University:
California State University, Northridge
Course:
COMP 482 | Algorithm Design and Analysis
Academic year:

2024
Views:

329

Pages:

6
Author:

Samuel K.

COMP 482 Sample Problems: Big-O-⌦-⇥ Problem 1. (10 points) For each of the following statements determine whether it is true of false. (+1 for a correct answer, -0.5 for an incorrect answer, 0 for a blank). If f (n) = O(g(n)) and g(n) = O(f (n)) then f (n) = g(n). If f (n) =pO(n2 ) then f (n) = O(n3 lg(n)). n = O(2 lg(n) ) 2 n = O(2lg(n )/2 ) If f (n) = O(g(n)) then g(n) = ⌦(f (n)). If f (n) = ⌦(g(n)) and g(n) = ⌦(h(n)) then f (n) = ⌦(h(n)). If f (n) = O(r(n)) and g(n) = O(s(n)) then f (n)/g(n) = O(r(n)/s(n)). If f (n) = O(n lg(n)) then f (n) = O(n2 ). If f (n) = ⌦(n lg(n)) then it must be the case that f (n) = ⌦(n2 ). F T F T T T F T F Problem p 2. (10 points) State the definition of f (n) = O(g(n)) and use the definition to show 3 lg(n) n2 10 + 4 lg((2n)n ) = O(n lg n)). Problem p3. (10 points) State the definition of f (n) = ⌦(g(n)) and use the definition to show 3 lg(n + 4) n2 10 + 4 lg((2n)n ) = ⌦(n lg n)). Problem 4. (10 points) Use the definition of f (n) = ⌦(g(n))) to show: If f (n) = ⌦(g(n)) and h(n) = ⌦(g(n)) then f (n) + h(n) = ⌦(g(n)). Problem p 5. (10 points) State the definition of f (n) = ⌦(g(n)) and use the definition to show 4 lg(n) n2 12 + 3 lg((2n)n ) = ⌦(n lg n)). Problem 6. (10 points) Use the definition of f (n) = O(g(n))) to show: If f (n) = O(g(n)) and g(n) = O(h(n)) then f (n) g(n) = O(h(n)). Problem 7. (10 points) Use the definition of f (n) = O(g(n))) to show: If f (n) = O(g(n)) and h(n) = O(g(n)) then f (n) + h(n) = O(g(n)). Problem 8. (10 points) State the definition of f (n) = O(g(n)) and f (n) = ⌦(h(n)) and use these definitions to show: If f (n) = O(g(n))) and f (n) = ⌦(h(n)) then h(n) = O(g(n)). Problem 9. (10 points) State the definition of f (n) = O(g(n)) and use the definition to show that p 3 (4n 2) lg(n) + 2n + 5 lg(n) = O (lg(nn )) Problem 10. (10 points) State the definition of f (n) = ⌦(g(n)) and use the definition to prove: If f (n) = ⌦(g(n)) and g(n) = ⌦(h(n)) then f (n) · g(n) = ⌦((h(n))2 ). Problem 11. (10 points) State the definition of f (n) = ⌦(g(n)) and use the definition to prove: If f (n) = ⌦(g(n)) and g(n) = ⌦(h(n)) then f (n) = ⌦(h(n)). Problem 12. (10 points) Show that f (n) = (4n2 + n 17) lg(n3 2n + 12) is ⇥(n2 lg(n)). Problem 13. (10 points) Give an example where f (n) = ⇥(g(n)) and h(n) = ⇥(g(n)), but f (n) T heta(g(n)). h(n) 6= f(n) = f (n) = O(g(n) C · g(n) -n if I = no constants f(n) no22 f(u) noz0 (102 = 0 [3log(n) 2 + 4 nlog (2n) blognon + 4(2) mlogn [Inlogn + Enlogn - C = 11 no < In < n < f(n) C = nlogn · g(n)fn if I 1 (n2 = no constants < h3 logne <.... in <2nd23nhc nz 3log(n) n2 = > 2 - - 10 n2 Unlog(2n) Inlog() = 4 (102 noz0 f(n) 31g(n + 4) = f(r) = average = f(n) (nign) - 1 f(n) (g(n) f (n)] that IInlogn ↳ & logh such no ? 2 + + = such r(nign) 41g((2n)n) Unlog (2n) nx] that " not Yn f(n) f(n) g(n) c = g(n) - - g(n) - g(n) (2(h(n)) + (, = C(zh(n) [( , (2 22 + g(n)) + = · C , (2 (f(n) < (2 h(n) = 0(h(n) = + f(n) < 0 (g(n) g (n) < (h(n)) if f (n) 1 C (2h(n)g(n) 0 ,. = (2 - if I constants 2 , & 30 I constants (2 M , =0 +(n) 0 = u, g(u)-(2h(n) Un = n2 g(h) h(n) (2)h(n) + [C(h(n) If(n) - g(n)) (h(n)) < = = o(n(n)) nam , f(n) = h(r) 0(g(n) if o(g(n) if = f (n) h(n) + f(n) J < = ( = c = c + 30$1210 such , (2301210 constants f(u) n , fin) = (2 h(n) . Vnznz to - = . g(n) ( , = constants Ena now , -n + (n) f(n) 3 = (4n-2) log(n) un2 logh 23 = 3 h - 3(2)nlogn logn [Gnlogn = f(u) · + + = C - C f(n) f(u) = = en e · , . 2n + Slog(n) Zelogn Inlogn + = that 0(log(n) blog (n) = f(n) (g(n) 0(nlogn) no z2 Snlogn + Inlogn = g(n) 13 · (2h(n) , (2 · no · = 2 h(n) [h(n) (h(n >2) <(h(n)) (g(n)) (g(n) + C =C , C => + such Ro Inlogn + Bnlogn g(n) (301010 constants J 0(g(n)) if = = = er((h(n)(2) f(u) = <, g(n) & g(n)-(h(r) f(n) n2 = g(n) = f(n) but f(n) 5 n2 = h(n) + 2n + = nz + 3 O(g(n)) f(n)-h(n) - h(n) , + h(n) 0 = nz + 2n = 2n + 2 = 2n + 2 = O(g(n) (g(n)) + 5 - ice (nz + 3) = 0(g(n) + : 0(nz)

COMP 482 Sample Ch 2

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock