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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <algorithm>
#include <climits>
#include <cmath>
#include <deque>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <regex>
#include <set>
#include <unordered_map>
#include <vector>

const int mod = 1e9 + 7;
const int kmax = 510000;
const int last_days[] = { 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 };

long long fact[kmax], fact_inv[kmax], inv[kmax];

void init_comb()
{
    fact[0] = fact[1] = 1;
    fact_inv[0] = fact_inv[1] = 1;
    inv[1] = 1;
    for (int i = 2; i < kmax; i++) {
        fact[i] = fact[i - 1] * i % mod;
        inv[i] = mod - inv[mod % i] * (mod / i) % mod;
        fact_inv[i] = fact_inv[i - 1] * inv[i] % mod;
    }
}

long long comb(int n, int r)
{
    if (n < r) {
        return 0;
    }

    if (n < 0 || r < 0) {
        return 0;
    }

    return fact[n] * (fact_inv[r] * fact_inv[n - r] % mod) % mod;
}

template <typename T, T N>
class UnionFind {
    T parent_[N];
    T rank_[N];
    T size_[N];

public:
    UnionFind();
    T Root(T idx);
    bool IsSame(T x, T y);
    void Unite(T x, T y);
    T GetSize(T idx);
};

template <typename T, T N>
UnionFind<T, N>::UnionFind()
{
    for (T i = 0; i < N; i++) {
        parent_[i] = i;
        rank_[i] = 0;
        size_[i] = 1;
    }
}

template <typename T, T N>
T UnionFind<T, N>::Root(T idx)
{
    return parent_[idx] == idx ? idx : parent_[idx] = Root(parent_[idx]);
}

template <typename T, T N>
bool UnionFind<T, N>::IsSame(T x, T y)
{
    return Root(x) == Root(y);
}

template <typename T, T N>
void UnionFind<T, N>::Unite(T x, T y)
{
    x = Root(x);
    y = Root(y);
    if (x == y) {
        return;
    }

    if (rank_[x] < rank_[y]) {
        parent_[x] = y;
        size_[y] += size_[x];
    } else {
        parent_[y] = x;
        size_[x] += size_[y];
        if (rank_[x] == rank_[y]) {
            rank_[x]++;
        }
    }
}

template <typename T, T N>
T UnionFind<T, N>::GetSize(T idx)
{
    return size_[Root(idx)];
}

template <typename T>
T pow_mod(T n, T p, T m)
{
    if (p == 0) {
        return 1;
    }

    if (p % 2 == 0) {
        T t = pow_mod(n, p / 2, m);
        return t * t % m;
    }
    return n * pow_mod(n, p - 1, m) % mod;
}

template <typename T>
T nCr_mod(T n, T r)
{
    T x = 1;
    for (T i = n - r + 1; i <= n; i++) {
        x *= i;
        x %= mod;
    }

    T a = 1;
    for (T i = 1; i <= r; i++) {
        a *= i;
        a %= mod;
    }

    T y = pow_mod(a, mod - 2) % mod;

    return x * y % mod;
}

template <typename T>
bool is_prime(T n)
{
    if (n == 1) {
        return false;
    }

    for (T i = 2; i * i <= n; i++) {
        if (n % i == 0) {
            return false;
        }
    }

    return true;
}

template <typename T>
bool is_leap(T y)
{
    return (y % 4 == 0 && y % 100 != 0) || y % 400 == 0;
}

template <typename T>
void next_day(T& y, T& m, T& d)
{
    d++;

    if (d > last_days[m - 1] + (m == 2 && is_leap(y) ? 1 : 0)) {
        d = 1;
        m++;
    }

    if (m > 12) {
        y++;
        m = 1;
    }
}

template <typename T>
T fib(T n)
{
    T a = 0, b = 1;
    for (T i = 0; i < n; i++) {
        T t = a;
        a = b;
        b = a + t;
    }

    return a;
}

// Note that the order of this function is O(n**n).
template <typename T>
std::vector<size_t> calculate_ranks(const std::vector<T>& v)
{
    std::vector<T> sorted = v;

    std::sort(sorted.begin(), sorted.end());

    std::map<T, long long> m;
    for (auto i = 0LU; i < v.size(); i++) {
        m.insert(std::make_pair(sorted[i], i));
    }

    std::vector<size_t> rank(v.size());

    for (auto i = 0LU; i < v.size(); i++) {
        rank[i] = m.find(v[i])->second + 1;
    }

    return rank;
}

template <typename T>
std::map<T, T> prime_factors_and_num(T n)
{
    std::map<T, T> m;
    for (T i = 2; i <= n; i++) {
        while (n % i == 0) {
            m[i]++;
            n /= i;
        }
    }

    return m;
}

inline long long calculate_sum(const std::vector<long long>& v)
{
    return std::accumulate(v.begin(), v.end(), 0LL);
}

template <typename T, T N>
class Graph {
    std::vector<std::vector<T>> weights_;
    std::vector<T> predecessors_;
    std::vector<T> shortest_path_estimate_;
    std::vector<std::vector<T>> edges_;

    void InitializeSingleSource(T start);
    void Relax(T v, T u);

public:
    Graph(std::vector<std::vector<T>> weights, std::vector<std::vector<T>> edges);
    Graph(std::vector<std::vector<T>> weights, T inf);

    T BellmanFord(T start, T end);
    T Dijkstra(T start, T end);
};

template <typename T, T N>
Graph<T, N>::Graph(std::vector<std::vector<T>> weights, std::vector<std::vector<T>> edges)
{
    weights_ = weights;
    predecessors_ = std::vector<T>(N, -1);
    shortest_path_estimate_ = std::vector<T>(N, mod);
    edges_ = edges;
}

template <typename T, T N>
Graph<T, N>::Graph(std::vector<std::vector<T>> weights, T inf)
{
    weights_ = weights;
    predecessors_ = std::vector<T>(N, -1);
    shortest_path_estimate_ = std::vector<T>(N, inf);
    edges_ = std::vector<std::vector<T>>(N, std::vector<T>());

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            if (weights_[i][j] != inf) {
                edges_[i].push_back(j);
            }
        }
    }
}

template <typename T, T N>
void Graph<T, N>::InitializeSingleSource(T start)
{
    for (int i = 0; i < N; i++) {
        shortest_path_estimate_[i] = mod;
        predecessors_[i] = -1;
    }

    shortest_path_estimate_[start] = 0;
}

template <typename T, T N>
void Graph<T, N>::Relax(T u, T v)
{
    if (shortest_path_estimate_[v] > shortest_path_estimate_[u] + weights_[u][v]) {
        shortest_path_estimate_[v] = shortest_path_estimate_[u] + weights_[u][v];
        predecessors_[v] = u;
    }
}

template <typename T, T N>
T Graph<T, N>::BellmanFord(T start, T end)
{
    InitializeSingleSource(start);
    for (int i = 0; i < N - 1; i++) {
        for (int j = 0; j < N; j++) {
            for (auto edge : edges_[j]) {
                Relax(j, edge);
            }
        }
    }

    for (int i = 0; i < N; i++) {
        for (auto edge : edges_[i]) {
            if (shortest_path_estimate_[edge] > shortest_path_estimate_[i] + weights_[i][edge]) {
                fprintf(stderr, "Graph contains negative circle!\n");
                exit(1);
            }
        }
    }

    return shortest_path_estimate_[end];
}

template <typename T, T N>
T Graph<T, N>::Dijkstra(T start, T end)
{
    InitializeSingleSource(start);
    std::set<T> s;

    auto compare_d = [=](const T& x, const T& y) {
        return shortest_path_estimate_[x] > shortest_path_estimate_[y];
    };

    std::priority_queue<T, std::vector<T>, decltype(compare_d)> q(compare_d);
    for (int i = 0; i < N; i++) {
        q.push(i);
    }

    while (q.size()) {
        T u = q.top();
        q.pop();
        s.insert(u);
        for (auto v : edges_[u]) {
            Relax(u, v);
        }
    }

    return shortest_path_estimate_[end];
}

template <typename T>
std::vector<T> Divisor(T n)
{
    std::vector<T> ans;
    for (T i = 1; i * i <= n; i++) {
        if (n % i == 0) {
            ans.push_back(i);
            if (i * i != n) {
                ans.push_back(n / i);
            }
        }
    }

    return ans;
}

int main()
{
    long long h, w;
    std::cin >> h >> w;

    std::vector<std::string> s(h);
    for (int i = 0; i < h; i++) {
        std::cin >> s[i];
    }

    std::vector<std::vector<long long>> dp(h, std::vector<long long>(w, mod));

    dp[0][0] = s[0][0] == '#';

    for (int i = 0; i < h; i++) {
        for (int j = 0; j < w; j++) {
            if (i + 1 < h) {
                dp[i + 1][j] = std::min(dp[i + 1][j], dp[i][j] + (s[i][j] == '.' && s[i + 1][j] == '#'));
            }

            if (j + 1 < w) {
                dp[i][j + 1] = std::min(dp[i][j + 1], dp[i][j] + (s[i][j] == '.' && s[i][j + 1] == '#'));
            }
        }
    }

    std::cout << dp[h - 1][w - 1] << std::endl;
}