Proof:
f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0 (14logn)^2 + 1.6n = O(n) => (14logn)^2 + 1.6n <= O(n) Let's assume c = 2 => (14logn)^2 + 1.6n <= O(n) => (14logn)^2 + 1.6n <= 2(n) => (14logn)^2 <= 0.4(n) => 2.5(14logn)^2 <= (n) it's true for all n >= 100 so, (14logn)^2 + 1.6n = O(n) given c = 2 and n0 = 100, using the definition of Big-O
1. What is the best asymptotic ("big-O”) characterization of the following function: f(n) = (14logn)2 +...
Example 3: The Growth of Functionsand Asymptotic notation a) Show that x is O(x )but that r is not O(x b) Give as good a big-O estimate as possible for each of the following (A formal proof is not required, but give your reasoning): log,n! 7n n +nlo 3n2 +2n+4 . (n log, (log,n") 2 42" c) Which of the functions in part b) above has the fastest growth rate? d) Show that if f(x) is Ollog, x)where b>1, and...
Provide a closed-form expression for the asymptotic growth of n + n/2 + n/3 + … + 1. Determine the big-O growth of the function f(n-WTgn. Explain and show work.
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.
Which big-O expression best characterizes the worst case time complexity of the following code? public static int foo(int N) ( int count = 0; int i1; while (i <N) C for (int j = 1; j < N; j=j+2) { count++ i=i+2; return count; A. O(log log N) B. O(log N2) C. O(N log N) D. O(N2)
1. Give the big-O characterization of the following loops, in terms of parameter n, and justify your answer: a) for (int i=1; i<=n, i++) {for (int j=1; j<=n; j++) {a constant-time operation}} b) for (int i=1;i<=n, i++) {for (int i=1; j<=i; j++) {a constant-time operation}} c) for (int i=1;i<=n*n, i++) {for (int j=1; j<=n; j++) {a constant-time operation }} d) for (int i=1; i<=n*n, i++) {for (int j=1; j<=i; j++) {a constant-time operation }} e) for (int i=1; i<=n, i++)...
Q-6e: Determine the big-O expression for the following T(N) function: T(1) = 1 T(N) = 2T(N – 1)+1 O 0(1) O O(log N) OO(N2) O O(N log N) O 0(2) OO(N)
Need help with 1,2,3 thank you. 1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether...
Give a good big-Oh characterization in terms of n of the running time of the following. Provide brief justification for your answer (in terms of finding a k and n_0). 4n^5 + 3n^3 + 7 15n^12 + 3n log n + 2n 3n log n + 2log n + n 12n*3^n + 50n
Choose the equivalent Big Oh notation for the functions given below. If there is more than one option, circle the tightest asymptotic bound. function f(n) = 5n - 10 belongs to a) O(1) b) O(n) c) O(n2) d) O(log n) function f(n) = 4n2 + 4n + 1 belongs to a) O(1) b) O(n) c) O(n2) d) O(log n) function f(n) = n2 + 100 log n belongs to a) O(1) b) O(n) c) O(n2) d) O(log n)
F(n)=n log n what is the most restrictive polynomial time asymptotic upperbound O(n^k) what is k?