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: Accepted)

#![allow(unused_imports)]
#![allow(dead_code)]
#![allow(unused_mut)]
// #![allow(non_camel_case_types)]

use crate::rust::libs::util::{io::*,util::*,lower_bound::*};
use crate::rust::libs::math::{mod_int::*,prime_factor::*};
use crate::rust::libs::data_structure::{union_find::*,segment_tree::*};
use crate::rust::libs::algebraic_structure::monoid::{plus::*,max::*,min::*};
use std::ops::Range;
use  std::collections::*;
const MOD: u32 = 1_000_000_007;
const INF:i64=1i64<<60;
const_mod!(P,MOD);
fn main() {
    input!{
        h:usize,w:usize,
        s:[chars;h],
    }
    let mut dp=make_vec!(INF,h,w);
    dp[0][0]=0;
    for i in 0..h{
        for j in 0..w{
            if i!=h-1{
                dp[i+1][j]=dp[i+1][j].min(dp[i][j]+btoi!(s[i+1][j]!=s[i][j]));
            }
            if j!=w-1{
                dp[i][j+1]=dp[i][j+1].min(dp[i][j]+btoi!(s[i][j+1]!=s[i][j]));
            }
        }
    }
    if s[0][0]=='.'&&s[h-1][w-1]=='.'{println!("{}",dp[h-1][w-1]/2);}
    else if s[0][0]!=s[h-1][w-1]{println!("{}",dp[h-1][w-1]/2+1);}
    else {println!("{}",dp[h-1][w-1]/2+1);}
}




pub mod rust {

pub mod libs {

pub mod algebraic_structure {

pub mod monoid {

pub mod max {

use crate::rust::libs::algebraic_structure::monoid::monoid::Monoid;
use std::option::Option;
#[derive(Debug,Copy,Clone)]
pub struct Max<T>{
    val:Option<T>,
}
impl<T:Ord+Copy> Monoid for Max<T>{
    type MonoidType=T;
    fn new()->Self{
        Self{
            val:None,
        }
    }
    fn make(t:T)->Self{
        Self{
            val:Some(t),
        }
    }
    fn f(&self,t:&Self)->Self{
        if self.val.is_none() {*t}
        else if t.val.is_none() {*self}
        else{
            Self{
                val:Some(T::max(self.val.unwrap(),t.val.unwrap())),
            }
        }
    }
    fn get(self)->Option<T>{
        self.val
    }
}

}  // mod max

pub mod min {

use crate::rust::libs::algebraic_structure::monoid::monoid::Monoid;
use std::option::Option;
#[derive(Debug,Copy,Clone)]
pub struct Min<T>{
    val:Option<T>,
}
impl<T:Ord+Copy> Monoid for Min<T>{
    type MonoidType=T;
    fn new()->Self{
        Self{
            val:None,
        }
    }
    fn make(t:T)->Self{
        Self{
            val:Some(t),
        }
    }
    fn f(&self,t:&Self)->Self{
        if self.val.is_none() {*t}
        else if t.val.is_none() {*self}
        else{
            Self{
                val:Some(T::min(self.val.unwrap(),t.val.unwrap())),
            }
        }
    }
    fn get(self)->Option<T>{
        self.val
    }
}

}  // mod min

pub mod monoid {

pub trait Monoid{
    type MonoidType;
    fn new()->Self;
    fn make(t:Self::MonoidType)->Self;
    fn get(self)->Option<Self::MonoidType>;
    fn f(&self,t:&Self)->Self;
}

}  // mod monoid

pub mod plus {

use crate::rust::libs::algebraic_structure::monoid::monoid::Monoid;
use std::option::Option;
use std::ops::Add;
#[derive(Debug,Copy,Clone)]
pub struct Plus<T>{
    val:Option<T>,
}
impl<T:Add<T,Output=T>+Copy> Monoid for Plus<T>{
    type MonoidType=T;
    fn new()->Self{
        Self{
            val:None,
        }
    }
    fn make(t:T)->Self{
        Self{
            val:Some(t),
        }
    }
    fn f(&self,t:&Self)->Self{
        if self.val.is_none() {*t}
        else if t.val.is_none() {*self}
        else{
            Self{
                val:Some(self.val.unwrap()+t.val.unwrap()),
            }
        }
    }
    fn get(self)->Option<T>{
        self.val
    }
}

}  // mod plus

}  // mod monoid

}  // mod algebraic_structure

pub mod data_structure {

pub mod segment_tree {

use crate::rust::libs::algebraic_structure::monoid::monoid::Monoid;
use std::ops::Range;
pub struct SegmentTree<T>{
    n:usize,
    v:Vec<T>,
}

impl<T:Monoid+Copy+std::fmt::Debug> SegmentTree<T>where T::MonoidType:Copy{
    pub fn new(n:usize)->Self{
        Self{
            n:n,
            v:vec![T::new();n*2],
        }
    }
    pub fn make(v:&[T::MonoidType])->Self{
        let n=v.len();
        let mut tmp=vec![T::new();n*2];
        for i in 0..n{
            tmp[v.len()+i]=T::make(v[i]);
        }
        for i in (1..n).rev(){
            tmp[i]=tmp[i*2].f(&tmp[i*2+1]);
        }
        Self{
            n:n,
            v:tmp,
        }
    }
    pub fn get(&self,ran:Range<usize>)->Option<T::MonoidType>{
        let mut l=ran.start+self.n;
        let mut r=ran.end+self.n;
        let mut s=T::new();
        let mut t=T::new();
        while l<r{
            if (l&1)==1{s=s.f(&self.v[l]);l+=1;}
            if (r&1)==1{r-=1;t=self.v[r].f(&t);}
            l>>=1;r>>=1;
        }
        s.f(&t).get()
    }
    pub fn set(&mut self,t:usize,val:T::MonoidType){
        let mut tp=t+self.n;
        self.v[tp]=self.v[tp].f(&T::make(val));
        while tp>1{
            tp=tp>>1;
            self.v[tp]=self.v[tp*2].f(&self.v[tp*2+1]);
        }
    }
    pub fn out(self){
        println!("{:?}",self.v);
    }
}

}  // mod segment_tree

pub mod union_find {

pub struct UF{
    par:Vec<usize>,
    rank:Vec<usize>,
}
impl UF{
    pub fn new(n:usize)->UF{
        let mut v=vec![0;n];
        for i in 0..n{
            v[i]=i;
        }
        UF{
            par:v,
            rank:vec![1;n],
        }
    }
    pub fn root(&mut self,x:usize)->usize{
        if x==self.par[x]{
            x
        }else{
            let par=self.par[x];
            let res=self.root(par);
            self.par[x]=res;
            res
        }
    }
    pub fn same(&mut self,a:usize,b:usize)->bool{
        self.root(a)==self.root(b)
    }
    pub fn unite(&mut self,a:usize,b:usize)->bool{
        let ap=self.root(a);
        let bp=self.root(b);
        if ap==bp{
            return false;
        }
        if self.rank[ap]<self.rank[bp]{
            self.par[bp]=ap;
            self.rank[ap]+=self.rank[bp];
        }else{
            self.par[ap]=bp;
            self.rank[bp]+=self.rank[ap];
        }
        return true;
    }
    pub fn size(&mut self,a:usize)->usize{
        let ap=self.root(a);
        self.rank[ap]
    }
}

}  // mod union_find

}  // mod data_structure

pub mod math {

pub mod mod_int {

use std::ops::{Add,AddAssign,Sub,SubAssign,Mul,MulAssign,Div,DivAssign,Neg};
use std::str::FromStr;
use std::num::ParseIntError;
use std::iter::{Sum,Product};

pub fn inv_mod(a:u64,m:u64)->u64{
    let m=m as i64;
    let mut a=a as i64;
    let mut b=m as i64;
    let mut u=1i64;
    let mut v=0i64;
    while b>0 {
        let t=a/b;
        a-=t*b;
        u-=t*v;
        std::mem::swap(&mut a,&mut b);
        std::mem::swap(&mut u,&mut v);
    }
    let ans =(if u>=0{u%m}else{m+(u%m)%m})as u64;
    ans
}

pub trait Mod:Sized{
    fn m()->u32;
    fn m64()->u64;
    fn mi64()->i64;
}
#[macro_export]
macro_rules! const_mod{
    ($st:ident,$m:expr)=>{
        struct $st{}
        impl Mod for $st {
            fn m()->u32{$m}
            fn m64()->u64{$m as u64}
            fn mi64()->i64{$m as i64}
        }
        //const MAX=10000000;
        type Fp=ModInt<P>;
        //const fact_table:[Fp;MAX]=(0..MAX)
    }
}
pub struct ModInt<M:Mod>{a:u32,_p:std::marker::PhantomData<M>}

impl<M: Mod> ModInt<M>{
    pub fn new(a:u32)->Self{
        ModInt{a,_p:std::marker::PhantomData}
    }
    pub fn newu64(a:u64)->Self{
        ModInt{a:(a%M::m64())as u32,_p:std::marker::PhantomData}
    }
    pub fn newi64(a:i64)->Self{
        ModInt{a:((a%M::mi64()+M::mi64())%M::mi64()) as u32,_p:std::marker::PhantomData}
    }
    pub fn value(&self)->u32{self.a}
    pub fn pow(&self,p:u64)->Self{
        let mut exp=p;
        let mut now=*self;
        let mut ans=ModInt::new(1);
        while exp>0{
            if (exp&1)==1{ans*=now;}
            now*=now;
            exp>>=1;
        }
        ans
    }
    // pub fn comb(k:i32)->Self{
    //     Self::new(k)
    // }
    pub fn inv(&self)->Self{Self::new(inv_mod(self.a as u64,M::m64())as u32)}
}

impl<M:Mod>Clone for ModInt<M>{
    fn clone(&self)->Self{ModInt::new(self.a)}
}
impl<M:Mod>Copy for ModInt<M>{}
impl<M:Mod>From<i64>for ModInt<M>{
    fn from(i:i64)->Self{Self::newi64(i)}
}
impl<M:Mod>From<i32>for ModInt<M>{
    fn from(i:i32)->Self{Self::newi64(i as i64)}
}

impl<M:Mod>FromStr for ModInt<M>{
    type Err = ParseIntError;
    fn from_str(i:&str)->Result<Self, Self::Err>{
        let res=i.parse::<i64>()?;
        Ok(Self::newi64(res))
    }
}
impl<M:Mod>Sum for ModInt<M>{
    fn sum<I>(iter: I) -> Self where I: Iterator<Item = Self>,{
        iter.fold(Self::new(0), |b, i| b + i)
    }
}

impl<M:Mod>Product for ModInt<M>{
    fn product<I>(iter: I) -> Self where I: Iterator<Item = Self>,{
        iter.fold(Self::new(0), |b, i| b * i)
    }
}
impl<M:Mod>Add for ModInt<M>{
    type Output=Self;
    fn add(self,rhs:Self)->Self{
        let a=self.a+rhs.a;
        ModInt::new(if a>=M::m(){a-M::m()}else{a})
    }
}
impl<M:Mod>Sub for ModInt<M>{
    type Output=Self;
    fn sub(self,rhs:Self)->Self{
        ModInt::new(if self.a<rhs.a{M::m()+self.a-rhs.a}else{self.a-rhs.a})
    }
}
impl<M:Mod>Mul for ModInt<M>{
    type Output=Self;
    fn mul(self,rhs:Self)->Self{
        ModInt::newu64(self.a as u64*rhs.a as u64)
    }
}
impl<M:Mod>Div for ModInt<M>{
    type Output=Self;
    fn div(self,rhs:Self)->Self{
        self*rhs.inv()
    }
}
impl<M:Mod>Neg for ModInt<M> {
    type Output = Self;
    fn neg(self) -> Self {
        ModInt::new(M::m()-self.value())
    }
}
impl<M:Mod>AddAssign for ModInt<M>{fn add_assign(&mut self,rhs:Self){*self=*self+rhs;}}
impl<M:Mod>SubAssign for ModInt<M>{fn sub_assign(&mut self,rhs:Self){*self=*self-rhs;}}
impl<M:Mod>MulAssign for ModInt<M>{fn mul_assign(&mut self,rhs:Self){*self=*self*rhs;}}
impl<M:Mod>DivAssign for ModInt<M>{fn div_assign(&mut self,rhs:Self){*self=*self/rhs;}}

impl<M:Mod>std::fmt::Debug for ModInt<M>{
    fn fmt(&self,f:&mut std::fmt::Formatter)->Result<(),std::fmt::Error>{
        write!(f,"M{}",self.a)
    }
}
impl<M:Mod> std::fmt::Display for ModInt<M> {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> Result<(), std::fmt::Error> {
        self.value().fmt(f)
    }
}

}  // mod mod_int

pub mod prime_factor {

use std::collections::BTreeMap;
pub fn prime_factor(n:usize)->BTreeMap<usize,usize>{
    let mut ret:BTreeMap<usize,usize>=BTreeMap::<usize,usize>::new();
    let mut m=n;
    let mut i=2;
    while i*i<=m {
        while m%i==0{
            let tmp=ret.entry(i).or_insert(0);
            *tmp+=1;
            m/=i;
        }
        i+=1;
    }
    ret
}

}  // mod prime_factor

}  // mod math

pub mod util {

pub mod io {

#[macro_export]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let mut s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

#[macro_export]
macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};

    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
    ($iter:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

#[macro_export] 
macro_rules! read_value {
    ($iter:expr, [tree;$len:expr]) => {
        {
            let mut g:Vec<Vec<usize>>=vec![Vec::new();$len];
            for (s,t) in read_value!($iter,[(usize1,usize1);$len-1]){
                g[s].push(t);
                g[t].push(s);
            }
            g
        }
    };
    ($iter:expr, [graph;$vertex:expr,$edge:expr]) => {
        {
            let mut g:Vec<Vec<usize>>=vec![Vec::new();$vertex];
            for (s,t) in read_value!($iter,[(usize1,usize1);$edge]){
                g[s].push(t);
                g[t].push(s);
            }
            g
        }
    };
    ($iter:expr, [directed;$vertex:expr,$edge:expr]) => {
        {
            let mut g:Vec<Vec<usize>>=vec![Vec::new();$vertex];
            for (s,t) in read_value!($iter,[(usize1,usize1);$edge]){
                g[s].push(t);
            }
            g
        }
    };
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };

    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };

    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };

    ($iter:expr, bytes) => {
        read_value!($iter, String).into_bytes()
    };

    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };

    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}


}  // mod io

pub mod lower_bound {

pub trait LowerBound<T> {
    fn lower_bound(&self,v:T)->usize;
}

impl<T:Ord> LowerBound<T> for Vec<T>{
    fn lower_bound(&self,x:T)->usize{
        let mut l:usize=0;
        let mut r:usize=self.len()+1;
        while r-l>1{
            let m=(l+r)/2;
            if self[m-1]<x {l=m;}
            else {r=m;}
        }
        r-1
    }
}
pub trait UpperBound<T> {
    fn upper_bound(&self,v:T)->usize;
}

impl<T:Ord> UpperBound<T> for Vec<T>{
    fn upper_bound(&self,x:T)->usize{
        let mut l:usize=0;
        let mut r:usize=self.len()+1;
        while r-l>1{
            let m=(l+r)/2;
            if self[m-1]<=x {l=m;}
            else {r=m;}
        }
        r-1
    }
}

}  // mod lower_bound

pub mod util {

use std::ops::Add;
pub trait CumSum<T> {
    fn cumsum(&self)->Vec<T>;
}

impl<T:Add<Output=T>+Default+Copy> CumSum<T> for Vec<T>{
    fn cumsum(&self)->Vec<T>{
        let mut ret=Vec::new();
        ret.push(T::default());
        for i in 0..self.len(){
            ret.push(ret[i]+self[i]);
        }
        ret
    }
}

pub fn type_of<T>(_: T) -> String{
    let a = std::any::type_name::<T>();
    return a.to_string();
}

#[macro_export]
macro_rules! make_vec{
    ( $val:expr , $head:expr)=>{
        vec![$val;$head]
    };
    ( $val:expr , $head:expr , $($tail:expr),+ )=>{
        {
            vec![make_vec!($val,$($tail),+);$head]
        }
    };
}
#[macro_export]
macro_rules! btoi{
    ( $val:expr)=>{
        if $val{1}else{0}
    };
}
pub trait Isomorphism<T> {
    fn isomorphism(&self)->Vec<T>;
}

// impl<T:Add<Output=T>+Default+Copy> Isomorphism<T> for Vec<T>{
//     fn isomorphism(&self)->Vec<T>{
        
//     }
// }

// pub fn mod_pow(x:i64,y:i64,z:i64){
//     let mut ret:T=1;
//     while y{
//         if y%2==1{
//             ret*=y;
//             ret%=z;
//         }
//         y/=2;
//     }
//     ret
// }

}  // mod util

}  // mod util

}  // mod libs

}  // mod rust