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The first line contains an integer $t$ ($1\leq t \leq 10^3$) — the number of test cases. |
The only line of each test case contains four integers $n$, $m$, $l$, $f$ ($2\leq n\leq m \leq 2\cdot 10^7$, $1\leq l,f \leq 10^9$, $\bf{n=m}$): $n$, $m$ gives the upper bound on the number of Lelle and Flamm's wins, $l$ and $f$ determine the final score of the performance. |
Unusual additional constraint: it is guaranteed that, for each test, there are no pairs of test cases with the same pair of $n$, $m$. |
For each test case, output a single integer — the maximum total score of a successful performance. |
In the first test case, a possible performance is as follows: |
* Flamm wins, $\gcd(0,1)=1$. * Lelle wins, $\gcd(1,1)=1$. * Flamm wins, $\gcd(1,2)=1$. * Flamm wins, $\gcd(1,3)=1$. * Lelle wins, $\gcd(2,3)=1$. * Lelle and Flamm agree to stop the match. |
The final score is $2\cdot2+3\cdot5=19$. |
In the third test case, a possible performance is as follows: |
* Flamm wins, $\gcd(0,1)=1$. * Lelle wins, $\gcd(1,1)=1$. * Lelle wins, $\gcd(2,1)=1$. * Lelle wins, $\gcd(3,1)=1$. * Lell |
This is the hard version of the problem. In this version, it is not guaranteed that $n=m$, and the time limit is higher. You can make hacks only if both versions of the problem are solved. |
In the court of the Blue King, Lelle and Flamm are having a performance match. The match consists of several rounds. In each round, either Lelle or Flamm wins. |
Let $W_L$ and $W_F$ denote the number of wins of Lelle and Flamm, respectively. The Blue King considers a match to be successful if and only if: |
* after every round, $\gcd(W_L,W_F)\le 1$; * at the end of the match, $W_L\le n, W_F\le m$. |
Note that $\gcd(0,x)=\gcd(x,0)=x$ for every non-negative integer $x$. |
Lelle and Flamm can decide to stop the match whenever they want, and the final score of the performance is $l \cdot W_L + f \cdot W_F$. |
Please help Lelle and Flamm coordinate their wins and losses such that the performance is successful, and the total score of the performance is maximized. |
The first line contains an integer $t$ ($1\leq t \leq 10^3$) — the number of test cases. |
The only line of each test case contains four integers $n$, $m$, $l$, $f$ ($2\leq n\leq m \leq 2\cdot 10^7$, $1\leq l,f \leq 10^9$): $n$, $m$ give the upper bound on the number of Lelle and Flamm's wins, $l$ and $f$ determine the final score of the performance. |
Unusual additional constraint: it is guaranteed that, for each test, there are no pairs of test cases with the same pair of $n$, $m$. |
For each test case, output a single integer — the maximum total score of a successful performance. |
In the first test case, a possible performance is as follows: |
* Flamm wins, $\gcd(0,1)=1$. * Lelle wins, $\gcd(1,1)=1$. * Flamm wins, $\gcd(1,2)=1$. * Flamm wins, $\gcd(1,3)=1$. * Flamm wins, $\gcd(1,4)=1$. * Lelle and Flamm agree to stop the match. |
The final score is $1\cdot2+4\cdot5=22$. |
Consider a grid graph with $n$ rows and $n$ columns. Let the cell in row $x$ and column $y$ be $(x,y)$. There exists a directed edge from $(x,y)$ to $(x+1,y)$, with non-negative integer value $d_{x,y}$, for all $1\le x < n, 1\le y \le n$, and there also exists a directed edge from $(x,y)$ to $(x,y+1)$, with non-negative integer value $r_{x,y}$, for all $1\le x \le n, 1\le y < n$. |
Initially, you are at $(1,1)$, with an empty set $S$. You need to walk along the edges and eventually reach $(n,n)$. Whenever you pass an edge, its value will be inserted into $S$. Please maximize the MEX$^{\text{∗}}$ of $S$ when you reach $(n,n)$. |
$^{\text{∗}}$The MEX (minimum excluded) of an array is the smallest non- negative integer that does not belong to the array. For instance: |
* The MEX of $[2,2,1]$ is $0$, because $0$ does not belong to the array. * The MEX of $[3,1,0,1]$ is $2$, because $0$ and $1$ belong to the array, but $2$ does not. * The MEX of $[0,3,1,2]$ is $4$, because $0, 1, 2$, and $3$ belong to the array, but $4$ does not. |
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le100$). The description of the test cases follows. |
The first line of each test case contains a single integer $n$ ($2\le n\le20$) — the number of rows and columns. |
Each of the next $n-1$ lines contains $n$ integers separated by single spaces — the matrix $d$ ($0\le d_{x,y}\le 2n-2$). |
Each of the next $n$ lines contains $n-1$ integers separated by single spaces — the matrix $r$ ($0\le r_{x,y}\le 2n-2$). |
It is guaranteed that the sum of all $n^3$ does not exceed $8000$. |
For each test case, print a single integer — the maximum MEX of $S$ when you reach $(n,n)$. |
In the first test case, the grid graph and one of the optimal paths are as follows: |
 |
In the second test case, the grid graph and one of the optimal paths are as follows: |
 |
Turtle gives you a string $s$, consisting of lowercase Latin letters. |
Turtle considers a pair of integers $(i, j)$ ($1 \le i < j \le n$) to be a pleasant pair if and only if there exists an integer $k$ such that $i \le k < j$ and both of the following two conditions hold: |
* $s_k \ne s_{k + 1}$; * $s_k \ne s_i$ or $s_{k + 1} \ne s_j$. |
Besides, Turtle considers a pair of integers $(i, j)$ ($1 \le i < j \le n$) to be a good pair if and only if $s_i = s_j$ or $(i, j)$ is a pleasant pair. |
Turtle wants to reorder the string $s$ so that the number of good pairs is maximized. Please help him! |
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows. |
The first line of each test case contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the string. |
The second line of each test case contains a string $s$ of length $n$, consisting of lowercase Latin letters. |
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$. |
For each test case, output the string $s$ after reordering so that the number of good pairs is maximized. If there are multiple answers, print any of them. |
In the first test case, $(1, 3)$ is a good pair in the reordered string. It can be seen that we can't reorder the string so that the number of good pairs is greater than $1$. bac and cab can also be the answer. |
In the second test case, $(1, 2)$, $(1, 4)$, $(1, 5)$, $(2, 4)$, $(2, 5)$, $(3, 5)$ are good pairs in the reordered string. efddd can also be the answer. |
This is an easy version of this problem. The differences between the versions are the constraint on $m$ and $r_i < l_{i + 1}$ holds for each $i$ from $1$ to $m - 1$ in the easy version. You can make hacks only if both versions of the problem are solved. |
Turtle gives you $m$ intervals $[l_1, r_1], [l_2, r_2], \ldots, [l_m, r_m]$. He thinks that a permutation $p$ is interesting if there exists an integer $k_i$ for every interval ($l_i \le k_i < r_i$), and if he lets $a_i = \max\limits_{j = l_i}^{k_i} p_j, b_i = \min\limits_{j = k_i + 1}^{r_i} p_j$ for every integer $i$ from $1$ to $m$, the following condition holds: |
$$\max\limits_{i = 1}^m a_i < \min\limits_{i = 1}^m b_i$$ |
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