Question 35

To minimize the number of elements replaced to make $A = [2, 5, 3, 1, 4, 2, 6]$ sorted in non-decreasing order, we maximize the number of elements kept unchanged, which must form a non-decreasing subsequence of $A$.

Inspecting $A = [2, 5, 3, 1, 4, 2, 6]$:

• The longest non-decreasing subsequence has length $L = 4$: specifically $[2, 3, 4, 6]$ (at indices $1, 3, 5, 7$).
• Keeping $[2, _, 3, _, 4, _, 6]$ fixed, we can replace the remaining $7 - 4 = 3$ elements ($5, 1, 2$) with integers $2, 3, 4$ respectively to obtain the non-decreasing array $[2, 2, 3, 3, 4, 4, 6]$.
• No non-decreasing subsequence of length $5$ exists.

Therefore, the distance of the array is $7 - 4 = 3$.