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.*;

 
public class Main{
	int R;
	int C;
	int N;
	HashMap<Integer, Integer> map;
	ArrayDeque<Pair> queue;
	
	class Pair{
		int r;
		int c;
		int value;
		public Pair(int r, int c, int value){
			this.r = r;
			this.c = c;
			this.value = value;
		}
	}
	public void solve(){
		R = nextInt();
		C = nextInt();
		N = nextInt();
		map = new HashMap<>(N * 10);
		queue = new ArrayDeque<>();
		
		for(int i = 0; i < N; i++){
			int r = nextInt();
			int c = nextInt();
			int a = nextInt();
			if(!(checkAdd(r, c, a))){
				out.println("No");
				return;
			}
		}
		while(!queue.isEmpty()){
			Pair p = queue.removeFirst();
			if(!checkAdd(p.r, p.c, p.value)){
				out.println("No");
				
				return;
			}
		}
		out.println("Yes");
	}
	public int key(int r, int c){
		return (r << 16) | c;
	}
	
	public boolean checkAdd(int r, int c, int a){
		if(map.containsKey(key(r, c))){
			return map.get(key(r, c)) == a; 
		}
		
		map.put(key(r, c), a);
		return check(r, c) && check(r - 1, c - 1) && check(r, c - 1) && check(r - 1, c);
	}
	
	public boolean check(int r, int c){
		if(map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			return map.get(key(r, c)) + map.get(key(r + 1, c + 1)) == map.get(key(r, c + 1)) + map.get(key(r + 1, c));
		}else if(map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			int a = map.get(key(r, c)) + map.get(key(r + 1, c + 1)) - map.get(key(r, c + 1));
			if(a >= 0){
				queue.addLast(new Pair(r + 1, c, a));
				return true;
			}else return false;

		}else if(map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			int a = map.get(key(r, c)) + map.get(key(r + 1, c + 1)) - map.get(key(r + 1, c));
			if(a >= 0){
				queue.addLast(new Pair(r, c + 1, a));
				return true;
			}else return false;

		}else if(map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;

		}else if(map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			int a = map.get(key(r, c + 1)) + map.get(key(r + 1, c)) - map.get(key(r, c));
			if(a >= 0){
				queue.addLast(new Pair(r + 1, c + 1, a));
				return true;
			}else return false;
			
			
		}else if(map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;
			
		}else if(map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			return true;
			
		}else if(map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;
			
		}else if(!map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			int a = map.get(key(r, c + 1)) + map.get(key(r + 1, c)) - map.get(key(r + 1, c + 1));
			if(a >= 0){
				queue.addLast(new Pair(r, c, a));
				return true;
			}else return false;
			
			
		}else if(!map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;

		}else if(!map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			return true;

		}else if(!map.containsKey(key(r, c)) &&  map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;

		}else if(!map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			return true;

		}else if(!map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  &&  map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;

		}else if(!map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  &&  map.containsKey(key(r + 1, c))){
			return true;

		}else if(!map.containsKey(key(r, c)) && !map.containsKey(key(r + 1, c + 1))  && !map.containsKey(key(r, c + 1))  && !map.containsKey(key(r + 1, c))){
			return true;

		}
		return true;
	}
	
	private static PrintWriter out;
	public static void main(String[] args){
		out = new PrintWriter(System.out);
		new Main().solve();
		out.flush();
	}
	
	
	
	public static int nextInt(){
		int num = 0;
		String str = next();
		boolean minus = false;
		int i = 0;
		if(str.charAt(0) == '-'){
			minus = true;
			i++;
		}
		int len = str.length();
		for(;i < len; i++){
			char c = str.charAt(i);
			if(!('0' <= c && c <= '9')) throw new RuntimeException();
			num = num * 10 + (c - '0');
		}
		return minus ? -num : num;
	}
	
	public static long nextLong(){
		long num = 0;
		String str = next();
		boolean minus = false;
		int i = 0;
		if(str.charAt(0) == '-'){
			minus = true;
			i++;
		}
		int len = str.length();
		for(;i < len; i++){
			char c = str.charAt(i);
			if(!('0' <= c && c <= '9')) throw new RuntimeException();
			num = num * 10l + (c - '0');
		}
		return minus ? -num : num;
	}
	public static String next(){
		int c;
		while(!isAlNum(c = read())){}
		StringBuilder build = new StringBuilder();
		build.append((char)c);
		while(isAlNum(c = read())){
			build.append((char)c);
		}
		return build.toString();
	}
	
	
	private static byte[] inputBuffer = new byte[1024];
	private static int bufferLength = 0;
	private static int bufferIndex = 0;
	private static int read(){
		if(bufferLength < 0) throw new RuntimeException();
		if(bufferIndex >= bufferLength){
			try{
				bufferLength = System.in.read(inputBuffer);
				bufferIndex = 0;
			}catch(IOException e){
				throw new RuntimeException(e);
			}
			if(bufferLength <= 0) return (bufferLength = -1);
		}
		return inputBuffer[bufferIndex++];
	}
	
	private static boolean isAlNum(int c){
		return '!' <= c && c <= '~';
	}
}