Dynamic Scheduling

AtCoder

IDabc363_g

Time8000ms

Memory256MB

Difficulty—

You are given two sequences of length $N$: $D=(D_1, D_2, \dots, D_N)$ and $P=(P_1, P_2, \dots, P_N)$. Process $Q$ queries in the order they are given. Each query is given in the following format: * `c x y`: Change $D_c$ to $x$ and $P_c$ to $y$. Then, solve the following problem and print the answer. > There are $N$ jobs numbered $1$ to $N$. > Starting from today (consider this as day $1$), you will choose and complete one job per day for $N$ days. > If you complete job $i$ on or before day $D_i$, you will receive a reward of $P_i$. (If you do not complete it by day $D_i$, you get nothing.) > Find the maximum total reward you can achieve by choosing the optimal order of completing the jobs. ## Constraints * $1 \leq N \leq 10^5$ * $1 \leq Q \leq 10^5$ * $1 \leq D_i \leq N$ * $1 \leq P_i \leq 10^9$ * $1 \leq c \leq N$ * $1 \leq x \leq N$ * $1 \leq y \leq 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format. Here, $\mathrm{query}_i$ denotes the $i$\-th query. $N$ $Q$ $D_1$ $D_2$ $\dots$ $D_N$ $P_1$ $P_2$ $\dots$ $P_N$ $\mathrm{query}_1$ $\mathrm{query}_2$ $\vdots$ $\mathrm{query}_Q$ Each query is given in the following format. $c$ $x$ $y$ [samples]

Samples

Input #1

Output #1

10
13

The first query is as follows:

*   Update $D_3$ to $1$ and $P_3$ to $4$. Now, $D = (1, 2, 1)$ and $P = (3, 6, 4)$.
*   In the subproblem, one optimal procedure is to complete job $3$ on day $1$, job $2$ on day $2$, and job $1$ on day $3$. The total reward is $10$, so print $10$.

The second query is as follows:

*   Update $D_2$ to $3$ and $P_2$ to $9$. Now, $D = (1, 3, 1)$ and $P = (3, 9, 4)$.
*   In the subproblem, one optimal procedure is to complete job $3$ on day $1$, job $1$ on day $2$, and job $2$ on day $3$. The total reward is $13$, so print $13$.

Input #2

5 1
1 2 3 4 5
1000000000 1000000000 1000000000 1000000000 1000000000
1 1 1000000000

Output #2

5000000000

Input #3

10 10
6 2 4 1 5 1 6 6 5 3
45 65 71 52 86 52 48 60 40 98
5 6 5
8 4 34
6 7 83
1 3 21
7 5 85
7 4 51
8 2 81
2 7 54
6 1 5
8 6 30

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Dynamic Scheduling",
    "description": {
      "content": "You are given two sequences of length $N$: $D=(D_1, D_2, \\dots, D_N)$ and $P=(P_1, P_2, \\dots, P_N)$. Process $Q$ queries in the order they are given. Each query is given in the following format: *  ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc363_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given two sequences of length $N$: $D=(D_1, D_2, \\dots, D_N)$ and $P=(P_1, P_2, \\dots, P_N)$.\nProcess $Q$ queries in the order they are given. Each query is given in the following format:\n\n*  ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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