Question

For each of the following problems, simplify and express your answer as Θ(nk) or Θ(nk(log n wherever possible. If the asymptotic running time is exponential, then just give exponential lower bounds. Random(n) generates a random number between 1 and n with uniform distribution (every integer between 1 and n is equally likely.) CoinFlip) returns heads or tails with equal probability.

Answer 1

Given,

Random generate a random number between 1 to n. So, it has to trigger a event to select a number between 1 to n with complexity on constant value.

The function is recursive in nature.

1. The first if condition is initial condition to trap value and end recursion and evaluate the value.

2. Second for block runs from 1 to n take O(n) time and increments the value of s as a sum.

3. Third for block runs const but the line no 9 feed the value into recursion chain with value n bringing near O(log n)

(a). The asymptotic worst case running time of Func5 will be O(n²).

(b). The asymptotic expected running time of Func5 will be O(n log n)

For better understanding we can take example value and go through program step wise and get the value and aggregate the value in terms of n, neglecting constant values.