Flip Machines

AtCoder

IDabc313_f

Time2000ms

Memory256MB

Difficulty—

There are $N$ cards numbered $1$ through $N$. Each face of a card has an integer written on it; card $i$ has $A_i$ on its front and $B_i$ on its back. Initially, all cards are face up. There are $M$ machines numbered $1$ through $M$. Machine $j$ has two (not necessarily distinct) integers $X_j$ and $Y_j$ between $1$ and $N$. If you power up machine $j$, it flips card $X_j$ with the probability of $\frac{1}{2}$, and flips card $Y_j$ with the remaining probability of $\frac{1}{2}$. This probability is independent for each power-up. Snuke will perform the following procedure. 1. Choose a set $S$ consisting of integers from $1$ through $M$. 2. For each element in $S$ in ascending order, power up the machine with that number. Among Snuke's possible choices of $S$, find the maximum expected value of the sum of the integers written on the face-up sides of the cards after the procedure. ## Constraints * $1\leq N \leq 40$ * $1\leq M \leq 10^5$ * $1\leq A_i,B_i \leq 10^4$ * $1\leq X_j,Y_j \leq N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $A_1$ $B_1$ $\vdots$ $A_N$ $B_N$ $X_1$ $Y_1$ $\vdots$ $X_M$ $Y_M$ [samples]

Samples

Input #1

Output #1

19.500000

If $S$ is chosen to be an empty set, no machine is powered up, so the expected sum of the integers written on the face-up sides of the cards after the procedure is $3+10+5=18$.
If $S$ is chosen to be $\lbrace 1 \rbrace$, machine $1$ is powered up.

*   If card $X_1 = 1$ is flipped, the expected sum of the integers written on the face-up sides of the cards after the procedure is $10+10+5=25$.
*   If card $Y_1 = 2$ is flipped, the expected sum of the integers written on the face-up sides of the cards after the procedure is $3+6+5=14$.

Thus, the expected value is $\frac{25+14}{2} = 19.5$.
Therefore, the maximum expected value of the sum of the integers written on the face-up sides of the cards after the procedure is $19.5$.

Input #2

Output #2

100.000000

Different machines may have the same $(X_j,Y_j)$.

Input #3

Output #3

45945.000000

API Response (JSON)

{
  "problem": {
    "name": "Flip Machines",
    "description": {
      "content": "There are $N$ cards numbered $1$ through $N$. Each face of a card has an integer written on it; card $i$ has $A_i$ on its front and $B_i$ on its back. Initially, all cards are face up. There are $M$ m",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc313_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ cards numbered $1$ through $N$. Each face of a card has an integer written on it; card $i$ has $A_i$ on its front and $B_i$ on its back. Initially, all cards are face up.\nThere are $M$ m...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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