Manhattan Max Matching

AtCoder

IDagc034_d

Time5000ms

Memory256MB

Difficulty—

Snuke is playing with red and blue balls, placing them on a two-dimensional plane. First, he performed $N$ operations to place red balls. In the $i$\-th of these operations, he placed $RC_i$ red balls at coordinates $(RX_i,RY_i)$. Then, he performed another $N$ operations to place blue balls. In the $i$\-th of these operations, he placed $BC_i$ blue balls at coordinates $(BX_i,BY_i)$. The total number of red balls placed and the total number of blue balls placed are equal, that is, $\sum_{i=1}^{N} RC_i = \sum_{i=1}^{N} BC_i$. Let this value be $S$. Snuke will now form $S$ pairs of red and blue balls so that every ball belongs to exactly one pair. Let us define the _score_ of a pair of a red ball at coordinates $(rx, ry)$ and a blue ball at coordinates $(bx, by)$ as $|rx-bx| + |ry-by|$. Snuke wants to maximize the sum of the scores of the pairs. Help him by finding the maximum possible sum of the scores of the pairs. ## Constraints * $1 \leq N \leq 1000$ * $0 \leq RX_i,RY_i,BX_i,BY_i \leq 10^9$ * $1 \leq RC_i,BC_i \leq 10$ * $\sum_{i=1}^{N} RC_i = \sum_{i=1}^{N} BC_i$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $RX_1$ $RY_1$ $RC_1$ $RX_2$ $RY_2$ $RC_2$ $\vdots$ $RX_N$ $RY_N$ $RC_N$ $BX_1$ $BY_1$ $BC_1$ $BX_2$ $BY_2$ $BC_2$ $\vdots$ $BX_N$ $BY_N$ $BC_N$ [samples]

Samples

Input #1

Output #1

8

If we pair the red ball at coordinates $(0,0)$ and the blue ball at coordinates $(2,2)$, the score of this pair is $|0-2| + |0-2|=4$. Then, if we pair the red ball at coordinates $(3,2)$ and the blue ball at coordinates $(5,0)$, the score of this pair is $|3-5| + |2-0|=4$. Making these two pairs results in the total score of $8$, which is the maximum result.

Input #2

Output #2

16

Snuke may have performed multiple operations at the same coordinates.

Input #3

10
582463373 690528069 8
621230322 318051944 4
356524296 974059503 6
372751381 111542460 9
392867214 581476334 6
606955458 513028121 5
882201596 791660614 9
250465517 91918758 3
618624774 406956634 6
426294747 736401096 5
974896051 888765942 5
726682138 336960821 3
715144179 82444709 6
599055841 501257806 6
390484433 962747856 4
912334580 219343832 8
570458984 648862300 6
638017635 572157978 10
435958984 585073520 7
445612658 234265014 6

Output #3

45152033546

API Response (JSON)

{
  "problem": {
    "name": "Manhattan Max Matching",
    "description": {
      "content": "Snuke is playing with red and blue balls, placing them on a two-dimensional plane. First, he performed $N$ operations to place red balls. In the $i$\\-th of these operations, he placed $RC_i$ red balls",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc034_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke is playing with red and blue balls, placing them on a two-dimensional plane.\nFirst, he performed $N$ operations to place red balls. In the $i$\\-th of these operations, he placed $RC_i$ red balls...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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