Let $A$ be an array containing integer values. The distance of $A$ is defined as the minimum number of elements in $A$ that must be replaced with another integer so that the resulting array is sorted in non-decreasing order. The distance of the array $[2, 5, 3, 1, 4, 2, 6]$ is _________

To minimize the number of replacements needed to make the array sorted in non-decreasing order, we should maximize the number of elements that remain unchanged.

Since we can replace elements with any integer (and the condition is non-decreasing, i.e., $x_1 \le x_2 \le \dots \le x_n$), the elements that remain unchanged must form a longest non-decreasing subsequence (LNDS) of $A$.

Given the array:

where $n = 7$.

Let us find the length of the longest non-decreasing subsequence:

• A valid non-decreasing subsequence is $[2, 3, 4, 6]$ (indices $0, 2, 4, 6$), which has length $4$.

• No non-decreasing subsequence of length $5$ or more exists.

Therefore:

For instance, replacing the elements at index $1$ (value $5$), index $3$ (value $1$), and index $5$ (value $2$) with $2, 3, 4$ respectively gives the non-decreasing array $[2, 2, 3, 3, 4, 4, 6]$.

Thus, the distance of the array is 3.