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.*;
import java.util.concurrent.atomic.AtomicReference;

public class Main {

    final static boolean DEBUG = false;

    final void solve() throws Exception {
        nextInt();
        nextInt();
        int n = nextInt();
        int[][] a = new int[n][3];
        for (int i = 0; i < n; ++i) {
            a[i][0] = nextInt();
            a[i][1] = nextInt();
            a[i][2] = nextInt();
        }

        Arrays.sort(a, new Comparator<int[]>() {
            @Override
            public int compare(int[] a, int[] b) {
                return a[0] != b[0] ?
                        a[0] - b[0] :
                        a[1] - b[1];
            }
        });

        boolean yes = true;

        rects = new ArrayList<>(n);
        xx = new HashMap<>(n);
        yy = new HashMap<>(n);
        xy = new HashMap<>(n);

        for (int x = 0, y = 0; x < a.length; ++x) {
            for (; y < n && (a[y][0] <= a[x][0] || a[y][0] == a[x][0] + 1 && a[y][1] < a[x][1] - 1); ++y) ;

            if (!(y >= n || a[x][0] + 1 != a[y][0] || a[x][1] + 1 < a[y][1])) {
                if (a[x][0] + 1 == a[y][0] && a[x][1] - 1 <= a[y][1] && a[y][1] <= a[x][1] + 1) {
                    if (!check0(a[x], a[y])) {
                        yes = false;
                        break;
                    }
                    if (y + 1 < n &&
                            a[x][0] + 1 == a[y + 1][0] &&
                            a[x][1] - 1 <= a[y + 1][1] &&
                            a[y + 1][1] <= a[x][1] + 1) {
                        if (!check0(a[x], a[y + 1])) {
                            yes = false;
                            break;
                        }
                        if (y + 2 < n &&
                                a[x][0] + 1 == a[y + 2][0] &&
                                a[x][1] - 1 <= a[y + 2][1] &&
                                a[y + 2][1] <= a[x][1] + 1) {
                            if (!check0(a[x], a[y + 2])) {
                                yes = false;
                                break;
                            }
                        }
                    }
                }
            }

            if (x + 1 < n && a[x][0] == a[x + 1][0]) {
                if (!check1(a[x], a[x + 1])) {
                    yes = false;
                    break;
                }
            }
        }

        if (yes) {
            Collections.sort(rects, new Comparator<int[]>() {
                @Override
                public int compare(int[] a, int[] b) {
                    return a[0] != b[0] ? a[0] - b[0] : a[1] - b[1];
                }
            });

            int pos0 = 0;
            int pos1 = 0;

            final int[][] d = new int[2][2];

            for (int[] rect : rects) {

                for (; pos0 < n && (a[pos0][0] < rect[0] || (a[pos0][0] == rect[0] && a[pos0][1] < rect[1])); ++pos0) ;
                for (; pos1 < n && (a[pos1][0] < rect[0] + 1 || a[pos1][0] == rect[0] + 1 && a[pos1][1] < rect[1]); ++pos1)
                    ;

                if (pos0 < n && pos1 < n && a[pos0][0] == rect[0] && a[pos1][0] == rect[0] + 1) {
                    d[0][0] = -1;
                    d[0][1] = -1;
                    d[1][0] = -1;
                    d[1][1] = -1;
                    
                    if (a[pos0][1] == rect[1]) {
                        d[0][0] = a[pos0][2];
                        if (pos0 + 1 < n && a[pos0 + 1][0] == rect[0] && a[pos0 + 1][1] == rect[1] + 1) {
                            d[0][1] = a[pos0 + 1][2];
                        }
                    } else if (a[pos0][1] == rect[1] + 1) {
                        d[0][1] = a[pos0][2];
                    }

                    if (a[pos1][1] == rect[1]) {
                        d[1][0] = a[pos1][2];
                        if (pos1 + 1 < n && a[pos1 + 1][0] == rect[0] + 1 && a[pos1 + 1][1] == rect[1] + 1) {
                            d[1][1] = a[pos1 + 1][2];
                        }
                    } else if (a[pos1][1] == rect[1] + 1) {
                        d[1][1] = a[pos1][2];
                    }

                    if (d[0][0] != -1 && d[1][1] != -1) {
                        if (d[0][1] != -1) {
                            if (d[0][0] + d[1][1] - d[0][1] < 0) {
                                yes = false;
                                break;
                            }
                            if (d[1][0] != -1 && d[1][0] != d[0][0] + d[1][1] - d[0][1]) {
                                yes = false;
                                break;
                            }
                            if (!check2(rect[0] + 1, rect[1], d[0][0] + d[1][1] - d[0][1])) {
                                yes = false;
                                break;
                            }
                        }
                        if (d[1][0] != -1) {
                            if (d[0][0] + d[1][1] - d[1][0] < 0) {
                                yes = false;
                                break;
                            }
                            if (d[0][1] != -1 && d[0][1] != d[0][0] + d[1][1] - d[1][0]) {
                                yes = false;
                                break;
                            }
                            if (!check2(rect[0], rect[1] + 1, d[0][0] + d[1][1] - d[1][0])) {
                                yes = false;
                                break;
                            }
                        }
                    }
                    if (d[0][1] != -1 && d[1][0] != -1) {
                        if (d[0][0] != -1) {
                            if (d[0][1] + d[1][0] - d[0][0] < 0) {
                                yes = false;
                                break;
                            }
                            if (d[1][1] != -1 && d[1][1] != d[0][1] + d[1][0] - d[0][0]) {
                                yes = false;
                                break;
                            }
                            if (!check2(rect[0] + 1, rect[1] + 1, d[0][1] + d[1][0] - d[0][0])) {
                                yes = false;
                                break;
                            }
                        }

                        if (d[1][1] != -1) {
                            if (d[0][1] + d[1][0] - d[1][1] < 0) {
                                yes = false;
                                break;
                            }

                            if (d[0][0] != -1 && d[0][0] != d[0][1] + d[1][0] - d[1][1]) {
                                yes = false;
                                break;
                            }
                            if (!check2(rect[0], rect[1], d[0][1] + d[1][0] - d[1][1])) {
                                yes = false;
                                break;
                            }
                        }
                    }
                }
            }
        }

        if (yes) {
            for (int[] p : a) {
                int pos;
                Integer diff;

                pos = (p[0] << 1) + (p[1] & 1);
                diff = xx.get(pos);
                if (diff != null && p[2] + diff < 0) {
                    yes = false;
                    break;
                }

                pos = (p[1] << 1) + (p[0] & 1);
                diff = yy.get(pos);
                if (diff != null && p[2] + diff < 0) {
                    yes = false;
                    break;
                }
            }
        }

        print(yes ? "Yes" : "No");
        println();
    }

    final boolean check0(final int[] a, final int[] b) {

        if (a[0] + 1 == b[0]) {
            if (a[1] - 1 == b[1]) {
                rects.add(new int[]{a[0], a[1] - 1, a[2] + b[2]});
            } else if (a[1] == b[1]) {
                int pos = (a[0] << 1) + (a[1] & 1);
                int diff = a[2] - b[2];
                Integer value = xx.get(pos);
                if (value != null && value != diff) {
                    return false;
                }
                if (value == null) {
                    xx.put(pos, diff);
                }
                pos ^= 1;
                diff = -diff;
                value = xx.get(pos);
                if (value != null && value != diff) {
                    return false;
                }
                if (value == null) {
                    xx.put(pos, diff);
                }
            } else if (a[1] + 1 == b[1]) {
                rects.add(new int[]{a[0], a[1], a[2] + b[2]});
            }
        }

        return true;
    }

    final boolean check1(final int[] a, final int[] b) {
        if (a[0] == b[0] && a[1] + 1 == b[1]) {
            int pos = (a[1] << 1) + (a[0] & 1);
            int diff = a[2] - b[2];
            Integer value = yy.get(pos);
            if (value != null && value != diff) {
                return false;
            }
            if (value == null) {
                yy.put(pos, diff);
            }
            pos ^= 1;
            diff = -diff;
            value = yy.get(pos);
            if (value != null && value != diff) {
                return false;
            }
            if (value == null) {
                yy.put(pos, diff);
            }
        }

        return true;
    }

    final boolean check2(final int x, final int y, final int value) {
        long key = (((long) x) << 32) | y;
        Integer hasValue = xy.get(key);
        if (hasValue != null && hasValue != value) {
            return false;
        }
        if (hasValue == null) {
            xy.put(key, value);
        }
        return true;
    }

    ArrayList<int[]> rects;
    HashMap<Integer, Integer> xx;
    HashMap<Integer, Integer> yy;

    HashMap<Long, Integer> xy;

    final static BufferedReader in;
    final static PrintWriter out;

    static final boolean isWhiteSpace(final int c) {
        return c == ' ' || c == '\n' || c == '\r' || c == -1;
    }

    static final int read() throws Exception {
        return in.read();
    }

    static final int nextInt() throws Exception {
        int result = 0;
        boolean negative = false;
        int c = read();
        while (isWhiteSpace(c)) {
            c = read();
        }
        if (c == '-') {
            negative = true;
            c = read();
        }
        while (!isWhiteSpace(c)) {
            result = result * 10 + (c - '0');
            c = read();
        }
        return negative ? -result : result;
    }

    static final long nextLong() throws Exception {
        long result = 0;
        boolean negative = false;
        int c = read();
        while (isWhiteSpace(c)) {
            c = read();
        }
        if (c == '-') {
            negative = true;
            c = read();
        }
        while (!isWhiteSpace(c)) {
            result = result * 10 + (c - '0');
            c = read();
        }
        return negative ? -result : result;
    }

    static final StringBuilder tmpString = new StringBuilder(1 << 20);

    static final String nextString() throws Exception {
        tmpString.setLength(0);
        int c = read();
        while (isWhiteSpace(c)) {
            c = read();
        }
        while (!isWhiteSpace(c)) {
            tmpString.append((char) c);
            c = read();
        }
        return tmpString.toString();
    }

    static final char nextChar() throws Exception {
        int c = read();
        while (isWhiteSpace(c)) {
            c = read();
        }
        while (!isWhiteSpace(c)) {
            return (char) c;
        }
        return (char) 0;
    }

    static final String readLine() throws Exception {
        return in.readLine();
    }

    static final void print(final int i) {
        out.print(i);
    }

    static final void print(final long l) {
        out.print(l);
    }

    static final void print(final String s) {
        out.print(s);
    }

    static final void println() {
        out.println();
    }

    static {
        try {
            if (DEBUG) {
                String fileName = Main.class.getSimpleName();
                in = new BufferedReader(new FileReader(fileName + ".in"));
                out = new PrintWriter(new BufferedWriter(new FileWriter(fileName + ".out")));
            } else {
                in = new BufferedReader(new InputStreamReader(System.in));
                out = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
            }
        } catch (Exception ex) {
            throw new RuntimeException(ex);
        }
    }

    static final AtomicReference<Exception> exs = new AtomicReference<Exception>();

    public static void main(String[] args) throws Exception {
        Thread th = new Thread(null, new Runnable() {
            @Override
            public void run() {
                try {
                    new Main().solve();
                    out.close();
                    return;
                } catch (Exception ex) {
                    exs.set(ex);
                }
            }
        }, "Test", 64 << 20);
        th.start();
        th.join();
        Exception ex = exs.get();
        if (ex != null) {
            throw ex;
        }
    }
}