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In the first test case, the following sequence of operations is possible: |
* perform the operation on $v=3$, then the values on the vertices will be $[0, 1, 1, 1]$; * perform the operation on $v=1$, then the values on the vertices will be $[1, 0, 0, 0]$. |
I see satyam343. I'm shaking. Please more median problems this time. I love those. Please satyam343 we believe in you. |
— satyam343's biggest fan |
You are given an array $a$ of length $n$ and an integer $k$. You are also given a binary array $b$ of length $n$. |
You can perform the following operation at most $k$ times: |
* Select an index $i$ ($1 \leq i \leq n$) such that $b_i = 1$. Set $a_i = a_i + 1$ (i.e., increase $a_i$ by $1$). |
Your score is defined to be $\max\limits_{i = 1}^{n} \left( a_i + \operatorname{median}(c_i) \right)$, where $c_i$ denotes the array of length $n-1$ that you get by deleting $a_i$ from $a$. In other words, your score is the maximum value of $a_i + \operatorname{median}(c_i)$ over all $i$ from $1$ to $n$. |
Find the maximum score that you can achieve if you perform the operations optimally. |
For an arbitrary array $p$, $\operatorname{median}(p)$ is defined as the $\left\lfloor \frac{|p|+1}{2} \right\rfloor$-th smallest element of $p$. For example, $\operatorname{median} \left( [3,2,1,3] \right) = 2$ and $\operatorname{median} \left( [6,2,4,5,1] \right) = 4$. |
The first line contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. |
Each test case begins with two integers $n$ and $k$ ($2 \leq n \leq 2 \cdot 10^5$, $0 \leq k \leq 10^9$) — the length of the $a$ and the number of operations you can perform. |
The following line contains $n$ space separated integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$) — denoting the array $a$. |
The following line contains $n$ space separated integers $b_1, b_2, \ldots, b_n$ ($b_i$ is $0$ or $1$) — denoting the array $b$. |
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 maximum value of score you can get on a new line. |
For the first test case, it is optimal to perform $5$ operations on both elements so $a = [8,8]$. So, the maximum score we can achieve is $\max(8 + \operatorname{median}[8], 8 + \operatorname{median}[8]) = 16$, as $c_1 = [a_2] = [ |
MOOOOOOOOOOOOOOOOO |
— Bessie the Cow, The Art of Racing on Islands |
Two of Farmer John's cows, Bessie and Elsie, are planning to race on $n$ islands. There are $n - 1$ main bridges, connecting island $i$ to island $i + 1$ for all $1 \leq i \leq n - 1$. Additionally, there are $m$ alternative bridges. Elsie can use both main and alternative bridges, while Bessie can only use main bridge... |
Initially, Elsie starts on island $1$, and Bessie starts on island $s$. The cows alternate turns, with Bessie making the first turn. Suppose the cow is on island $i$. During a cow's turn, if there are any bridges connecting island $i$ to island $j$, then the cow can move to island $j$. Then, island $i$ collapses, and a... |
* If there are no bridges connecting island $i$ to another island, then island $i$ collapses, and this cow is eliminated from the race. * If the other cow is also on island $i$, then after this cow moves to another island, the island collapses, and the other cow is eliminated from the race. * After an island or... |
The race ends once either cow reaches island $n$. It can be shown that regardless of the cows' strategies, at least one cow reaches island $n$. Bessie wins if and only if she reaches island $n$ first. |
For each $1 \leq s \leq n - 1$, determine whether Bessie wins if she starts the race on island $s$. Assume both cows follow optimal strategies to ensure their own respective victories. |
The first line contains $t$ ($1 \leq t \leq 10^4$) – the number of test cases. |
The first line of each test case contains $n$ and $m$ ($2 \leq n \leq 2 \cdot 10^5$, $0 \leq m \leq 2 \cdot 10^5$) – the number of islands and the number of alternative bridges. |
The next $m$ lines of each test case conta |
Drink water. |
— Sun Tzu, The Art of Becoming a Healthy Programmer |
This is the easy version of the problem. The only difference is that $x=n$ in this version. You must solve both versions to be able to hack. |
You are given two integers $n$ and $x$ ($x=n$). There are $n$ balls lined up in a row, numbered from $1$ to $n$ from left to right. Initially, there is a value $a_i$ written on the $i$-th ball. |
For each integer $i$ from $1$ to $n$, we define a function $f(i)$ as follows: |
* Suppose you have a set $S = \\{1, 2, \ldots, i\\}$. |
* In each operation, you have to select an integer $l$ ($1 \leq l < i$) from $S$ such that $l$ is not the largest element of $S$. Suppose $r$ is the smallest element in $S$ which is greater than $l$. |
* If $a_l > a_r$, you set $a_l = a_l + a_r$ and remove $r$ from $S$. * If $a_l < a_r$, you set $a_r = a_l + a_r$ and remove $l$ from $S$. * If $a_l = a_r$, you choose either the integer $l$ or $r$ to remove from $S$: * If you choose to remove $l$ from $S$, you set $a_r = a_l + a_r$ and remove $l$ f... |
* $f(i)$ denotes the number of integers $j$ ($1 \le j \le i$) such that it is possible to obtain $S = \\{j\\}$ after performing the above operations exactly $i - 1$ times. |
For each integer $i$ from $x$ to $n$, you need to find $f(i)$. |
The first line contains $t$ ($1 \leq t \leq 10^4$) — the number of test cases. |
The first line of each test case contains two integers $n$ and $x$ ($1 \leq n \leq 2 \cdot 10^5; x = n$) — the number of balls and the smallest index $i$ for which you need to find $f(i)$. |
The second line of each test case contains $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$) — the initial number written on each ball. |
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$. |
For each test case, output $n-x+1$ space separated integers on a new line, where the $j$-th integer should represent $f(x+j-1)$. |
In |
As a computer science student, Alex faces a hard challenge — showering. He tries to shower daily, but despite his best efforts there are always challenges. He takes $s$ minutes to shower and a day only has $m$ minutes! |
He already has $n$ tasks planned for the day. Task $i$ is represented as an interval $(l_i$, $r_i)$, which means that Alex is busy and can not take a shower in that time interval (at any point in time strictly between $l_i$ and $r_i$). No two tasks overlap. |
Given all $n$ time intervals, will Alex be able to shower that day? In other words, will Alex have a free time interval of length at least $s$? |
 |
In the first test case, Alex can shower for the first $3$ minutes of the day and not miss any of the tasks. |
The first line contains a single integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. |
The first line of each test case contains three integers $n$, $s$, and $m$ ($1 \leq n \leq 2 \cdot 10^5$; $1 \leq s, m \leq 10^9$) — the number of time intervals Alex already has planned, the amount of time Alex takes to take a shower, and the amount of minutes a day has. |
Then $n$ lines follow, the $i$-th of which contains two integers $l_i$ and $r_i$ ($0 \leq l_i < r_i \leq m$) — the time interval of the $i$-th task. No two tasks overlap. |
Additional constraint on the input: $l_i > r_{i-1}$ for every $i > 1$. |
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