Problem:
Score : 800 points

Problem Statement

There is a grid with R rows and C columns.
We call the cell in the r-th row and c-th column (r，c).

Mr. Takahashi wrote non-negative integers into N of the cells, that is, he wrote a non-negative integer a_i into (r_i，c_i) for each i (1≤i≤N).
After that he fell asleep.

Mr. Aoki found the grid and tries to surprise Mr. Takahashi by writing integers into all remaining cells.
The grid must meet the following conditions to really surprise Mr. Takahashi.

Condition 1: Each cell contains a non-negative integer.

Condition 2: For any 2×2 square formed by cells on the grid, the sum of the top left and bottom right integers must always equal to the sum of the top right and bottom left integers.

Determine whether it is possible to meet those conditions by properly writing integers into all remaining cells.

Constraints

2≤R，C≤10^5

1≤N≤10^5

1≤r_i≤R

1≤c_i≤C

(r_i，c_i) ≠ (r_j，c_j) (i≠j)

a_i is an integer.

0≤a_i≤10^9

Input

The input is given from Standard Input in the following format:

R C
N
r_1 c_1 a_1
r_2 c_2 a_2
:
r_N c_N a_N

Output

Print Yes if it is possible to meet the conditions by properly writing integers into all remaining cells.
Otherwise, print No.

Sample Input 1

2 2
3
1 1 0
1 2 10
2 1 20

Sample Output 1

Yes

You can write integers as follows.

Sample Input 2

2 3
5
1 1 0
1 2 10
1 3 20
2 1 30
2 3 40

Sample Output 2

No

There are two 2×2 squares on the grid, formed by the following cells:

Cells (1，1), (1，2), (2，1) and (2，2)

Cells (1，2), (1，3), (2，2) and (2，3)

You have to write 40 into the empty cell to meet the condition on the left square, but then it does not satisfy the condition on the right square.

Sample Input 3

2 2
3
1 1 20
1 2 10
2 1 0

Sample Output 3

No

You have to write -10 into the empty cell to meet condition 2, but then it does not satisfy condition 1.

Sample Input 4

3 3
4
1 1 0
1 3 10
3 1 10
3 3 20

Sample Output 4

Yes

You can write integers as follows.

Sample Input 5

2 2
4
1 1 0
1 2 10
2 1 30
2 2 20

Sample Output 5

No

All cells already contain a integer and condition 2 is not satisfied.

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

#define REP(i,n) for(int i=0;i<n;i++)
#define rep(i,n) for(int i=0;i<n;i++)
#define INF 1<<29
#define LINF LLONG_MAX/3
#define MP make_pair
#define PB push_back
#define EB emplace_back
#define ALL(v) (v).begin(),(v).end()
#define all(v) (v).begin(),(v).end()
#define debug(x) cerr<<#x<<":"<<x<<endl
#define debug2(x,y) cerr<<#x<<","<<#y":"<<x<<","<<y<<endl
#define CININIT cin.tie(0),ios::sync_with_stdio(false)
template<typename T> ostream& operator<<(ostream& os,const vector<T>& vec){ os << "["; for(const auto& v : vec){ os << v << ","; } os << "]"; return os; }

typedef long long ll;
typedef unsigned long long ull;
typedef pair<int,int> pii;
typedef vector<int> vi;
typedef vector<vi> vvi;

ll R,C;
ll N;
vector<ll> r,c,a;

int main(){
    CININIT;
    cin>>R>>C>>N;
    r.resize(N);
    c.resize(N);
    a.resize(N);
    vector<int> needx,needy;

    rep(i,N){
        cin>>r[i]>>c[i]>>a[i];
        r[i]--;
        c[i]--;
        a[i]++;
        needx.PB(r[i]);
        if(r[i]+1<R) needx.emplace_back(r[i]+1);
        if(r[i]-1>=0) needx.emplace_back(r[i]-1);

        needy.emplace_back(c[i]);
        if(c[i]+1<C) needy.emplace_back(c[i]+1);
        if(c[i]-1>=0) needy.emplace_back(c[i]-1);
    }


    sort(ALL(needx));
    sort(ALL(needy));
    needx.erase(unique(needx.begin(),needx.end()),needx.end());
    needy.erase(unique(needy.begin(),needy.end()),needy.end());


    vi rr(N),cc(N);
    vi aa(N);

    for(int i=0;i<N;i++){
        rr[i] = find(all(needx),r[i]) - needx.begin();
        cc[i] = find(all(needy),c[i]) - needy.begin();
        aa[i] = a[i];
    }

    ll RR = needx.size();
    ll CC = needy.size();

    vector<vector<ll>> vec(RR,vector<ll>(CC,0));
    for(int i=0;i<N;i++){
        vec[rr[i]][cc[i]] = aa[i];
    }


    __int128 hoge = RR*CC;
    hoge=1000000LL/hoge;

    auto beg(vec);

    ll counter = 0;
    while(true){
        if(++counter==hoge){
            cout <<"No" << endl;
            return 0;
        }
        for(int i=0;i<RR-1;i++){
            for(int j=0;j<CC-1;j++){
                if(vec[i][j]>0){
                    ll sum1 = vec[i][j]+vec[i+1][j+1];
                    ll sum2 = vec[i+1][j]+vec[i][j+1];
                    if(sum1 == sum2) continue;

                    if(vec[i+1][j+1]!=0){
                        if(sum2 > sum1){
                            cout <<"No" << endl;
                            return 0;
                        }else{
                            vec[i+1][j+1]=sum2-sum1;
                        }
                    }

                    if(vec[i+1][j]!=0 and vec[i][j+1]!=0){
                        if(sum1 > sum2){
                            cout << "No" << endl;
                            return 0;
                        }else{
                            if(vec[i+1][j]>0 and vec[i][j+1]==0){
                                vec[i][j+1] = sum1 - sum2;
                            }else if(vec[i+1][j]==0 and vec[i][j+1]>0){
                                vec[i+1][j] = sum1 - sum2;
                            }
                        }
                    }
                }
            }
        }
        if(vec == beg){
            break;
        }
    }
    cout << "Yes" << endl;
}