1. (10 pts total) For parts (1a) and (1b), justify your answers in terms of deterministic QuickSort, and for part (1c), refer to Randomized QuickSort. In both cases, refer to the versions of the algorithms given in the lecture notes for Week 3. (a) (3 points) What is the asymptotic running time of QuickSort when every element of the input A is identical, i.e., for 1 i,j n, A[i] = A[j]? Prove your answer is correct. (b) (3 points) Let the input array A be [2,1,1,4,5,4,6,3,3,0]. What is the number of times a comparison is made to the element with value 3 (note the minus sign)? (c) (4 points) How many calls are made to random-int in (i) the worst case and (ii) the best case? Give your answers in asymptotic notation.

Randomized-Quicksort(A,p,r) {

if (p

q = RPartition(A,p,r)

Randomized-Quicksort(A,p,q-1)

Randomized-Quicksort(A,q+1,r)

}}.

RPartition(A,p,r) {

i = random-int(p,r) // NEW: choose uniformly random integer on [p..r]

swap(A[i],A[r]) // NEW: swap corresponding element with last element

x = A[r] // pivot is now a uniformly random element

i = p-1 // code from deterministic Partition

for (j=p; j<=r-1;j++) {

if A[j]<=x { // same as before

i++ //

swap(A[i],A[j]) //

}}

swap(A[i+1],A[r]) //

return i+1 //

},