Arrange the functions below in non-decreasing order such that if f_i appears bef
ID: 3882815 • Letter: A
Question
Arrange the functions below in non-decreasing order such that if f_i appears before f_j, then f_i (n) elementof O(f_j (n)). f_1 (n) = 10^20 f_2 (n) = (lg n)^4 f_3 (n) = 4^n f_4 (n) = n lg n f_5 (n) = n^3 - 100n^2 f_6 (n) = n + lg n f_7 (n) = lg lg n f_8 (n) = n^0.1 f_9 (n) = lg n^5 Group the following functions into classes so that two functions f(n) and g(n) are in the same class if and only if f(n) elementof theta (g(n)). List the classes in increasing order of magnitude of its members. A class may consist of one or more members. f_1 (n) = 6000 f_2 (n) = (lg n)^6 f_3 (n) = 3^n f_4 (n) = lg n f_5 (n) = n + lg n f_6 (n) = n^3 f_7 (n) = n^2 lg n f_8 (n) = n^2 - 100n f_9 (n) = 4n + squareroot n f_10 (n) = lg lg n^2 f_11 (n) = n^0.3 f_12 (n) = n^2 f_13 (n) = lg n^2 f_14 (n) = squareroot n^2 + 4 f_15 (n) = 2^nExplanation / Answer
1.
Here is the ascending order in terms of complexity:
f1(n) = 1020 (Constant)
f7(n) = lg lg n (Log-of-log)
f9(n) = lg n5 = 5lg n (log)
f2(n) = (lg n)4 = (Log-exponential complexity)
f8(n) = n0.1 (Sublinear complexity, n-root, n=0.1).
f6(n) = n+lg n (Linear)
f4(n) = nlgn (Linear*Log)
f5(n) = n3-100n2 (Polynomial)
f3(n) = 4n (Exponential)
2.
Class 1: f1(n) = 6000 (The constant complexity).
Class 2: f10(n) = lg lg n2 = lg (2lg n) (Log of log complexity).
Class 3: f4(n) = lg n, f13(n) = lg n2 = 2logn (Logarithmic complexity).
Class 4: f2(n) = (lg n)6 (Log-exponential complexity).
Class 5: f11(n) = n0.3 (Sublinear complexity, n-root, n=0.3).
Class 6: f5(n) = n+lg n, f9(n) = 4*n+n, f14(n) = (n2+4) (Linear complexity).
Class 7: f8(n) = n2-100n, f12(n) = n2 (Polynomial complexity).
Class 8: f7(n) = n2*lg n (Polynomial*Log complexity).
Class 9: f6(n) = n3 (Polynomial complexity).
Class 10: f3(n) = 2^n, f15(n) = 3^n. (Exponential complexity).