Introduction to Algorithms third Edition exercise 7.3

7.3-1

In order to make the worst case randomization, we are concerned about the expected running time of the algorithm. 7.3-2

In the worst case, each time random gets the index of the largest or smallest element in the Subarray, each partition will have two sub-arrays of 0 and n−1 0 and n-1, so each layer node will only call the random, so T (n) =t (n−1) +θ (1) =θ (n) t (n) = T (n-1) +\theta (1) =\theta (n)
In the best case, every time random gets exactly the subscript of the element centered in the sub-array, each partition appears ⌊n/2⌋\lfloor N/2\rfloor and ⌊n/2⌋−1 \lfloor N/2 \rfloor-1 Two sub-arrays, so T (n) =2t (N/2) +θ (1) =θ (n) T (n) =2t (N/2) +\theta (1) =\theta (n)