 is O(2^n), Loop4 is O(log n), and Loop5 is O(1).

Step-by-step explanation:

The running time of the Loop1 method shown in Algorithm 1.21 is characterized by the summation of the numbers from 1 to n. Using the formula for the sum of an arithmetic series, the running time can be expressed in big-Oh notation as O(n^2).

The Loop2 method in Algorithm 1.21 has a running time characterized by the multiplication of the numbers from 1 to n. Using the formula for the factorial of n, the running time can be expressed in big-Oh notation as O(n!).

Similarly, the Loop3 method has a running time characterized by the exponentiation of 2 to the power of n. Therefore, the running time can be expressed in big-Oh notation as O(2^n).

The Loop4 method has a running time characterized by the logarithm of n. Using the formula for the logarithm of n, the running time can be expressed in big-Oh notation as O(log n).

The Loop5 method has a running time characterized by a constant time complexity. Therefore, the running time can be expressed in big-Oh notation as O(1).