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: Accepted)

import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.io.IOException;
import java.io.InputStream;

/**
 * Built using CHelper plug-in Actual solution is at the top
 * 
 * @author daltao
 */
public class Main {
    public static void main(String[] args) throws Exception {
        Thread thread = new Thread(null, new TaskAdapter(), "daltao", 1 << 27);
        thread.start();
        thread.join();
    }

    static class TaskAdapter implements Runnable {
        @Override
        public void run() {
            InputStream inputStream = System.in;
            OutputStream outputStream = System.out;
            FastInput in = new FastInput(inputStream);
            PrintWriter out = new PrintWriter(outputStream);
            TaskD solver = new TaskD();
            solver.solve(1, in, out);
            out.close();
        }
    }
    static class TaskD {
        public void solve(int testNumber, FastInput in, PrintWriter out) {
            int r = in.readInt();
            int c = in.readInt();
            int n = in.readInt();
            int[][] rcas = new int[n][3];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < 3; j++) {
                    rcas[i][j] = in.readInt();
                }
            }

            out.println(check(r, c, rcas) ? "Yes" : "No");
        }

        public boolean check(int r, int c, int[][] rcas) {
            int n = rcas.length;
            Arrays.sort(rcas, (a, b) -> a[0] == b[0] ? a[1] - b[1] : a[0] - b[0]);

            Node[] cols = new Node[c + 1];
            for (int i = 1; i <= c; i++) {
                cols[i] = new Node();
            }

            for (int i = 1; i < n; i++) {
                int ci = rcas[i][1];

                for (int j = i - 1; j >= 0 && rcas[i][0] == rcas[j][0]; j--) {
                    int cj = rcas[j][1];
                    if (cols[ci].find() == cols[cj].find()) {
                        if (cols[ci].diff - cols[cj].diff != rcas[i][2] - rcas[j][2]) {
                            return false;
                        }
                    } else {
                        Node.merge(cols[cj], cols[ci], rcas[i][2] - rcas[j][2]);
                    }
                    break;
                }
            }

            Arrays.sort(rcas, (a, b) -> a[1] == b[1] ? a[0] - b[0] : a[1] - b[1]);
            Node[] rows = new Node[r + 1];
            for (int i = 1; i <= r; i++) {
                rows[i] = new Node();
            }

            for (int i = 1; i < rcas.length; i++) {
                int ri = rcas[i][0];

                for (int j = i - 1; j >= 0 && rcas[i][1] == rcas[j][1]; j--) {
                    int rj = rcas[j][0];
                    if (rows[ri].find() == rows[rj].find()) {
                        if (rows[ri].diff - rows[rj].diff != rcas[i][2] - rcas[j][2]) {
                            return false;
                        }
                    } else {
                        Node.merge(rows[rj], rows[ri], rcas[i][2] - rcas[j][2]);
                    }
                    break;
                }
            }

            for (int i = 0; i < n; i++) {
                int rr = rcas[i][0];
                int cc = rcas[i][1];
                long v = rcas[i][2];

                Node minRow = rows[rr].find().min;
                v = v - rows[rr].diff + minRow.diff;
                Node minCol = cols[cc].find().min;
                v = v - cols[cc].diff + minCol.diff;
                if (v < 0) {
                    return false;
                }
            }

            return true;
        }

    }
    static class FastInput {
        private final InputStream is;
        private byte[] buf = new byte[1 << 13];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastInput(InputStream is) {
            this.is = is;
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

    }
    static class Node {
        Node p = this;
        int rank;
        long diff;
        Node min = this;

        public Node find() {
            if (p.p == p) {
                return p;
            }
            p.find();
            diff += p.diff;
            p = p.find();
            return p;
        }

        public static Node min(Node a, Node b) {
            a.find();
            b.find();
            return a.diff <= b.diff ? a : b;
        }

        public static void merge(Node a, Node b, long d) {
            a.find();
            b.find();
            // Rb + Db = Ra + Da + d
            // Rb = Ra + Da + d - Db
            d = a.diff + d - b.diff;
            a = a.find();
            b = b.find();
            if (a == b) {
                return;
            }
            if (a.rank == b.rank) {
                a.rank++;
            }
            if (a.rank > b.rank) {
                b.p = a;
                b.diff = d;
                a.min = min(a.min, b.min);

            } else {
                a.p = b;
                a.diff = -d;
                b.min = min(a.min, b.min);
            }
        }

    }
}