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.util.*;
import java.io.*;

public class Main{
    static final Reader sc = new Reader();
    static final PrintWriter out = new PrintWriter(System.out,false);

    public static void main(String[] args) throws Exception {
    	int R = sc.nextInt();
        int C = sc.nextInt();
        int n = sc.nextInt();
        int[] r = new int[n];
        int[] c = new int[n];
        int[] a = new int[n];
        ArrayList<HashMap<Integer,Integer>> maplist = new ArrayList<HashMap<Integer,Integer>>();
        for(int i=0;i<R;i++){
            maplist.add(new HashMap<Integer,Integer>());
        }
        for(int i=0;i<n;i++){
            r[i] = sc.nextInt()-1;
            c[i] = sc.nextInt()-1;
            a[i] = sc.nextInt();
            HashMap<Integer,Integer> mapl = maplist.get(r[i]);
            mapl.put(c[i],a[i]);
        }
        ArrayList<Two> l2 = new ArrayList<Two>();
        ArrayList<ArrayList<One>> l1 = new ArrayList<ArrayList<One>>();
        for(int i=0;i<R;i++){
            ArrayList<One> x = new ArrayList<One>();
            l1.add(x);
        }
        for(int i=0;i<n;i++){
            if(r[i]!=R-1){
                HashMap<Integer,Integer> mapl = maplist.get(r[i]+1);
                if(mapl.containsKey(c[i])){
                    Two t = new Two(r[i],c[i],a[i],r[i]+1,c[i],mapl.get(c[i]));
                    l2.add(t);
                }
                else{
                    ArrayList<One> x = l1.get(r[i]);
                    One o = new One(r[i],c[i],a[i]);
                    x.add(o);
                }
            }
            else{
                ArrayList<One> x = l1.get(R-1);
                One o = new One(r[i],c[i],a[i]);
                x.add(o);
            }
        }
        Point[] ten = new Point[R];
        for(int i=0;i<R;i++){
            ten[i] = new Point(-1,-1);
        }
        boolean flag = true;
        for(int j=0;j<l2.size();j++){
            Two t = l2.get(j);
            //out.println(t.x+" "+t.y+" "+t.z+" "+t.xx+" "+t.yy+" "+t.zz);
            if(ten[t.x].x==-1){
                ArrayList<One> x = l1.get(t.x);
                for(int i=0;i<x.size();i++){
                    One o = x.get(i);
                    if((t.y%2==0 && o.y%2==0) || (t.y%2==1 && o.y%2==1)){
                        if(o.z-t.z+t.zz<0){
                            flag = false;
                        }
                    }
                    else{
                        if(o.z+t.zz-t.z<0){
                            flag = false;
                        }
                    }
                }
                x = l1.get(t.xx);
                for(int i=0;i<x.size();i++){
                    One o = x.get(i);
                    if((t.y%2==0 && o.y%2==0) || (t.y%2==1 && o.y%2==1)){
                        if(t.z-t.zz+o.z<0){
                            flag = false;
                        }
                    }
                    else{
                        if(o.z+t.z-t.zz<0){
                            flag = false;
                        }
                    }   
                }
                Point p = new Point(t.y,t.z-t.zz);
                ten[t.x] = p;
            }
            else{
                if((ten[t.x].x%2==0 && t.y%2==0) || (ten[t.x].x%2==1 && t.y%2==1)){
                    if(ten[t.x].y!=t.z-t.zz){
                        flag = false;
                    }
                }
                else{
                    if(ten[t.x].y!=t.zz-t.z){
                        flag = false;
                    }
                }
            }
            if(!flag){
                break;
            }
        }
        if(flag){
            out.println("Yes");
        }
        else{
            out.println("No");
        }
        out.flush();
        sc.close();
        out.close();
    }

    static void trace(Object... o) { System.out.println(Arrays.deepToString(o));}
}

class Point{
    int x;
    int y;
    Point(int x,int y){
        this.x = x;
        this.y = y;
    }
}

class Two{
    int x;
    int y;
    int z;
    int xx;
    int yy;
    int zz;
    Two(int x,int y,int z,int xx,int yy,int zz){
        this.x = x;
        this.y = y;
        this.z = z;
        this.xx = xx;
        this.yy = yy;
        this.zz = zz;
    }
}

class One{
    int x;
    int y;
    int z;
    One(int x,int y,int z){
        this.x = x;
        this.y = y;
        this.z = z;
    }
}

class Reader {

    private final InputStream in;
    private final byte[] buf = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    public Reader() { this(System.in);}
    public Reader(InputStream source) { this.in = source;}

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

    private int readByte() { if (hasNextByte()) return buf[ptr++]; else return -1;}

    private boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}

    private void skip() { while(hasNextByte() && !isPrintableChar(buf[ptr])) ptr++;}

    public boolean hasNext() {skip(); 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();
        boolean minus = false;
        long num = readByte();

        if(num == '-'){
            num = 0;
            minus = true;
        }else if (num < '0' || '9' < num){
            throw new NumberFormatException();
        }else{
            num -= '0';
        }

        while(true){
            int b = readByte();
            if('0' <= b && b <= '9')
                num = num * 10 + (b - '0');
            else if(b == -1 || !isPrintableChar(b))
                return minus ? -num : num;
            else
                throw new NoSuchElementException();
        }
    }

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

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

    public char nextChar() {
        if (!hasNext()) throw new NoSuchElementException();
        return (char)readByte();
    }

    public String nextLine() {
        while (hasNextByte() && (buf[ptr] == '\n' || buf[ptr] == '\r')) ptr++;
        if (!hasNextByte()) throw new NoSuchElementException();

        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (b != '\n' && b != '\r') {
            sb.appendCodePoint(b);
            b = readByte();
        }

        return sb.toString();
    }

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

    public char[] nextCharArray(int n) {
        char[] res = new char[n];
        for (int i=0; i<n; i++) res[i] = nextChar();
        return res;
    }

    public void close() {try{ in.close();}catch(IOException e){ e.printStackTrace();}};
}