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Consider the following function that takes as input a sequence $A$ of integers with n elements, $A$$\left [ A1 \right ]$, $\left [ A2 \right ], \ldots, \left [ An \right ]$and an integer $k$ and returns an integer value. The function length$(S)$ returns the length of sequence $S$. Comments start with //.

function mystery(A, k){
n = length(A);
if (k > n) return A[n];
v = A[1];
AL = [ A[j] : 1 <= j <= n, A[j] < v ]; // AL has elements < v in A
Av = [ A[j] : 1 <= j <= n, A[j] == v ]; // Av has elements = v in A
AR = [ A[j] : 1 <= j <= n, A[j] > v ]; // AR has elements > v in A
if (length(AL) >= k) return mystery(AL,k);
if (length(AL) + length(Av) >= k) return v;
return mystery(AR, k - (length(AL) + length(Av)));
}
1. Explain what the function computes.
2. What is the worst-case complexity of this algorithm in terms of the length of the input sequence $A$?
3. Give an example of a worst-case input for this algorithm.
edited | 115 views

+1 vote
a. Function computes kth smallest element (if k<number of elements in the array, else it returns max element).
b. Time complexity = $O(k$*$n)$, where n is the number of elements in the array.
c. Worst case input: Array consists of a distinct increasing sequence of numbers and from this sequence, if we need to find nth smallest (i.e. max element) then it will take O$(n$*$n)$ time.