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.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.*;
 
/*
                   _ooOoo_
                  o8888888o
                  88" . "88
                  (| -_- |)
                  O\  =  /O
               ____/`---'\____
             .'  \\|     |//  `.
            /  \\|||  :  |||//  \
           /  _||||| -:- |||||-  \
           |   | \\\  -  /// |   |
           | \_|  ''\---/''  |   |
           \  .-\__  `-`  ___/-. /
         ___`. .'  /--.--\  `. . __
      ."" '<  `.___\_<|>_/___.'  >'"".
     | | :  `- \`.;`\ _ /`;.`/ - ` : | |
     \  \ `-.   \_ __\ /__ _/   .-` /  /
======`-.____`-.___\_____/___.-`____.-'======
                   `=---='
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
            pass System Test!
*/

public class Main {
  private static class Task {
    class Edge {
      int to;
      long w;
      Edge(int to, long w) {
        this.to = to;
        this.w = w;
      }
    }

    void solve(FastScanner in, PrintWriter out) throws Exception {
      int R = in.nextInt();
      int C = in.nextInt();
      int N = in.nextInt();

      PriorityQueue<int[]>[] rows = new PriorityQueue[R];
      for (int i = 0; i < R; i++) {
        rows[i] = new PriorityQueue<>((o1, o2) -> Integer.compare(o1[0], o2[0]));
      }
      PriorityQueue<int[]>[] columns = new PriorityQueue[C];
      for (int i = 0; i < C; i++) {
        columns[i] = new PriorityQueue<>((o1, o2) -> Integer.compare(o1[0], o2[0]));
      }

      int[][] rca = new int[N][3];
      long[] minR = new long[R];
      Arrays.fill(minR, Integer.MAX_VALUE);
      long[] minC = new long[C];
      Arrays.fill(minC, Integer.MAX_VALUE);
      for (int i = 0; i < N; i++) {
        int r = in.nextInt() - 1;
        int c = in.nextInt() - 1;
        int a = in.nextInt();
        rca[i][0] = r;
        rca[i][1] = c;
        rca[i][2] = a;

        rows[r].add(new int[]{c, a});
        columns[c].add(new int[]{r, a});

        minR[r] = Math.min(minR[r], a);
        minC[r] = Math.min(minC[c], a);
      }

      int startR = -1, startC = -1;
      int min = Integer.MAX_VALUE;
      for (int i = 0; i < N; i++) {
        if (min > rca[i][2]) {
          min = rca[i][2];
          startR = rca[i][0];
          startC = rca[i][1];
        }
      }

      if (!check(columns, R, minR) || !check(rows, C, minC)) {
        out.println("No");
      } else {
        out.println("Yes");
      }
    }

    boolean check(PriorityQueue<int[]>[] columns, int R, long[] minR) {
      ArrayList<Edge>[] graph = new ArrayList[R];
      for (int r = 0; r < R; r++) {
        graph[r] = new ArrayList<>();
      }
      for (PriorityQueue<int[]> q : columns) {
        if (q.size() < 2) continue;
        int prevR = q.peek()[0];
        int prevA = q.peek()[1];
        q.poll();
        while (!q.isEmpty()) {
          int r = q.peek()[0];
          int a = q.peek()[1];
          q.poll();
          graph[prevR].add(new Edge(r, a - prevA));
          graph[r].add(new Edge(prevR, prevA - a));
        }
      }

      long[] dist = new long[R];
      Arrays.fill(dist, Long.MAX_VALUE);
      ArrayDeque<Integer> deque = new ArrayDeque<>();
      for (int start = 0; start < R; start++) {
        if (dist[start] != Long.MAX_VALUE) continue;
        deque.add(start);
        dist[start] = minR[start];
        while (!deque.isEmpty()) {
          int v = deque.poll();
          for (Edge edge : graph[v]) {
            if (dist[edge.to] == Long.MAX_VALUE) {
              dist[edge.to] = dist[v] + edge.w;
              deque.add(edge.to);
            }
            if (dist[edge.to] != dist[v] + edge.w) return false;
            if (dist[edge.to] < 0) return false;
          }
        }
      }
      return true;
    }
  }

  /**
   * ここから下はテンプレートです。
   */
  public static void main(String[] args) throws Exception {
    OutputStream outputStream = System.out;
    FastScanner in = new FastScanner();
    PrintWriter out = new PrintWriter(outputStream);
    Task solver = new Task();
    solver.solve(in, out);
    out.close();
  }
  private static class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int bufferLength = 0;

    private boolean hasNextByte() {
      if (ptr < bufferLength) {
        return true;
      } else {
        ptr = 0;
        try {
          bufferLength = in.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (bufferLength <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) return buffer[ptr++];
      else return -1;
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    private void skipUnprintable() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
    }

    boolean hasNext() {
      skipUnprintable();
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) throw new NoSuchElementException();
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    long nextLong() {
      if (!hasNext()) throw new NoSuchElementException();
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    double nextDouble() {
      return Double.parseDouble(next());
    }

    double[] nextDoubleArray(int n) {
      double[] array = new double[n];
      for (int i = 0; i < n; i++) {
        array[i] = nextDouble();
      }
      return array;
    }

    double[][] nextDoubleMap(int n, int m) {
      double[][] map = new double[n][];
      for (int i = 0; i < n; i++) {
        map[i] = nextDoubleArray(m);
      }
      return map;
    }

    public int nextInt() {
      return (int) nextLong();
    }

    public int[] nextIntArray(int n) {
      int[] array = new int[n];
      for (int i = 0; i < n; i++) array[i] = nextInt();
      return array;
    }

    public long[] nextLongArray(int n) {
      long[] array = new long[n];
      for (int i = 0; i < n; i++) array[i] = nextLong();
      return array;
    }

    public String[] nextStringArray(int n) {
      String[] array = new String[n];
      for (int i = 0; i < n; i++) array[i] = next();
      return array;
    }

    public char[][] nextCharMap(int n) {
      char[][] array = new char[n][];
      for (int i = 0; i < n; i++) array[i] = next().toCharArray();
      return array;
    }

    public int[][] nextIntMap(int n, int m) {
      int[][] map = new int[n][];
      for (int i = 0; i < n; i++) {
        map[i] = nextIntArray(m);
      }
      return map;
    }
  }
}