Increment Decrement Again

AtCoder

IDarc182_d

Time2000ms

Memory256MB

Difficulty—

An integer sequence where no two adjacent elements are the same is called a **good sequence**. You are given two good sequences of length $N$: $A=(A_1,A_2,\dots,A_N)$ and $B=(B_1,B_2,\dots,B_N)$. Each element of $A$ and $B$ is between $0$ and $M-1$, inclusive. You can perform the following operations on $A$ any number of times, possibly zero: * Choose an integer $i$ between $1$ and $N$, inclusive, and perform one of the following: * Set $A_i \leftarrow (A_i + 1) \bmod M$. * Set $A_i \leftarrow (A_i - 1) \bmod M$. Here, $(-1) \bmod M = M - 1$. However, you cannot perform an operation that makes $A$ no longer a good sequence. Determine if it is possible to make $A$ equal to $B$, and if it is possible, find the minimum number of operations required to do so. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $2 \leq M \leq 10^6$ * $0\leq A_i,B_i< M(1\leq i\leq N)$ * $A_i\ne A_{i+1}(1\leq i\leq N-1)$ * $B_i\ne B_{i+1}(1\leq i\leq N-1)$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $A_1$ $A_2$ $\dots$ $A_N$ $B_1$ $B_2$ $\dots$ $B_N$ [samples]

Samples

Input #1

3 9
2 0 1
4 8 1

Output #1

3

You can achieve the goal in three operations as follows:

*   Set $A_1 \leftarrow (A_1 + 1) \bmod M$. Now $A = (3, 0, 1)$.
*   Set $A_2 \leftarrow (A_2 - 1) \bmod M$. Now $A = (3, 8, 1)$.
*   Set $A_1 \leftarrow (A_1 + 1) \bmod M$. Now $A = (4, 8, 1)$.

It is impossible to achieve the goal in two or fewer operations, so the answer is 3.
For example, you cannot set $A_2 \leftarrow (A_2 + 1) \bmod M$ in the first operation, because it would make $A = (2, 1, 1)$, which is not a good sequence.

Input #2

3 9
1 8 2
1 8 2

Output #2

0

$A$ and $B$ might be equal from the beginning.

Input #3

24 182
128 115 133 52 166 92 164 119 143 99 54 162 86 2 59 166 24 78 81 5 109 67 172 99
136 103 136 28 16 52 2 85 134 64 123 74 64 28 85 161 19 74 14 110 125 104 180 75

Output #3

API Response (JSON)

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    "name": "Increment Decrement Again",
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      "content": "An integer sequence where no two adjacent elements are the same is called a **good sequence**. You are given two good sequences of length $N$: $A=(A_1,A_2,\\dots,A_N)$ and $B=(B_1,B_2,\\dots,B_N)$. Each",
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      "statement_type": "Markdown",
      "content": "An integer sequence where no two adjacent elements are the same is called a **good sequence**.\nYou are given two good sequences of length $N$: $A=(A_1,A_2,\\dots,A_N)$ and $B=(B_1,B_2,\\dots,B_N)$. Each...",
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