{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

f00-hwex - CS 373 Combinatorial Algorithms Fall 2000...

This preview shows pages 1–3. Sign up to view the full content.

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

View Full Document
CS 373 Homework 0 (due 8/31/00) Fall 2000 2. (a) Prove that any positive integer can be written as the sum of distinct powers of 2 . For example: 42 = 2 5 + 2 3 + 2 1 , 25 = 2 4 + 2 3 + 2 0 , 17 = 2 4 + 2 0 . [Hint: “Write the number in binary” is not a proof; it just restates the problem.] (b) Prove that any positive integer can be written as the sum of distinct nonconsecutive Fi- bonacci numbers—if F n appears in the sum, then neither F n +1 nor F n 1 will. For exam- ple: 42 = F 9 + F 6 , 25 = F 8 + F 4 + F 2 , 17 = F 7 + F 4 + F 2 . (c) Prove that any integer (positive, negative, or zero) can be written in the form i ± 3 i , where the exponents i are distinct non-negative integers. For example: 42 = 3 4 3 3 3 2 3 1 , 25 = 3 3 3 1 + 3 0 , 17 = 3 3 3 2 3 0 . 3. Solve the following recurrences. State tight asymptotic bounds for each function in the form Θ( f ( n )) for some recognizable function f ( n ) . You do not need to turn in proofs (in fact, please don’t turn in proofs), but you should do them anyway just for practice. If no base cases are given, assume something reasonable but nontrivial. Extra credit will be given for more exact solutions.
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