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.IOException;
import java.io.InputStream;
import java.util.*;

public class Main {

    private static final long INF = Integer.MAX_VALUE;
    private static final int R = 0;
    private static final int C = 1;
    private Map<Long, Integer> idMap = new HashMap<>();
    private GraphL g = new GraphL();
    private Map<Integer, Long> nodeCost = new HashMap<>();
    private boolean ans = true;

    private long iToL(int fst, int snd) {
        return ((long) (fst) << 32) | (long) snd & 0xffffffffL;
    }

    private int getFst(long intPair) {
        return (int) (intPair >> 32);
    }

    private int getSnd(long intPair) {
        return (int) (intPair & 0xffffffffL);
    }

    private int toId(int fst, int snd) {
        long lid = iToL(fst, snd);
        Integer id = idMap.get(lid);
        if (id == null) {
            idMap.put(lid, id = idMap.size());
            g.assoc.add(new ArrayList<>());
        }
        return id;
    }

    private void solve() {
        FastScanner sc = new FastScanner();
        int rMax = sc.nextInt();
        int cMax = sc.nextInt();
        int n = sc.nextInt();
        for (int i = 0; i < n; i++) {
            int r = toId(R, sc.nextInt());
            int c = toId(C, sc.nextInt());
            int a = sc.nextInt();
            g.addEdge(r, c, a);
            g.addEdge(c, r, a);
        }
        for (Map.Entry<Long, Integer> e : idMap.entrySet()) {
            int i = e.getValue();
            if (nodeCost.containsKey(i)) continue;
            int rc = getFst(e.getKey());
            Pair p = dfs(i, 0, rc);
            ans &= p.fst + p.snd >= 0;
        }
        System.out.println(ans ? "Yes" : "No");
    }

    private Pair dfs(int i, long c, int rc) {
        nodeCost.put(i, c);
        Pair p = new Pair(rc == R ? c : INF, rc == C ? c : INF);
        for (long next : g.getNext(i)) {
            int j = getFst(next);
            long a = getSnd(next);
            if (nodeCost.containsKey(j)) {
                long d = nodeCost.get(j);
                ans &= d + c == a;
            } else {
                Pair q = dfs(j, a - c, 1 - rc);
                p.fst = Math.min(p.fst, q.fst);
                p.snd = Math.min(p.snd, q.snd);
            }
        }
        ans &= true;
        return p;
    }

    public static void main(String[] args) {
        new Main().solve();
    }
}

class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

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

    public boolean hasNext() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
        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();
    }

    public 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();
        }
    }

    public int nextInt() {
        long nl = nextLong();
        if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
        return (int) nl;
    }

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

class Pair {

    long fst;
    long snd;

    Pair(long fst, long snd) {
        this.fst = fst;
        this.snd = snd;
    }

}

class Triple {

    public int fst;
    public int snd;
    public int trd;

    public Triple(int fst, int snd, int trd) {
        this.fst = fst;
        this.snd = snd;
        this.trd = trd;
    }

    public Triple(List<Integer> list) {
        this.fst = list.get(0);
        this.snd = list.get(1);
        this.trd = list.get(2);
    }

    public Triple(int[] array) {
        this.fst = array[0];
        this.snd = array[1];
        this.trd = array[2];
    }

    @Override
    public boolean equals(Object o) {
        if (this == o) return true;
        if (o == null || getClass() != o.getClass()) return false;

        Triple triple = (Triple) o;

        return fst == triple.fst && snd == triple.snd && trd == triple.trd;

    }

    @Override
    public int hashCode() {
        int result = fst;
        result = 31 * result + snd;
        result = 31 * result + trd;
        return result;
    }

    @Override
    public String toString() {
        return "(" + fst + ", " + snd + ", " + trd + ")";
    }
}

class GraphL {
    static long iToL(int fst, int snd) {
        return ((long) (fst) << 32) | (long) snd & 0xffffffffL;
    }

    static int getFst(long intPair) {
        return (int) (intPair >> 32);
    }

    static int getSnd(long intPair) {
        return (int) (intPair & 0xffffffffL);
    }

    List<List<Long>> assoc = new ArrayList<>();

    List<Long> getNext(int s) {
        return assoc.get(s);
    }

    void addEdge(int s, int t, int w) {
        assoc.get(s).add(iToL(t, w));
    }
}