Algorithm Design & Complexity Analysis - IT Assignment Help

Assignment Task

Task    

Problem 1 - at most 1/2 page). Determine if the following statements are true or false AND provide a formal proof using either limits or the definitions of the big-O, big-Omega, and big-Theta notations. For instance, to prove that f (n) ∈ O(g(n)) or f (n) ∉ O(g(n)), using the definitions of big-O, we need to demonstrate the existence of a constant c and a sufficient large n0 such that f (n) ≤ c g(n) for all n ≥ n0, or showing that there are no such values. Using limits, for instance, we need to show that limn→∞ f (n)/g(n) < ∞ for f (n) ∈ O(g(n)), or showing that limn→∞ f (n)/g(n) = ∞ for f (n) ∉ O(g(n)). Note that there will be no marks if only true/false answer is given. 

• a) [] p4n2 +2n ∈ Θ(n). 
• b) [] n100 ∈ O(en). 
• c) [] loge(n2) ∈ ?(pn). 

Problem 2 (- at most 2 sentences). Arrange the following functions in an in creasing order of their growth rates. 

• f1(n) = 1000n, 
• f2(n) = (logn)2, 
• f3(n) = n!, 
• f4(n) = nlogn, 
• f5(n) = 2n. 

Problem 3 (- at most 1 page). In the following questions we assume that m grows more slowly than logn. 

a) Design another algorithm (description + pseudo code complexity analysis) to find m smallest elements (possibly repeated) in an array of real numbers of size n with time complexity O(nlogm). Extra data structures are allowed. 

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