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.*;
import java.util.*;
import java.util.function.Function;

public class Main {

    static int H, W, N;
    static Cord[] C;

    public static void main(String[] args) {
        FastScanner sc = new FastScanner(System.in);
        H = sc.nextInt();
        W = sc.nextInt();
        N = sc.nextInt();
        C = new Cord[N];
        for (int i = 0; i < N; i++) {
            C[i] = new Cord(sc.nextInt()-1, sc.nextInt()-1, sc.nextInt());
        }

        System.out.println(solve() ? "Yes" : "No");
    }

    static class Cord {
        int r, c, a;

        public Cord(int r, int c, int a) {
            this.r = r;
            this.c = c;
            this.a = a;
        }
    }

    static boolean solve() {
        Arrays.sort(C, Comparator.<Cord>comparingInt(c -> c.r).thenComparingInt(c -> c.c));
        long UNDEF = Long.MAX_VALUE;
        UnionFind uf = new UnionFind(W);

        List<Edge> edges = new ArrayList<>();
        int idx = 0;
        for (int r = 0; r < H; r++) {
            while(idx < N-1 && C[idx].r == r ) {
                if( C[idx+1].r == r ) {
                    uf.unite(C[idx].c, C[idx+1].c);
                    edges.add( new Edge(C[idx].c, C[idx+1].c, C[idx+1].a - C[idx].a) );
                    edges.add( new Edge(C[idx+1].c, C[idx].c, C[idx].a - C[idx+1].a) );
                }
                idx++;
            }
        }

        Edge[][] G = adjD(W, edges);
        long[] DIFF = new long[W];
        Arrays.fill(DIFF, UNDEF);
        ArrayDeque<Integer> q = new ArrayDeque<>();
        for (int c = 0; c < W; c++) {
            if( DIFF[c] != UNDEF ) continue;
            if( G[c].length == 0 ) continue;

            DIFF[c] = 0;
            q.add(c);
            while( !q.isEmpty() ) {
                int a = q.poll();
                for (Edge e : G[a]) {
                    int b = e.a == a ? e.b : e.a;

                    if( DIFF[b] == UNDEF ) {
                        DIFF[b] = DIFF[a] + e.d;
                        q.add(b);

                    } else if( DIFF[b] != DIFF[a] + e.d) {
                        return false;
                    }
                }
            }
        }

        // 連結のrootの中で最低
        long[] MIN_DIFF = new long[W];
        Arrays.fill(MIN_DIFF, UNDEF);
        for (int c = 0; c < W; c++) {
            if( DIFF[c] == UNDEF ) continue;
            int r = uf.root(c);
            MIN_DIFF[r] = Math.min(MIN_DIFF[r], DIFF[c]);
        }

        for (int i = 0; i < N; i++) {
            long sa = C[i].a - DIFF[C[i].c]; // 差
            int r = uf.root(C[i].c);

            if( MIN_DIFF[r] + sa < 0 ) {
                return false;
            }
        }

        return true;
    }

    static Edge[][] adjD(int n, List<Edge> es) {
        Edge[][] adj = new Edge[n][];
        int[] cnt = new int[n];

        for (Edge e : es) {
            cnt[e.a]++;
        }
        for (int i = 0; i < n; i++) {
            adj[i] = new Edge[cnt[i]];
        }
        for (Edge e : es) {
            adj[e.a][--cnt[e.a]] = e;
        }
        return adj;
    }

    static class Edge {
        int a, b;
        int d;

        public Edge(int a, int b, int d) {
            this.a = a;
            this.b = b;
            this.d = d;
        }
    }

    static class UnionFind {

        private final int[] parent;
        private final int[] rank;

        public UnionFind(int n) {
            parent = new int[n];
            rank = new int[n];
            for (int i = 0; i < n; i++) {
                parent[i] = i;
                rank[i] = 0;
            }
        }

        public int root(int i) {
            if( parent[i] == i ) {
                return i;
            } else {
                return parent[i] = root(parent[i]);
            }
        }

        public int unite(int i, int j) {
            int ri = root(i);
            int rj = root(j);
            if( ri == rj ) return ri;

            if( rank[ri] < rank[rj] ) {
                parent[ri] = rj;
                return rj;

            } else {
                parent[rj] = ri;
                if( rank[ri] == rank[rj] ) {
                    rank[ri]++;
                }
                return ri;
            }
        }

        public boolean isSame(int a, int b) {
            return root(a) == root(b);
        }
    }


    @SuppressWarnings("unused")
    static class FastScanner {
        private BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
            tokenizer = null;
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        String nextLine() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    return reader.readLine();
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken("\n");
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

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

        int[] nextIntArray(int n, int delta) {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt() + delta;
            return a;
        }

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

    static <A> void writeLines(A[] as, Function<A, String> f) {
        PrintWriter pw = new PrintWriter(System.out);
        for (A a : as) {
            pw.println(f.apply(a));
        }
        pw.flush();
    }

    static void writeLines(int[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (int a : as) pw.println(a);
        pw.flush();
    }

    static void writeLines(long[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (long a : as) pw.println(a);
        pw.flush();
    }

    static int max(int... as) {
        int max = Integer.MIN_VALUE;
        for (int a : as) max = Math.max(a, max);
        return max;
    }

    static int min(int... as) {
        int min = Integer.MAX_VALUE;
        for (int a : as) min = Math.min(a, min);
        return min;
    }

    static void debug(Object... args) {
        StringJoiner j = new StringJoiner(" ");
        for (Object arg : args) {
            if (arg instanceof int[]) j.add(Arrays.toString((int[]) arg));
            else if (arg instanceof long[]) j.add(Arrays.toString((long[]) arg));
            else if (arg instanceof double[]) j.add(Arrays.toString((double[]) arg));
            else if (arg instanceof Object[]) j.add(Arrays.toString((Object[]) arg));
            else j.add(arg.toString());
        }
        System.err.println(j.toString());
    }
}