Paired Parentheses

AtCoder

IDasaporo2_c

Time2000ms

Memory256MB

Difficulty—

You are given two sequences $a$ and $b$, both of length $2N$. The $i$\-th elements in $a$ and $b$ are $a_i$ and $b_i$, respectively. Using these sequences, Snuke is doing the job of calculating the _beauty_ of pairs of **balanced sequences of parentheses** (defined below) of length $2N$. The beauty of a pair $(s,t)$ is calculated as follows: * Let $X=0$. * For each $i$ between $1$ and $2N$ (inclusive), increment $X$ by $a_i$ if $s_i = t_i$, and increment $X$ by $b_i$ otherwise. * The beauty of $(s,t)$ is the final value of $X$. You will be given $Q$ queries. Process them in order. In the $i$\-th query, update the value of $a_{p_i}$ to $x_i$, and the value of $b_{p_i}$ to $y_i$. Then, find the maximum possible beauty of a pair of balanced sequences of parentheses. In this problem, only the sequences below are defined to be balanced sequences of parentheses. * An empty string * The concatenation of `(`, $s$, `)` in this order, where $s$ is a balanced sequence of parentheses * The concatenation of $s$, $t$ in this order, where $s$ and $t$ are balanced sequences of parentheses ## Constraints * $1 \leq N,Q \leq 10^{5}$ * $-10^{9} \leq a_i,b_i,x_i,y_i \leq 10^{9}$ * $1 \leq p_i \leq 2N$ * All input values are integers. ## Input Input is given from Standard Input in the following format: $N$ $Q$ $a_1$ $a_2$ $...$ $a_{2N}$ $b_1$ $b_2$ $...$ $b_{2N}$ $p_1$ $x_1$ $y_1$ $:$ $p_Q$ $x_Q$ $y_Q$ ## Partial Scores * In the test set worth $200$ points, $N \leq 5$ and $Q \leq 10$. * In the test set worth $300$ points, $Q = 1$. [samples]

Samples

Input #1

Output #1

15
15

*   The first query updates $a$ and $b$ to $a=(1,4,7,3),b=(4,6,3,3)$. The maximum possible beauty is $15$ for $(s,t) =($`()()`,`()()`$)$.
*   The second query updates $a$ and $b$ to $a=(1,4,2,3),b=(4,6,5,3)$. The maximum possible beauty is $15$ for $(s,t) =($`()()`,`(())`$)$.

Input #2

7 7
34 -20 -27 42 44 29 9 11 20 44 27 19 -31 -29
46 -50 -11 20 28 46 12 13 33 -22 -48 -27 35 -17
7 27 34
12 -2 22
4 -50 -12
3 -32 15
8 -7 23
3 -30 11
4 -2 23

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Paired Parentheses",
    "description": {
      "content": "You are given two sequences $a$ and $b$, both of length $2N$. The $i$\\-th elements in $a$ and $b$ are $a_i$ and $b_i$, respectively. Using these sequences, Snuke is doing the job of calculating the _b",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "asaporo2_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given two sequences $a$ and $b$, both of length $2N$. The $i$\\-th elements in $a$ and $b$ are $a_i$ and $b_i$, respectively. Using these sequences, Snuke is doing the job of calculating the _b...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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