Problem:
Score : 400 points

Problem Statement

Consider a grid with H rows and W columns of squares. Let (r, c) denote the square at the r-th row from the top and the c-th column from the left.
Each square is painted black or white.

The grid is said to be good if and only if the following condition is satisfied:

From (1, 1), we can reach (H, W) by moving one square right or down repeatedly, while always being on a white square.

Note that (1, 1) and (H, W) must be white if the grid is good.

Your task is to make the grid good by repeating the operation below. Find the minimum number of operations needed to complete the task. It can be proved that you can always complete the task in a finite number of operations.

Choose four integers r_0, c_0, r_1, c_1(1 \leq r_0 \leq r_1 \leq H, 1 \leq c_0 \leq c_1 \leq W). For each pair r, c (r_0 \leq r \leq r_1, c_0 \leq c \leq c_1), invert the color of (r, c) - that is, from white to black and vice versa.

Constraints

2 \leq H, W \leq 100

Input

Input is given from Standard Input in the following format:

H W
s_{11} s_{12} \cdots s_{1W}
s_{21} s_{22} \cdots s_{2W}
\vdots
s_{H1} s_{H2} \cdots s_{HW}

Here s_{rc} represents the color of (r, c) - # stands for black, and . stands for white.

Output

Print the minimum number of operations needed.

Sample Input 1

3 3
.##
.#.
##.

Sample Output 1

1

Do the operation with (r_0, c_0, r_1, c_1) = (2, 2, 2, 2) to change just the color of (2, 2), and we are done.

Sample Input 2

2 2
#.
.#

Sample Output 2

2

Sample Input 3

4 4
..##
#...
###.
###.

Sample Output 3

0

No operation may be needed.

Sample Input 4

5 5
.#.#.
#.#.#
.#.#.
#.#.#
.#.#.

Sample Output 4

4

Solution in Rust (1.42.0) (judge verdict: Wrong Answer)

#![allow(non_snake_case)]
#![allow(unused_imports)]
use ascii::AsciiChar;
use ascii::AsciiStr;
use itertools::Itertools as _;
use permutohedron::LexicalPermutation as _;
use petgraph::algo::dijkstra;
use petgraph::algo::min_spanning_tree;
use petgraph::data::FromElements;
use petgraph::graph::NodeIndex;
use petgraph::graph::UnGraph;
use petgraph::unionfind::UnionFind;
use proconio::fastout;
use proconio::input;
use proconio::marker::Chars;
use proconio::marker::Usize1;
use std::cmp::max;
use std::cmp::min;
use std::collections::HashMap;
use std::collections::HashSet;
use std::collections::VecDeque;
use superslice::Ext as _;

#[fastout]
fn main() {
    input! {
        H: usize,
        W: usize,
        mut Fieldss: [Chars; H],
    }

    // ルートを全探索するなら、(H+W-2)C(H-1)
    // H <= 13とかならいけるけど、100だとムリ
    // O(HWW)かO(HHW)かな

    // その地点に辿り着くための最小の操作の回数
    // ただし、「最後に掘った方向」に掘り進むのはノーコストでできるので、それを覚えておく必要がある
    // dp[h][w][1] : 直近で下方向に進んで(h,w)に辿り着く最短の回数
    // dp[h][w][0] : 直近で右方向に進んで(h,w)に辿り着く最短の回数
    let mx = H * W + 1;
    let mut dp = vec![vec![vec![mx; 2]; W]; H];
    let cnt0 = if Fieldss[0][0] == '#' { 1 } else { 0 };
    dp[0][0][0] = cnt0;
    dp[0][0][1] = cnt0;

    // h = 0、つまり右方向に進むのみ
    for w in 1..W {
        if Fieldss[0][w - 1] == '.' && Fieldss[0][w] == '#' {
            dp[0][w][0] = dp[0][w - 1][0] + 1;
        } else {
            dp[0][w][0] = dp[0][w - 1][0];
        }

        // dp[0][w][1] = 無限
    }

    for h in 1..H {
        if Fieldss[h - 1][0] == '.' && Fieldss[h][0] == '#' {
            dp[h][0][1] = dp[h - 1][0][1] + 1;
        } else {
            dp[h][0][1] = dp[h - 1][0][1];
        }

        // dp[h][0][0] = 無限
    }

    for h in 1..H {
        for w in 1..W {
            // →
            dp[h][w][0] = match (Fieldss[h][w - 1], Fieldss[h][w]) {
                ('.', '.') => min(dp[h][w - 1][0], dp[h][w - 1][1]),
                ('.', '#') => min(dp[h][w - 1][0], dp[h][w - 1][1]) + 1,
                ('#', '.') => min(dp[h][w - 1][0], dp[h][w - 1][1]),
                ('#', '#') => min(dp[h][w - 1][0], dp[h][w - 1][1] + 1),
                _ => mx + 1,
            };

            // ↓
            dp[h][w][1] = match (Fieldss[h - 1][w], Fieldss[h][w]) {
                ('.', '.') => min(dp[h - 1][w][0], dp[h - 1][w][1]),
                ('.', '#') => min(dp[h - 1][w][0], dp[h - 1][w][1]) + 1,
                ('#', '.') => min(dp[h - 1][w][0], dp[h - 1][w][1]),
                ('#', '#') => min(dp[h - 1][w][0] + 1, dp[h - 1][w][1]),
                _ => mx + 1,
            };
        }
    }

    // for dbg
    // for h in 0..H {
    //     for w in 0..W {
    //         eprint!("({},{}) ", dp[h][w][0], dp[h][w][1]);
    //     }
    //        eprintln!("");
    // }

    let ans = min(dp[H - 1][W - 1][0], dp[H - 1][W - 1][1]);
    println!("{}", ans);
}