Question

Two integers N<=10^5 and K<=N are given, where N is the size of array A[] and K is the length of continuous subsequence we can choose in our process.Each element A[i]<=10^9. Now suppose initially all the elements of array are unmarked. In each step we’ll choose any subsequence of length K and if this subsequence has unmarked elements then we will mark all the unmarked elements which are minimum in susequence. Now how to calculate minimum number of steps to mark all the elements?

For better understanding of problem see this example–

N=5 K=3
A[]=40 30 40 30 40

Step 1- Select interval [1,3] and mark A[1] and A[3]

Step2- Select interval [0,2] and mark A[0] and A[2]

Step 3- Select interval [2,4] and mark A[4]

Hence minimum number of steps here is 3.

My approach(which is not fast enough to pass)-

I am starting from first element of array and marking all the unmarked elements equal to it at distance <=K and incrementing steps by 1.

Answer 1

First consider how you’d answer the question for K == N (i.e. without any effective restriction on the length of subsequences). Your answer should be that the minimum number of steps is the number of distinct values in the array.

Then consider how this changes as K decreases; all that matters is how many copies of a K-length interval you need to cover the selection set {i: A[i] == n} for each value n present in A. The naive algorithm of walking a K-length interval along A, halting at each position A[i] not yet covered for that value of n is perfectly adequate.