Need help with 1,2,3 thank you. 1. Order of growth (20 points) Order the following functions...
Order the following functions by growth rate: N, squrerootN, N1.5, N2, NlogN, N log logN, Nlog2N, Nlog(N2), 2/N,2N, 2N/2, 37, N2 logN, N3. Indicate which functions grow at the same rate.
Order the following functions by asymptotic growth rate. 2n log n + 2n, 210, 2 log n, 3n + 100 log n, 4n, 2n, n2 + 10n, n3, n log n2
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) = 0.5n3 . Prove that f(n) = O(g(n)) using the definition of Big-O notation. (You need to find constants c and n0). b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use the definition of big-O notation to prove that f(n) = O(g(n)) (you need to find constants c and n0) and g(n) = O(f(n)) (you need to find constants c and n0). Conclude that...
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove your claim. 157. f(n) -100n+logn, gn) (logn)2. 158,介f(n) = logn, g(n) = log log(n2). 159. . f(n)-n2/log n, g(n) = n(log n)2. 160·介介f(n)-(log n)106.9(n)-n10-6 . 161. (n)logn, g(n) (log nlog n 162. f(n) n2, gn) 3. Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove...
3-3 Ordering by asymptotic growth rates a. Rank the following functions by order of growth; that is, find an arrangement 81,82, 830 of the functions satisfying gi = Ω(82), g2 Ω(83), , g29 = Ω(g30). Partition your list into equivalence classes such that functions f(n) and g(n) are in the same class if and only if f(n) = Θ(g(n)) Chaptr3 Growth of Functions 1n In Inn lg* g nn-2" n'ln Ig nIn n 2" nlgn 22+1 b. Give an example...
Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class. a)n^15log n + n^9 is O(n^9 log n) b) 15^7n^5 + 5n^4 + 8000000n^2 + n is Θ(n^3) c) n^n is Ω (n!) d) 0.01n^9 + 800000n^7 is O(n^9) e) n^14 + 0.0000001n^5 is Ω(n^13) f) n! is O(3n)
2. Asymptotic Notation (8 points) Show the following using the definitions of O, Ω, and Θ. (1) (2 points) 2n 3 + n 2 + 4 ∈ Θ(n 3 ) (2) (2 points) 3n 4 − 9n 2 + 4n ∈ Θ(n 4 ) (Hint: careful with the negative number) (3) (4 points) Suppose f(n) ∈ O(g1(n)) and f(n) ∈ O(g2(n)). Which of the following are true? Justify your answers using the definition of O. Give a counter example if...
1. (10 points) Write an efficient iterative (i.e., loop-based) function Fibonnaci(n) that returns the nth Fibonnaci number. By definition Fibonnaci(0) is 1, Fibonnaci(1) is 1, Fibonnaci(2) is 2, Fibonnaci(3) is 3, Fibonnaci(4) is 5, and so on. Your function may only use a constant amount of memory (i.e. no auxiliary array). Argue that the running time of the function is Θ(n), i.e. the function is linear in n. 2. (10 points) Order the following functions by growth rate: N, \N,...
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or disprove each of the following statements T a、 f(n) + g(n)=0(max(f(n), g(n))) 1. b. f(n) = 0(g(n)) implies g(n) = Ω(f(n)) T rc. f(n)- o F d. f(n) o(f(n)) 0(f (n)) f(n)=6((f(n))2)
Arrange the following functions in ascending order of asymptotic growth rate; that is if function g(n) immediately follows function f(n) in your list, then it should be the case that f(n) is O(g(n)): 2 Squareroot log n, 2^n, n^4/3, n(log n)^3, n log n, 2 2^n, 2^n^2. Justify your answer.