Consider the recurrence T(1) = 0, T(n) = T([n/2]) +T([n/2])+n. (a) Let D(n) = T(n +1) - T(n). It is a fact that D(1) = 2, D(n) = D([n/2])+1. Prove using the strong form of induction that for any n e N, if n > 1 then D(n) = [lg n] +2. (b) Then prove that T(n) - T(1) = E= D(k), and show that an immediate consequence is that T(n) = Skri (Llg k] + 2). (c) Now show that T(n) = Ex=1 (Llg k] + 2) implies that T(n) = O(n log(n)).

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