3. (i) Solve the following recurrence relation by expansion (substitution): for n ≥ 2, n a power of 2 T(n) = 8T() + n², T(1) = 1 (ii) Express T(n) in order, i.e., T(n) = (f(n)) for n ≥ 1, n a power of 2. (iii) Check your solution by plugging it back into the recurrence relation.

Q: f. T(n) = 2T (n/4) + √n.

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Q: apply the Master Theorem to show that T(n)=Θ(logn)

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Q: g. T(n) = T(n − 2) + n².

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Concept of Randomized Approximation Algorithms

Randomized algorithms are algorithms that use random numbers to determine what to do next at any point in their logic. This means that it uses randomization for operation in the logic. Due to this randomness, the time and space complexity of the standard algorithm is reduced.