Squared Norm Maximization

AtCoder

IDagc076_e

Time4000ms

Memory256MB

Difficulty—

You are given $N$ pairs of integers $(A_1,B_1),(A_2,B_2),\cdots,(A_N,B_N)$. For a subset $s$ of ${1,2,\cdots,N}$, we define its **score** as follows: * $(\sum_{i \in s} A_i)^2 + (\sum_{i \in s} B_i)^2 - 3 \sum_{i \in s} (A_i^2+B_i^2)$ Find the maximum possible value of the score. Solve $T$ cases for one input. ## Constraints * $1 \leq T \leq 250000$ * $1 \leq N \leq 250000$ * $\sum_{1 \leq i \leq N} |A_i| \leq 10^9$ * $\sum_{1 \leq i \leq N} |B_i| \leq 10^9$ * The sum of $N$ over the $T$ cases is at most $250000$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $case_1$ $case_2$ $\vdots$ $case_T$ Each test case is given in the following format: $N$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_N$ $B_N$ [samples]

Samples

Input #1

4
4
1 0
0 1
-1 0
0 -1
4
1 0
1 0
1 0
1 0
6
1 -9
-7 -2
0 -10
10 1
2 -8
1 -8
10
19271058 -140039457
-3441379 -134156148
227767903 -70474702
-837477 -74699717
153584327 80404336
115741008 -22141349
-190028077 -21940357
76149725 17176032
159846847 -296045185
-53332199 142922717

Output #1

0
4
296
178262497119089558

In the first case, if $s={}$, the score is $0$, which is the maximum.
In the second case, if $s={1,2,3,4}$, the score is $4$, which is the maximum.

API Response (JSON)

{
  "problem": {
    "name": "Squared Norm Maximization",
    "description": {
      "content": "You are given $N$ pairs of integers $(A_1,B_1),(A_2,B_2),\\cdots,(A_N,B_N)$. For a subset $s$ of ${1,2,\\cdots,N}$, we define its **score** as follows: *   $(\\sum_{i \\in s} A_i)^2 + (\\sum_{i \\in s} B_i",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc076_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given $N$ pairs of integers $(A_1,B_1),(A_2,B_2),\\cdots,(A_N,B_N)$.\nFor a subset $s$ of ${1,2,\\cdots,N}$, we define its **score** as follows:\n\n*   $(\\sum_{i \\in s} A_i)^2 + (\\sum_{i \\in s} B_i...",
      "is_translate": false,
      "language": "English"
    }
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}

Full JSON Raw Segments