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.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.Comparator;
import java.util.InputMismatchException;

public class Main {
	static InputStream is;
	static PrintWriter out;
	// 01
	// 010
	
	static String INPUT = "";
	
	static void solve()
	{
		int R = ni(), C = ni();
		int n = ni();
		int[][] co = new int[n][];
		for(int i = 0;i < n;i++){
			co[i] = new int[]{ni(), ni(), ni()};
		}
		long[] minc = new long[100001];
		long[] minr = new long[100001];
		DJSet dsc = new DJSet(100001);
		DJSet dsr = new DJSet(100001);
		try{
			{
				Arrays.sort(co, new Comparator<int[]>() {
					public int compare(int[] a, int[] b) {
						return a[0] - b[0];
					}
				});
				Arrays.fill(minc, Long.MAX_VALUE / 4);
				for(int i = 0;i < n;){
					int j = i;
					while(j < n && co[i][0] == co[j][0])j++;
					for(int k = i+1;k < j;k++){
						dsc.union(co[i][1], co[k][1], co[i][2]-co[k][2]);
					}
					i = j;
				}
				for(int i = 0;i <= 100000;i++){
					minc[dsc.root(i)] = Math.min(minc[dsc.root(i)], dsc.delta[i]);
				}
			}
			{
				Arrays.sort(co, new Comparator<int[]>() {
					public int compare(int[] a, int[] b) {
						return a[1] - b[1];
					}
				});
				Arrays.fill(minr, Long.MAX_VALUE / 4);
				for(int i = 0;i < n;){
					int j = i;
					while(j < n && co[i][1] == co[j][1])j++;
					for(int k = i+1;k < j;k++){
						dsr.union(co[i][0], co[k][0], co[i][2]-co[k][2]);
					}
					i = j;
				}
				for(int i = 0;i <= 100000;i++){
					minr[dsr.root(i)] = Math.min(minr[dsr.root(i)], dsr.delta[i]);
				}
			}
//			tr(dsr.delta);
//			tr(dsr.root(1));
//			tr(Arrays.copyOf(dsr.delta, 10));
//			tr(Arrays.copyOf(minr, 10));
//			tr(minr);
			for(int i = 0;i < n;i++){
				int rr = dsr.root(co[i][0]);
				int rc = dsc.root(co[i][1]);
				long val = (long)co[i][2] + minr[rr] - dsr.delta[co[i][0]] + minc[rc] - dsc.delta[co[i][1]];
				if(val < 0){
					out.println("No");
					return;
				}
			}
			out.println("Yes");
		}catch(IllegalStateException e){
			out.println("No");
			return;
		}
	}
	
	public static class DJSet {
		public int[] upper;
		public long[] delta;

		public DJSet(int n) {
			upper = new int[n];
			delta = new long[n];
			Arrays.fill(upper, -1);
		}

		public int root(int x) {
			if(upper[x] < 0)return x;
			int newr = root(upper[x]);
			delta[x] += delta[upper[x]];
			upper[x] = newr;
			return newr;
		}

		public boolean equiv(int x, int y) {
			return root(x) == root(y);
		}

		public boolean union(int x, int y, long xmy) {
			int rx = root(x);
			// v[x]-v[y]
			// v[rx]-v[ry]
			xmy += -delta[x];
			x = rx;
			int ry = root(y);
			xmy += delta[y];
			y = ry;
			if (x != y) {
				if (upper[y] < upper[x]) {
					int d = x;
					x = y;
					y = d;
					xmy = -xmy;
				}
				upper[x] += upper[y];
				delta[y] = -xmy;
				upper[y] = x;
			}else{
				if(xmy != 0)throw new IllegalStateException();
			}
			return x == y;
		}

		public int count() {
			int ct = 0;
			for (int u : upper)
				if (u < 0)
					ct++;
			return ct;
		}
	}

	
	public static void main(String[] args) throws Exception
	{
		long S = System.currentTimeMillis();
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		solve();
		out.flush();
		long G = System.currentTimeMillis();
		tr(G-S+"ms");
	}
	
	private static boolean eof()
	{
		if(lenbuf == -1)return true;
		int lptr = ptrbuf;
		while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
		
		try {
			is.mark(1000);
			while(true){
				int b = is.read();
				if(b == -1){
					is.reset();
					return true;
				}else if(!isSpaceChar(b)){
					is.reset();
					return false;
				}
			}
		} catch (IOException e) {
			return true;
		}
	}
	
	private static byte[] inbuf = new byte[1024];
	static int lenbuf = 0, ptrbuf = 0;
	
	private static int readByte()
	{
		if(lenbuf == -1)throw new InputMismatchException();
		if(ptrbuf >= lenbuf){
			ptrbuf = 0;
			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
			if(lenbuf <= 0)return -1;
		}
		return inbuf[ptrbuf++];
	}
	
	private static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
//	private static boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private static double nd() { return Double.parseDouble(ns()); }
	private static char nc() { return (char)skip(); }
	
	private static String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private static char[] ns(int n)
	{
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while(p < n && !(isSpaceChar(b))){
			buf[p++] = (char)b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}
	
	private static char[][] nm(int n, int m)
	{
		char[][] map = new char[n][];
		for(int i = 0;i < n;i++)map[i] = ns(m);
		return map;
	}
	
	private static int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private static int ni()
	{
		int num = 0, b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static long nl()
	{
		long num = 0;
		int b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
}