{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MidtermSolS04

MidtermSolS04 - UML CS 91.404 Midterm Exam Spring 2004...

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

View Full Document Right Arrow Icon
UML CS 91.404 Midterm Exam Spring, 2004 1 of 5 MIDTERM EXAM & SOLUTIONS 1. (20 points) What can you conclude? Given: ( ) ( ) n n n f lg 2 lg ) ( ) 2 Ω ( ) ) ( ) ( ) 3 lg lg 3 n n n f Θ 2 1 ) ( ) 1 n n f Ω a) (10 points) Can we conclude from statements (1)-(3) that ? )) ( ( ) ( 2 3 n f O n f Why or why not? Either prove or provide a counterexample. ) ( 1 n f 2 n ) ( 2 n f ) ( 3 n f () n n n n lg lg lg lg = SOLUTION : YES. First, observe that () . This can be verified by taking lg of both sides. Next, because lglgn > 2 for n > 16. From this we obtain the picture above. n n n n lg lg lg lg = () lg lg lg lg n n n n > = 2 n To show that , we first use the definition of Θ to conclude that . Since )) ( ( ) ( 2 3 n f O n f ) ( ) 3 lg lg n n f n ) ( ( ) ( lg lg 3 n n O n f Θ ( ) n n n n lg lg lg lg = , this implies, transitively, that . Now, () ) lg n n lg ( ) ( 3 O n f ( ) n n n lg 2 ) (lg ) ( Ω f implies, using
Background image of page 1

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

View Full Document Right Arrow Icon
UML CS 91.404 Midterm Exam Spring, 2004 2 of 5 transpose symmetry, that ( ) ( ) ) ( lg 2 lg n f O n n . Thus, transitively, . )) ( ( ) ( 2 3 n f O n f b) (10 points) Can we conclude from statements (1)-(3) that ? )) ( ( ) ( 3 1 n f O n f Why or why not? Either prove or provide a counterexample. SOLUTION : NO. Counter-example: and . n n = n n n f lg lg 3 ) ( = n f ) ( 1 2. (20 points) Recurrence In this problem, you will find a tight upper and lower bound on the closed- form solution for the following recurrence: = > + + = 1 1 1 3 8 5 6 ) ( 2
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 5

MidtermSolS04 - UML CS 91.404 Midterm Exam Spring 2004...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online