Ex - K-th beautiful Necklace

AtCoder

IDabc252_h

Time2000ms

Memory256MB

Difficulty—

We have $N$ gemstones. The color and beauty of the $i$\-th gemstone are $D_i$ and $V_i$, respectively. Here, the color of each gemstone is one of $1, 2, \ldots, C$, and there is at least one gemstone of each color. Out of the $N$ gemstones, we will choose $C$ with distinct colors and use them to make a necklace. (The order does not matter.) The beautifulness of the necklace will be the bitwise $\rm XOR$ of the chosen gemstones. Among all possible ways to make a necklace, find the beautifulness of the necklace made in the way with the $K$\-th greatest beautifulness. (If there are multiple ways with the same beautifulness, we count all of them.) What is bitwise $\rm XOR$?The bitwise $\rm XOR$ of 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 either $A$ or $B$, but not both, has $1$ in the $2^k$'s place, and $0$ otherwise. For example, $3 \oplus 5 = 6$. (In base two: $011 \oplus 101 = 110$.) ## Constraints * $1 \leq C \leq N \leq 70$ * $1 \leq D_i \leq C$ * $0 \leq V_i < 2^{60}$ * $1 \leq K \leq 10^{18}$ * There are at least $K$ ways to make a necklace. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $C$ $K$ $D_1$ $V_1$ $\vdots$ $D_N$ $V_N$ [samples]

Samples

Input #1

Output #1

5

There are four ways to make a necklace, as follows.

*   Choose the $1$\-st and $3$\-rd gemstones to make a necklace with the beautifulness of $4\ {\rm XOR}\ 2 =6$.
*   Choose the $1$\-st and $4$\-th gemstones to make a necklace with the beautifulness of $4\ {\rm XOR}\ 3 =7$.
*   Choose the $2$\-nd and $3$\-rd gemstones to make a necklace with the beautifulness of $6\ {\rm XOR}\ 2 =4$.
*   Choose the $2$\-nd and $4$\-th gemstones to make a necklace with the beautifulness of $6\ {\rm XOR}\ 3 =5$.

Thus, the necklace with the $3$\-rd greatest beautifulness has the beautifulness of $5$.

Input #2

Output #2

0

There are three ways to make a necklace, all of which result in the beautifulness of $0$.

Input #3

10 3 11
1 414213562373095048
1 732050807568877293
2 236067977499789696
2 449489742783178098
2 645751311064590590
2 828427124746190097
3 162277660168379331
3 316624790355399849
3 464101615137754587
3 605551275463989293

Output #3

766842905529259824

API Response (JSON)

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  "problem": {
    "name": "Ex - K-th beautiful Necklace",
    "description": {
      "content": "We have $N$ gemstones. The color and beauty of the $i$\\-th gemstone are $D_i$ and $V_i$, respectively.   Here, the color of each gemstone is one of $1, 2, \\ldots, C$, and there is at least one gemston",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have $N$ gemstones. The color and beauty of the $i$\\-th gemstone are $D_i$ and $V_i$, respectively.  \nHere, the color of each gemstone is one of $1, 2, \\ldots, C$, and there is at least one gemston...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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