Score : $600$ points

问题描述

我们有 $N$ 颗宝石。第 $i$ 颗宝石的颜色和美丽度分别为 $D_i$ 和 $V_i$。

这里的每颗宝石颜色均为 $1, 2, \ldots, C$ 中的一种，并且每种颜色至少有一颗宝石。

在 $N$ 颗宝石中，我们将选择 $C$ 颗颜色各不相同的宝石来制作一条项链。（顺序无关紧要。）项链的美观度将为所选宝石的按位异或值。

在所有可能的项链制作方式中，找出具有第 $K$ 大美观度的项链的美观度。 （如果有多种方式具有相同的美观度，我们将全部计算在内。）

什么是按位异或？

整数 $A$ 和 $B$ 的按位异或（bitwise XOR），记作 $A \oplus B$，定义如下：

• 当 $A \oplus B$ 用二进制表示时，在 $2^k$ 位（$k \geq 0$）上的数字是 $1$，如果 $A$ 或 $B$ 中恰好有一个在 $2^k$ 位上为 $1$，否则为 $0$。

例如，$3 \oplus 5 = 6$。（以二进制表示：$011 \oplus 101 = 110$。）

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$N$ $C$ $K$

$D_1$ $V_1$

$\vdots$

$D_N$ $V_N$

Output

Print the answer.

Sample Input 1

4 2 3
2 4
2 6
1 2
1 3

Sample 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 $5$.

Sample Input 2

3 1 2
1 0
1 0
1 0

Sample Output 2

0

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

Sample 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

Sample Output 3

766842905529259824

update @ 2024/3/10 10:46:56