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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.io.*;
import java.util.*;

public class Main {
    public static final int p = (int)1e9+7;
    public static final int di[] = new int[]{1, -1, 1, -1};
    public static final int dj[] = new int[]{1, -1, -1, 1};
    public static class Cell {

      int i, j;
      public Cell(int _i, int _j) {
        i = _i ; j = _j ; 
      }
      public int hashCode() {
        return i*p + j;
      }
      public boolean equals(Object o) {
        if (o == null) return false;
        if (!(o instanceof Cell)) return false;

        Cell that = (Cell)o;
        if (i != that.i || j != that.j) return false;
        return true;
      }
    }

    public boolean checkPair(Cell x, long xval, Cell y, HashMap<Cell, Long> all) {
      Cell f1 = new Cell(x.i, y.j);
      Cell f2 = new Cell(y.i, x.j);
      if (!all.containsKey(f1) || !all.containsKey(f2)) {
        return true;
      }
      
      long v1 = all.get(f1), v2 = all.get(f2);
      if (xval + all.get(y) == v1 + v2) return true;
      return false;
    }

    public boolean checkIfCanInsert(HashMap<Cell, Long> all, Cell x, long xval) {
      for (int k = 0 ; k < 4; k++) {
        int ni = x.i + di[k], nj = x.j + dj[k];
        Cell U = new Cell(ni, nj);
        if (!checkCell(U) || !all.containsKey(U)) continue;
        if (!checkPair(x, xval, U, all)) return false;
      }
      return true;
    }
    int rows, cols;
    public boolean checkCell(Cell x) {
      return x.i >= 1 && x.j >= 1 && x.i <= rows && x.j <= cols;
    }
    public void solve() throws IOException {
      int rows = nextInt(), cols = nextInt();
      this.rows = rows; this.cols = cols;
      int n = nextInt();
      HashMap<Cell, Long> all = new HashMap<>();
      Queue<Cell> q = new LinkedList<>();
      for (int i = 0 ; i < n ; i ++) {
        Cell x = new Cell(nextInt(), nextInt());
        all.put(x, nextLong());
        q.add(x);
      }
      for (Map.Entry<Cell, Long> e : all.entrySet()) {
        Cell x = e.getKey();
        long v = e.getValue();
        if (!checkIfCanInsert(all, x, v)) {
          out.println("No");
          return;
        }
      }
      while (!q.isEmpty()) {
        Cell f = q.poll();
        int i = f.i, j = f.j;
        //System.out.println("getting " + i + " " + j);
        for (int k = 0 ; k < 4; k++) {
          int ni = i + di[k], nj = j + dj[k];
          Cell U = new Cell(ni, nj);
          if (!checkCell(U) ) {
            continue;
          }
          if (all.containsKey(U)) {
            if (checkPair(f, all.get(f), U, all)) {
              continue;
            }
          }

          Cell f1 = new Cell(ni, j);
          Cell f2 = new Cell(i, nj);
          if (!all.containsKey(f1) || !all.containsKey(f2)) {
            continue;
          }
          
          long v1 = all.get(f1), v2 = all.get(f2);
          long v = all.get(f);
          long u = v1 + v2 - v;
          if (u < 0) {
            out.println("No");
            return;
          }
          //System.out.println("Adding " + U.i + " " + U.j + " " + u);
          if (!checkIfCanInsert(all, U, u)) {
            out.println("No");
            return;
          }
          all.put(U, u);
          q.add(U);
        }
      }
      for (Map.Entry<Cell, Long> e : all.entrySet()) {
        q.add(e.getKey());
      }
      while (!q.isEmpty()) {
        Cell f = q.poll();
        int i = f.i, j = f.j;
        //System.out.println("getting " + i + " " + j);
        for (int k = 0 ; k < 4; k++) {
          int ni = i + di[k], nj = j + dj[k];
          Cell U = new Cell(ni, nj);
          if (!checkCell(U) ) {
            continue;
          }
          if (all.containsKey(U)) {
            if (checkPair(f, all.get(f), U, all)) {
              continue;
            }
          }

          Cell f1 = new Cell(ni, j);
          Cell f2 = new Cell(i, nj);
          if (!all.containsKey(f1) || !all.containsKey(f2)) {
            continue;
          }
          
          long v1 = all.get(f1), v2 = all.get(f2);
          long v = all.get(f);
          long u = v1 + v2 - v;
          if (u < 0) {
            out.println("No");
            return;
          }
          //System.out.println("Adding " + U.i + " " + U.j + " " + u);
          if (!checkIfCanInsert(all, U, u)) {
            out.println("No");
            return;
          }
          all.put(U, u);
          q.add(U);
        }
      }
      for (Map.Entry<Cell, Long> e : all.entrySet()) {
        Cell x = e.getKey();
        long v = e.getValue();
        if (!checkIfCanInsert(all, x, v)) {
          out.println("No");
          return;
        }
      }
      out.println("Yes");
      
    }

    

    
    
    BufferedReader bf;
    StringTokenizer st;
    PrintWriter out;
    
    public String next() throws IOException {
        while (st == null || !st.hasMoreTokens()) 
            st = new StringTokenizer(bf.readLine());
        return st.nextToken();
    }
    public int nextInt() throws IOException {
        return Integer.parseInt(next());
    }
    
    public long nextLong() throws IOException {
        return Long.parseLong(next());
    }

    public Main() throws IOException {
        bf = new BufferedReader(new InputStreamReader(System.in));
        out = new PrintWriter(new OutputStreamWriter(System.out));      
        solve();
        bf.close();
        out.close();
    }
    
    public static void main(String args[]) throws IOException {
        new Main();
    }
}