Increment or XOR

AtCoder

IDagc061_e

Time2000ms

Memory256MB

Difficulty—

There is a non-negative integer $X$ initially equal to $S$. One can perform the following operations, in any order, any number of times: * Add $1$ to $X$. This has a cost of $A$. * Choose $i$ between $1$ and $N$, and replace $X$ with $X \oplus Y_i$. This has a cost of $C_i$. Here, $\oplus$ denotes bitwise $\mathrm{XOR}$. Find the smallest total cost to make $X$ equal to a given non-negative integer $T$, or report that this is impossible. What is bitwise $\mathrm{XOR}$?The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows: * When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of the digits in that place of $A$ and $B$ is $1$, and $0$ otherwise. For example, we have $3 \oplus 5 = 6$ (in base two: $011 \oplus 101 = 110$). Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$. ## Constraints * $1 \leq N \leq 8$ * $0 \leq S, T < 2^{40}$ * $0 \leq A \leq 10^5$ * $0 \leq Y_i < 2^{40}$ ($1 \leq i \leq N$) * $0 \leq C_i \leq 10^{16}$ ($1 \leq i \leq N$) * All values in the input are integers. ## Input Input is given from Standard Input in the following format: $N$ $S$ $T$ $A$ $Y_1$ $C_1$ $\vdots$ $Y_N$ $C_N$ [samples]

Samples

Input #1

1 15 0 1
8 2

Output #1

5

You can make $X$ equal to $T$ in the following manner:

*   Choose $i=1$ and replace $X$ with $X \oplus 8$ to get $X=7$. This has a cost of $2$.
*   Add $1$ to $X$ to get $X=8$. This has a cost of $1$.
*   Choose $i=1$ and replace $X$ with $X \oplus 8$ to get $X=0$. This has a cost of $2$.

Input #2

3 21 10 100
30 1
12 1
13 1

Output #2

Input #3

1 2 0 1
1 1

Output #3

\-1

Input #4

8 352217 670575 84912
239445 2866
537211 16
21812 6904
50574 8842
380870 5047
475646 8924
188204 2273
429397 4854

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Increment or XOR",
    "description": {
      "content": "There is a non-negative integer $X$ initially equal to $S$. One can perform the following operations, in any order, any number of times: *   Add $1$ to $X$. This has a cost of $A$. *   Choose $i$ bet",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc061_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a non-negative integer $X$ initially equal to $S$. One can perform the following operations, in any order, any number of times:\n\n*   Add $1$ to $X$. This has a cost of $A$.\n*   Choose $i$ bet...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments