This question deals with the quicksort algorithm.
(a) Suppose that we modify the partitioning algorithm so that it always partitions an input array of length n into two partitions in such a way that the length of the left partition is n - K and the length of the right partition is K - 1 (for some constant K, where K > 0). Let us refer to this partitioning algorithm as KPartition. Now consider the following variation of quicksort called KQuickSort:
function KQuickSort(int[] A, int low, int high)
n = high - low + 1;
if n < K then
insertionSort(A, low, high);
else
q = KPartition(A, low, high);
KQuickSort(A, low, q-1);
KQuickSort(A, q+1, high);
end
end
Let T(n) denote the worst-case number of steps to run KQuickSort on input arrays of size n. Complete the following recurrence relation for T(n). Assume that KPartition takes Θ(n) steps to partition an array of size n.
T(n) ≤
{ n < K;
T(n) + Θ(n) + T(n - K) + Θ(n - K) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1) + T(n - K - 1) + Θ(n - K - 1) + Θ(n) + Θ(n) + T(K - 1) + Θ(K - 1)