Up-Down Queries

AtCoder

IDarc168_f

Time5000ms

Memory256MB

Difficulty—

For an integer sequence $x=(x_1,x_2,\cdots,x_N)$ of length $N$ with each element being between $0$ and $M$, inclusive, we define $f(x)$ as follows: * Prepare an integer sequence $y=(y_1,y_2,\cdots,y_M)$ of length $M$. Initially, set all elements of $y$ to $0$. Then, for each $i=1,2,\cdots,N$, in order, perform the following operations: * For each integer $j$ ($1 \leq j \leq x_i$), replace the value of $y_j$ with $\max(y_j-1,0)$. * For each integer $j$ ($x_i < j \leq M$), replace the value of $y_j$ with $y_j+1$. * The sum of the elements of $y$ after all operations is the value of $f(x)$. You are given an integer sequence $A=(A_1,A_2,\cdots,A_N)$ of length $N$ with each element being between $0$ and $M$, inclusive. Process $Q$ queries: * The $i$\-th query: Given integers $X_i$ and $Y_i$, replace the value of $A_{X_i}$ with $Y_i$. Then, print the value of $f(A)$. ## Constraints * $1 \leq N,M,Q \leq 250000$ * $0 \leq A_i \leq M$ * $1 \leq X_i \leq N$ * $0 \leq Y_i \leq M$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $Q$ $A_1$ $A_2$ $\cdots$ $A_N$ $X_1$ $Y_1$ $X_2$ $Y_2$ $\vdots$ $X_Q$ $Y_Q$ [samples]

Samples

Input #1

Output #1

2
6

Consider the first query. We replace the value of $A_1$ with $4$, making $A=(4,2,3)$. Then, the value of $f(A)$ is calculated as follows:

*   Prepare $y=(0,0,0,0)$.
*   Perform the operation for $A_1=4$, resulting in $y=(0,0,0,0)$.
*   Perform the operation for $A_2=2$, resulting in $y=(0,0,1,1)$.
*   Perform the operation for $A_3=3$, resulting in $y=(0,0,0,2)$.
*   The sum of the elements of $y$, which equals $2$, is the value of $f(A)$.

Next, consider the second query. We replace the value of $A_3$ with $0$, making $A=(4,2,0)$. Then, the value of $f(A)$ is calculated as follows:

*   Prepare $y=(0,0,0,0)$.
*   Perform the operation for $A_1=4$, resulting in $y=(0,0,0,0)$.
*   Perform the operation for $A_2=2$, resulting in $y=(0,0,1,1)$.
*   Perform the operation for $A_3=0$, resulting in $y=(1,1,2,2)$.
*   The sum of the elements of $y$, which equals $6$, is the value of $f(A)$.

Input #2

7 2 9
2 0 2 2 0 1 0
1 1
3 0
4 0
4 1
6 1
3 2
2 0
3 2
2 0

Output #2

Input #3

20 200000 10
39664 143179 193565 153887 16141 91985 51452 155409 116777 190060 87620 64458 106481 51272 9108 100995 139248 18243 181424 6182
4 196305
13 59753
8 96194
6 57037
19 125781
16 142779
15 13967
10 17772
16 84763
12 17283

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Up-Down Queries",
    "description": {
      "content": "For an integer sequence $x=(x_1,x_2,\\cdots,x_N)$ of length $N$ with each element being between $0$ and $M$, inclusive, we define $f(x)$ as follows: *   Prepare an integer sequence $y=(y_1,y_2,\\cdots,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc168_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For an integer sequence $x=(x_1,x_2,\\cdots,x_N)$ of length $N$ with each element being between $0$ and $M$, inclusive, we define $f(x)$ as follows:\n\n*   Prepare an integer sequence $y=(y_1,y_2,\\cdots,...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments