Striped Horse

AtCoder

IDabc440_c

Time2000ms

Memory256MB

Difficulty—

> Ringo-san is trying to paint a horse with stripes to make it a zebra. There are $N$ squares numbered $1$ to $N$ arranged in a line. Initially, all squares are white, and the cost of painting square $i$ black is $C_i$. Consider performing the following procedure once to paint some squares black: * Choose a positive integer $x$ freely. * Paint square $i$ black for all integers $i$ satisfying $1 \leq i \leq N$ such that the remainder when $(i+x)$ is divided by $2W$ is less than $W$. Find the minimum total cost to perform this procedure. You are given $T$ test cases; solve each of them. ## Constraints * $1\leq T \leq 2\times 10^5$ * $1\leq N \leq 2\times 10^5$ * $1\leq W \leq 2\times 10^5$ * $1\leq C_i \leq 10^9$ * The sum of $N$ over the $T$ test cases is at most $2\times 10^5$. * The sum of $W$ over the $T$ test cases is at most $2\times 10^5$. * All input values are integers. ## Input The input is given from Standard Input in the following format, where $\mathrm{case}_i$ denotes the $i$\-th test case. $T$ $\mathrm{case}_1$ $\vdots$ $\mathrm{case}_T$ Each test case is given in the following format: $N$ $W$ $C_1$ $\dots$ $C_N$ [samples]

Samples

Input #1

4
8 2
1 10 10 1 1 10 10 1
8 3
1 10 10 1 1 10 10 1
8 4
1 10 10 1 1 10 10 1
4 100
100000 100000 100000 100000

Output #1

4
12
22
0

In the first test case, if the procedure is executed with $x=4$, squares $1,4,5,8$ are painted black, and the total cost is $4$. The total cost cannot be less than $4$, so this is the minimum.

API Response (JSON)

{
  "problem": {
    "name": "Striped Horse",
    "description": {
      "content": "> Ringo-san is trying to paint a horse with stripes to make it a zebra. There are $N$ squares numbered $1$ to $N$ arranged in a line.   Initially, all squares are white, and the cost of painting squa",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc440_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "> Ringo-san is trying to paint a horse with stripes to make it a zebra.\n\nThere are $N$ squares numbered $1$ to $N$ arranged in a line.  \nInitially, all squares are white, and the cost of painting squa...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments