Chef Learning Hungarian Algorithm

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### Read problems statements in [Hindi](http://www.codechef.com/download/translated/COOK105/hindi/HUNGALGO.pdf), [Mandarin Chinese](http://www.codechef.com/download/translated/COOK105/mandarin/HUNGALGO.pdf), [Russian](http://www.codechef.com/download/translated/COOK105/russian/HUNGALGO.pdf), [Vietnamese](http://www.codechef.com/download/translated/COOK105/vietnamese/HUNGALGO.pdf) and [Bengali](http://www.codechef.com/download/translated/COOK105/bengali/HUNGALGO.pdf) as well. Chef is studying computer science. Tomorrow, he has an exam on algorithms and data structures. One of the algorithms he is required to study is the Hungarian algorithm (you do not have to know this algorithm to solve this problem). In the Hungarian algorithm, you start with a matrix of positive integers with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$). The algorithm proceeds in steps. In the first step, for each row of the matrix, you should find the smallest element in this row (let's denote it by $A$) and subtract $A$ from each element of the matrix in this row. In the second step, for each column of the matrix, you should find the smallest element in this column (let's denote it by $A$) and subtract $A$ from each element in this column. The remaining steps of the algorithm are not relevant in this problem. Currently, Chef is practicing implementation of the first two steps of the algorithm. He wants you to check if he is doing these two steps correctly or not. You are given a matrix with $N$ rows and $N$ columns; let's denote the element of this matrix in the $i$th row and $j$th column by $a_{i, j}$. Determine if this matrix can be the result of applying the first two steps of the Hungarian algorithm on some matrix. ### Input  The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.  The first line of each test case contains a single integer $N$.  $N$ lines follow. For each $i$ ($1 \le i \le N$), the $i$th of the following $N$ lines contains $N$ spaceseparated integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, N}$. ### Output For each test case, print a single line containing the string `"YES"` (without quotes) if the given matrix can be a result of the first two steps or `"NO"` (without quotes) if it cannot. ### Constraints  $1 \le T \le 10,000$  $2 \le N \le 500$  $0 \le a_{i, j} \le 10^3$ for each valid $i, j$  the sum of $N^2$ over all test cases does not exceed $2 \cdot 10^6$ ### Example Input ``` 5 3 0 0 4 3 0 1 0 1 0 2 0 5 3 2 2 1 3 5 0 3 0 3 4 4 0 5 6 5 0 4 3 0 2 4 0 2 3 5 3 4 3 0 3 4 0 3 ``` ### Example Output ``` YES NO NO YES YES ``` ### Explanation **Example case 1:** Let's consider the matrix ``` 3 1 5 7 2 3 5 4 3 ``` and apply the first two steps of the Hungarian algorithm on it. In the first step:  the smallest integer in row $1$ is $1$  the smallest integer in row $2$ is $2$  the smallest integer in row $3$ is $3$ After subtracting, the matrix becomes ``` 2 0 4 5 0 1 2 1 0 ``` and the second step is performed on this matrix. In the second step:  the smallest integer in column $1$ is $2$  the smallest integer in column $2$ is $0$  the smallest integer in column $3$ is $0$ After subtracting, the resulting matrix is ``` 0 0 4 3 0 1 0 1 0 ``` and this is the matrix given on the input, so the answer is YES. **Example case 2:** It is impossible to find a matrix such that after applying the first two steps of the Hungarian algorithm, it becomes the matrix given on the input, so the answer is NO.Author:  kingofnumbers 
Editorial  https://discuss.codechef.com/problems/HUNGALGO 
Tags  cakewalk, cookoff, kingofnumbers, matrix 
Date Added:  19042019 
Time Limit:  1 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.6, PYPY, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, rust, SCALA, swift, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, kotlin, PERL6, TEXT, SCM chicken, PYP3, CLOJ, R, COB, FS 
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