1. Prove that log2(n) is O(n)
2. Prove that log(n!) is O(n log(n))
How would I prove that log2(32n) = O(n) (Big Oh of N) I got: 2 log2(3) n <= c * n , however, I do not know how to continue from this part. Thanks
(a) Prove that n log^3 n is O(n^2). Prove that n^3 is not O(n^2 log n). (b) The multi Pop (i) method pops i items from the top of a stack. Analyse the amortized complexity of the multiPop (i) method.
4. a) Solve the equation log2 (x - 5) - log: ( x - 2) +1 algebraically. b) Verify your answers are reasonable by graphing both y = log2 (x - 5) and y = log: (x-2) +| on the same set of axes.
solev log(x) + log: (x − 2) = log2 (x + 10)
1. For each of the following, prove using the definition of O): (a) 7n + log(n) = O(n) (b) n2 + 4n + 7 =0(na) (c) n! = ((n") (d) 21 = 0(221)
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Prove this using the definition R7: log(n*) is O(log n) for any fixed x > 0
Assuming we know from contour integration that r log(t Jo t (1+t) Deduce that 1 log2(t)dt-T3 o t1/2(1 +t) 2 Without further integration Assuming we know from contour integration that r log(t Jo t (1+t) Deduce that 1 log2(t)dt-T3 o t1/2(1 +t) 2 Without further integration
4. Suppose T (n) satisfies the recurrence equations T(n) = 2 * T( n/2 ) + 6 * n, n 2 We want to prove that T (n)-o(n * log(n)) T(1) = 3 (log (n) is log2 (n) here and throughout ). a. compute values in this table for T (n) and n*log (n) (see problem #7) T(n) | C | n * log(n) 2 4 6 b. based on the values in (a) find suitable "order constants" C and...
Using a recurrence relation, prove that the time complexity of the binary search is O(log n). You can use ^ operator to represent exponentiation operation. For example, 2^n represents 2 raised to the power of n.