Minimum Number of Operations to Satisfy Conditions

Pick a single value for each column (so every column is constant) such that adjacent columns have different values, minimizing the total number of cell changes; this becomes a min-cost coloring of a path where each column has up to 10 possible colors and costs equal mismatches. With m,n ≤ 1000 and values 0–9, the core solution is DP over columns (10 states per column) enforcing different neighboring colors.