Matrix Fraud

For this problem, a matrix is a binary matrix if every entry is $0$ or $1$. A binary matrix is a banded binary matrix if it satisfies the following properties:

1. Every row has at least one $1$.
2. Every column has at least one $1$.
3. All $1$s in each row are contiguous.
4. For row $i > 1$, let $s_i$ be the leftmost column with a $1$ entry and $t_i$ the rightmost. Then $s_i \ge s_{i-1}$ and $t_i \ge t_{i-1}$.

The Immoral Cartel of Pure Cozeners (ICPC) wants to publish results from a data set, but its matrix may not be banded. The group toggles cells, changing a $0$ to a $1$ or a $1$ to a $0$, so that the matrix becomes a banded binary matrix. Find the fewest toggles needed to turn the given matrix into a banded binary matrix.

The first line contains two integers $r$ and $c$ ($1 \le r \times c \le 2 \cdot 10^5$), the number of rows and columns. Each of the next $r$ lines contains a binary string of length $c$, which is a row of the matrix.

Print a single integer, the minimum number of entries to toggle to make the matrix a banded binary matrix.