Inc, Dec - Decomposition

AtCoder

IDarc123_d

Time2000ms

Memory256MB

Difficulty—

Given is a sequence of integers $A = (A_1, \ldots, A_N)$. Consider a pair of sequences of integers $B = (B_1, \ldots, B_N)$ and $C = (C_1, \ldots, C_N)$ that satisfies the following conditions: * $A_i = B_i + C_i$ holds for each $1\leq i\leq N$; * $B$ is non-decreasing, that is, $B_i\leq B_{i+1}$ holds for each $1\leq i\leq N-1$; * $C$ is non-increasing, that is, $C_i\geq C_{i+1}$ holds for each $1\leq i\leq N-1$. Find the minimum possible value of $\sum_{i=1}^N \bigl(\lvert B_i\rvert + \lvert C_i\rvert\bigr)$. ## Constraints * $1\leq N\leq 2\times 10^5$ * $-10^8\leq A_i\leq 10^8$ ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3
1 -2 3

Output #1

10

One pair of sequences $B = (B_1, \ldots, B_N)$ and $C = (C_1, \ldots, C_N)$ that yields the minimum value is:

*   $B = (0, 0, 5)$,
*   $C = (1, -2, -2)$.

Here we have $\sum_{i=1}^N \bigl(\lvert B_i\rvert + \lvert C_i\rvert\bigr) = (0+1) + (0+2) + (5+2) = 10$.

Input #2

4
5 4 3 5

Output #2

17

One pair of sequences $B$ and $C$ that yields the minimum value is:

*   $B = (0, 1, 2, 4)$,
*   $C = (5, 3, 1, 1)$.

Input #3

1
-10

Output #3

10

One pair of sequences $B$ and $C$ that yields the minimum value is:

*   $B = (-3)$,
*   $C = (-7)$.

API Response (JSON)

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      "content": "Given is a sequence of integers $A = (A_1, \\ldots, A_N)$.\nConsider a pair of sequences of integers $B = (B_1, \\ldots, B_N)$ and $C = (C_1, \\ldots, C_N)$ that satisfies the following conditions:\n\n*   $...",
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