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* write number: write the minimum positive integer that isn't present in the array $a$ to the element $a_i$, $i$ is the position where the pointer points at. Such operation can be performed only when $a_i = -1$. * carriage return: return the pointer to its initial position (i.e. $1$ for the first typewriter, $n$ for the second) * move pointer: move the pointer to the next position, let $i$ be the position the pointer points at before this operation, if Misuki is using the first typewriter, $i := i + 1$ would happen, and $i := i - 1$ otherwise. Such operation can be performed only if after the operation, $1 \le i \le n$ holds. |
Your task is to construct any permutation $p$ of length $n$, such that the minimum number of carriage return operations needed to make $a = p$ is the same no matter which typewriter Misuki is using. |
Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. The description of the test cases follows. |
The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the length of the permutation. |
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$. |
For each test case, output a line of $n$ integers, representing the permutation $p$ of length $n$ such that the minimum number of carriage return operations needed to make $a = p$ is the same no matter which typewriter Misuki is using, or $-1$ if it is impossible to do so. |
If there are multiple valid permutations, you can output any of them. |
In the |
This is an interactive problem. |
Misuki has chosen a secret tree with $n$ nodes, indexed from $1$ to $n$, and asked you to guess it by using queries of the following type: |
* "? a b" — Misuki will tell you which node $x$ minimizes $|d(a,x) - d(b,x)|$, where $d(x,y)$ is the distance between nodes $x$ and $y$. If more than one such node exists, Misuki will tell you the one which minimizes $d(a,x)$. |
Find out the structure of Misuki's secret tree using at most $15n$ queries! |
Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases. |
Each test case consists of a single line with an integer $n$ ($2 \le n \le 1000$), the number of nodes in the tree. |
It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$. |
A tree is an undirected acyclic connected graph. A tree with $n$ nodes will always have $n-1$ edges. |
In the example case, the answer to "? 1 2" is $1$. This means that there is an edge between nodes $1$ and $2$. |
The answer to "? 1 3" is $1$. This means that there is an edge between nodes $1$ and $3$. |
The answer to "? 1 4" is $3$. It can be proven that this can only happen if node $3$ is connected to both node $1$ and $4$. |
The edges of the tree are hence $(1,2)$, $(1,3)$ and $(3,4)$. |
This is the hard version of the problem. The difference between the two versions is the definition of deterministic max-heap, time limit, and constraints on $n$ and $t$. You can make hacks only if both versions of the problem are solved. |
Consider a perfect binary tree with size $2^n - 1$, with nodes numbered from $1$ to $2^n-1$ and rooted at $1$. For each vertex $v$ ($1 \le v \le 2^{n - 1} - 1$), vertex $2v$ is its left child and vertex $2v + 1$ is its right child. Each node $v$ also has a value $a_v$ assigned to it. |
Define the operation $\mathrm{pop}$ as follows: |
1. initialize variable $v$ as $1$; 2. repeat the following process until vertex $v$ is a leaf (i.e. until $2^{n - 1} \le v \le 2^n - 1$); 1. among the children of $v$, choose the one with the larger value on it and denote such vertex as $x$; if the values on them are equal (i.e. $a_{2v} = a_{2v + 1}$), you can choose any of them; 2. assign $a_x$ to $a_v$ (i.e. $a_v := a_x$); 3. assign $x$ to $v$ (i.e. $v := x$); 3. assign $-1$ to $a_v$ (i.e. $a_v := -1$). |
Then we say the $\mathrm{pop}$ operation is deterministic if there is a unique way to do such operation. In other words, $a_{2v} \neq a_{2v + 1}$ would hold whenever choosing between them. |
A binary tree is called a max-heap if for every vertex $v$ ($1 \le v \le 2^{n - 1} - 1$), both $a_v \ge a_{2v}$ and $a_v \ge a_{2v + 1}$ hold. |
A max-heap is deterministic if the $\mathrm{pop}$ operation is deterministic to the heap when we do it for the first and the second time. |
Initially, $a_v := 0$ for every vertex $v$ ($1 \le v \le 2^n - 1$), and your goal is to count the number of different deterministic max- heaps produced by applying the following operation $\mathrm{add}$ exactly $k$ times: |
* Choose an integer $v$ ($1 \le v \le 2^n - 1$) and, for every vertex $x$ on the path between $1$ and $v$, add $1$ to $a_x$. |
Two heaps are considered different if there is a node which has different values in the heaps. |
Since the answer might be large, print it modulo $p$. |
Each test con |
Given an array of integers $s_1, s_2, \ldots, s_l$, every second, cosmic rays will cause all $s_i$ such that $i=1$ or $s_i\neq s_{i-1}$ to be deleted simultaneously, and the remaining parts will be concatenated together in order to form the new array $s_1, s_2, \ldots, s_{l'}$. |
Define the strength of an array as the number of seconds it takes to become empty. |
You are given an array of integers compressed in the form of $n$ pairs that describe the array left to right. Each pair $(a_i,b_i)$ represents $a_i$ copies of $b_i$, i.e. $\underbrace{b_i,b_i,\cdots,b_i}_{a_i\textrm{ times}}$. |
For each $i=1,2,\dots,n$, please find the strength of the sequence described by the first $i$ pairs. |
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le10^4$). The description of the test cases follows. |
The first line of each test case contains a single integer $n$ ($1\le n\le3\cdot10^5$) — the length of sequence $a$. |
The next $n$ lines contain two integers each $a_i$, $b_i$ ($1\le a_i\le10^9,0\le b_i\le n$) — the pairs which describe the sequence. |
It is guaranteed that the sum of all $n$ does not exceed $3\cdot10^5$. |
It is guaranteed that for all $1\le i<n$, $b_i\neq b_{i+1}$ holds. |
For each test case, print one line containing $n$ integers — the answer for each prefix of pairs. |
In the first test case, for the prefix of length $4$, the changes will be $[0,0,1,0,0,0,1,1,1,1,1]\rightarrow[0,0,0,1,1,1,1]\rightarrow[0,0,1,1,1]\rightarrow[0,1,1]\rightarrow[1]\rightarrow[]$, so the array becomes empty after $5$ seconds. |
In the second test case, for the prefix of length $4$, the changes will be $[6,6,6,6,3,6,6,6,6,0,0,0,0]\rightarrow[6,6,6,6,6,6,0,0,0]\rightarrow[6,6,6,6,6,0,0]\rightarrow[6,6,6,6,0]\rightarrow[6,6,6]\rightarrow[6,6]\rightarrow[6]\rightarrow[]$, so the array becomes empty after $7$ seconds. |
This is the easy version of the problem. In this version, $n=m$ and the time limit is lower. 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. |
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