# Sample problem 3 in this problem you are not allowed

This preview shows pages 2–3. Sign up to view the full content.

Sample Problem 3 In this problem you are NOT allowed to use the theorems about Big-Oh stated in the lecture notes. Your proof should follow just from the definition of Big-Oh. Let f ( n ) = 1 10 n n and g ( n ) = 100 n + 1000 Prove that f ( n ) is not O ( g ( n )). Answer Proof by contradiction. Suppose that f ( n ) is O ( g ( n )), i.e., N, c > 0 such that n N we have 1 10 n n c (100 n + 1000) First we transform a bit the last inequality into an equivalent one: 1 10 n n c (100 n + 1000) iff n n 1000 cn + 10000 c iff n ( n - 1000 c ) 10000 c This last inequality is contradicted if n > 1 and n - 1000 c > 10000 c , that is n > 11000 2 c 2 . Therefore the statement n N 1 10 n n c (100 n + 1000) is contradicted by taking n = max( N, 1 , 11000 2 c 2 ) (The reason we have 1 there is because our def of BigOh does not require N, c > 1.) Sample Problem 4 Analyze the following fragment of code and give a Big-Oh characterization of its running time (that is, give the best upper bound on the worst-case running time of the algorithm that you can find). Explain your analysis. public static void doWork (int[] arr) { for (int i=1; i < 10; i++) { for (int j=0; j < 3*i; j++) { for (int k=0; k < arr.length; k++) { System.out.print(i+j+k); } } } Answer Let n = arr . length be the size of the input.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern