#P1389D. Segment Intersections
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ID: 5492
Type: RemoteJudge
2000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: (None)
Uploaded By:
Hydro
Tags> Segment Intersections
No submission language available for this problem.
Description
You are given two lists of segments $[al_1, ar_1], [al_2, ar_2], \dots, [al_n, ar_n]$ and $[bl_1, br_1], [bl_2, br_2], \dots, [bl_n, br_n]$.
Initially, all segments $[al_i, ar_i]$ are equal to $[l_1, r_1]$ and all segments $[bl_i, br_i]$ are equal to $[l_2, r_2]$.
In one step, you can choose one segment (either from the first or from the second list) and extend it by $1$. In other words, suppose you've chosen segment $[x, y]$ then you can transform it either into $[x - 1, y]$ or into $[x, y + 1]$.
Let's define a total intersection $I$ as the sum of lengths of intersections of the corresponding pairs of segments, i.e. $\sum\limits_{i=1}^{n}{\text{intersection_length}([al_i, ar_i], [bl_i, br_i])}$. Empty intersection has length $0$ and length of a segment $[x, y]$ is equal to $y - x$.
What is the minimum number of steps you need to make $I$ greater or equal to $k$?
The first line contains the single integer $t$ ($1 \le t \le 1000$) — the number of test cases.
The first line of each test case contains two integers $n$ and $k$ ($1 \le n \le 2 \cdot 10^5$; $1 \le k \le 10^9$) — the length of lists and the minimum required total intersection.
The second line of each test case contains two integers $l_1$ and $r_1$ ($1 \le l_1 \le r_1 \le 10^9$) — the segment all $[al_i, ar_i]$ are equal to initially.
The third line of each test case contains two integers $l_2$ and $r_2$ ($1 \le l_2 \le r_2 \le 10^9$) — the segment all $[bl_i, br_i]$ are equal to initially.
It's guaranteed that the sum of $n$ doesn't exceed $2 \cdot 10^5$.
Print $t$ integers — one per test case. For each test case, print the minimum number of step you need to make $I$ greater or equal to $k$.
Input
The first line contains the single integer $t$ ($1 \le t \le 1000$) — the number of test cases.
The first line of each test case contains two integers $n$ and $k$ ($1 \le n \le 2 \cdot 10^5$; $1 \le k \le 10^9$) — the length of lists and the minimum required total intersection.
The second line of each test case contains two integers $l_1$ and $r_1$ ($1 \le l_1 \le r_1 \le 10^9$) — the segment all $[al_i, ar_i]$ are equal to initially.
The third line of each test case contains two integers $l_2$ and $r_2$ ($1 \le l_2 \le r_2 \le 10^9$) — the segment all $[bl_i, br_i]$ are equal to initially.
It's guaranteed that the sum of $n$ doesn't exceed $2 \cdot 10^5$.
Output
Print $t$ integers — one per test case. For each test case, print the minimum number of step you need to make $I$ greater or equal to $k$.
Samples
3
3 5
1 2
3 4
2 1000000000
1 1
999999999 999999999
10 3
5 10
7 8
7
2000000000
0
Note
In the first test case, we can achieve total intersection $5$, for example, using next strategy:
- make $[al_1, ar_1]$ from $[1, 2]$ to $[1, 4]$ in $2$ steps;
- make $[al_2, ar_2]$ from $[1, 2]$ to $[1, 3]$ in $1$ step;
- make $[bl_1, br_1]$ from $[3, 4]$ to $[1, 4]$ in $2$ steps;
- make $[bl_2, br_2]$ from $[3, 4]$ to $[1, 4]$ in $2$ steps.
In the second test case, we can make $[al_1, ar_1] = [0, 1000000000]$ in $1000000000$ steps and $[bl_1, br_1] = [0, 1000000000]$ in $1000000000$ steps.
In the third test case, the total intersection $I$ is already equal to $10 > 3$, so we don't need to do any steps.