Asymptotic Notation Midterm 1 Review Problems, Study notes of Computer Science

Download Asymptotic Notation Midterm 1 Review Problems and more Study notes Computer Science in PDF only on Docsity! CMPS 101 Midterm 1 Review Problems 1. Let )(nf and )(ng be asymptotically non-negative functions which are defined on the positive integers. a. State the definition of ))(()( ngOnf = . b. State the definition of ))(()( ngnf ω= 2. State whether the following assertions are true or false. If any statements are false, give a related statement which is true. a. ))(()( ngOnf = implies ))(()( ngonf = . b. ))(()( ngOnf = if and only if ))(()( nfng Ω= . c. ))(()( ngnf Θ= if and only if Lngnf n = ∞→ ))(/)((lim , where ∞<< L0 . 3. Prove that ))()(())(())(( ngnfngnf ⋅Θ=Θ⋅Θ . In other words, if ))(()(1 nfnh Θ= and ))(()(2 ngnh Θ= , then ))()(()()( 21 ngnfnhnh ⋅Θ=⋅ . 4. Use limits to prove the following (these are some of the exercises at the end of the asymptotic growth rates handout): a. If )(nP is a polynomial of degree 0≥k , then )()( knnP Θ= . b. For any positive real numbers α and β : )( βα non = iff βα < , )( βα nn Θ= iff βα = , and )( βα ω nn = iff βα > . c. For any positive real numbers a and b: )( nn boa = iff ba < , )( nn ba Θ= iff ba = , and )( nn ba ω= iff ba > . d. ))(())(()( nfnfonf Θ=+ . 5. Use Stirling’s formula: ( ))/1(12! n e n nn n Θ+⋅
    ⋅= π , to prove that )log()!log( nnn Θ= . 6. Use Stirling’s formula to prove that

     Θ=

     nn n n42 . 7. Consider the following sketch of an algorithm called ProcessArray which performs some unspecified operation on a subarray ][ rpA
. ProcessArray(A, p, r) (Preconditions: 1≥p and ][Alengthr ≤ ) 1. do something which takes constant time. 2. if rp < 3.   +← 2 rp q 4. ProcessArray(A, p, q) 5. ProcessArray(A, q+1, r) Write a recurrence which gives the running time )(nT of this algorithm, when called on the full array ]1[ nA
. Give a tight asymptotic solution to this recurrence.