ORIE 4520: Stochastics at ScaleFall 2015Homework 1: SolutionsSid Banerjee ([email protected])Problem 1: (Practice with Asymptotic Notation)An essential requirement for understanding scaling behavior is comfort with asymptotic (or ‘big-O’)notation. In this problem, you will prove some basic facts about such asymptotics.Part (a)Given any two functionsf(·) andg(·), show thatf(n) +g(n) = Θ(max{f(n), g(n)}). Get answer to your question and much more

Part (b)An algorithmALGconsists of twotunablesub-algorithmsALGAandALGB, which have to beexecuted serially (i.e., one run ofALGinvolves first executingALGAfollowed byALGB). Moreover,given any functionf(n), we can tune the two algorithms such that one run ofALGAtakes timeO(f(n)) andALGBtakes timeO(n/f(n)). How should we choosefto minimize the overall runtimeof ALG (i.e., to ensure the runtime of ALG isO(h(n)) for the smallest-growing functionh)?How would your answer change ifALGAandALGBcould be executed in parallel, and we haveto wait for both to finish?

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Part (c)We are given a recursive algorithm which, given an input of sizen, splits it into 2 problems of sizen/2, solves each recursively, and then combines the two parts in timeO(n). Thus, ifT(n) denotesthe runtime for the algorithm on an input of sizen, then we have:T(n) = 2T(n/2) +O(n)Prove thatT(n) =O(nlogn).1