Convex Dombination

AtCoder

ID1202Contest_i

Time2000ms

Memory256MB

Difficulty—

You are given $N$ distinct integer grid points $(X_i, Y_i) \ (i = 1, 2, \dots, N)$ in the plane, each of which has an integer score $P_i$. You can choose an arbitrary number of grid points among them under the following condition. Find the maximum value of the sum of the scores of chosen grid points. **Condition:** A grid point **dominated** by a **convex combination** of the chosen grid points must be chosen. That is, for each $k = 1, 2, \dots, N$, if there exists a tuple $(\lambda_1, \lambda_2, \dots, \lambda_N)$ of $N$ **nonnegative** reals with the following conditions, then $(X_k, Y_k)$ is chosen: * $\lambda_i > 0 \implies (X_i, Y_i)$ is chosen. * $\sum_{i=1}^N \lambda_i = 1.$ * $\sum_{i=1}^N \lambda_i X_i \geq X_k.$ * $\sum_{i=1}^N \lambda_i Y_i \geq Y_k.$ ## Constraints * $1 \leq N \leq 200$ * $1 \leq X_i \leq 10^9 \ \ (1 \leq i \leq N)$ * $1 \leq Y_i \leq 10^9 \ \ (1 \leq i \leq N)$ * $(X_i, Y_i) \neq (X_j, Y_j) \ \ (i \neq j)$ * $|P_i| \leq 10^7 \ \ (1 \leq i \leq N)$ ## Input The input is given from Standard Input in the following format: $N$ $X_1 \ Y_1 \ P_1$ $X_2 \ Y_2 \ P_2$ $\vdots$ $X_N \ Y_N \ P_N$ [samples]

Samples

Input #1

Output #1

3

It is optimal to choose only $(X_2, Y_2) = (4, 1)$. If you choose $(X_1, Y_1) = (1, 4)$ in addition, then for example $\lambda_1 = 0.4$ and $\lambda_2 = 0.6$ satisfy
\\\[ \\lambda_1 + \\lambda_2 = 0.4 + 0.6 = 1\\\\\[1mm\] \\lambda_1 X_1 + \\lambda_2 X_2 = 0.4 \\times 1 + 0.6 \\times 4 = 2.8 \\geq 2 = X_3\\\\\[1mm\] \\lambda_1 Y_1 + \\lambda_2 Y_2 = 0.4 \\times 4 + 0.6 \\times 1 = 2.2 \\geq 2 = Y_3, \\\]
and hence $(X_3, Y_3) = (2, 2)$ must be chosen in addition, which decreases the sum of scores as a result.

Input #2

Output #2

4

It is optimal to choose all the grid points.

Input #3

Output #3

0

It is sometimes optimal to choose no grid point.

API Response (JSON)

{
  "problem": {
    "name": "Convex Dombination",
    "description": {
      "content": "You are given $N$ distinct integer grid points $(X_i, Y_i) \\ (i = 1, 2, \\dots, N)$ in the plane, each of which has an integer score $P_i$. You can choose an arbitrary number of grid points among them ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "1202Contest_i"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given $N$ distinct integer grid points $(X_i, Y_i) \\ (i = 1, 2, \\dots, N)$ in the plane, each of which has an integer score $P_i$. You can choose an arbitrary number of grid points among them ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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