7-2 Quicksort with equal element values¶
The analysis of the expected running time of randomized quicksort in Section 7.4.2 assumes that all element values are distinct. This problem examines what happens when they are not.
a. Suppose that all element values are equal. What is randomized quicksort’s running time in this case?
b. The PARTITION procedure returns an index $q$ such that each element of $A[p:q-1]$ is less than or equal to $A[q]$ and each element of $A[q+1:r]$ is greater than $A[q]$. Modify the PARTITION procedure to produce a procedure PARTITION'(A,p,r), which permutes the elements of $A[p:r]$ and returns two indices $q$ and $t$, where $p\leq q \leq t \leq r$, such that
- all element of $A[q:t]$ are equal,
- each element of $A[p:q-1]$ is less than $A[q]$, and
- each element of $A[t+1:r]$ is greater than $A[q]$.
Like PARTITION, your PARTITION' procedure should take $\Theta(r-p)$ time.
c. Modify the RANDOMIZED-PARTITION procedure to call PARTITION', and name the new procedure RANDOMIZED-PARTITION'. Then modify the QUICKSORT procedure to produce a procedure QUICKSORT'(A,p,r) that calls RANDOMIZED-PARTITION' and recurses only on partitions where elements are not known to be equal to each other.
d. Using QUICKSORT', adjust the analysis in Section 7.4.2 to avoid the assumption that all elements are distinct.