Minimum Operations to Make the Array K-Increasing

Partition the array into k independent subsequences of indices i, i+k, i+2k, ...; each subsequence must be non-decreasing for the array to be K-increasing, so the minimum operations equals the sum over subsequences of (length − length of their longest non-decreasing subsequence). Compute each subsequence's LNDS (e.g., with patience/binary-search) to get the result efficiently.