Xor Cards

AtCoder

IDabc249_g

Time2000ms

Memory256MB

Difficulty—

There are $N$ cards numbered $1, \dots, N$. Card $i \, (1 \leq i \leq N)$ has an integer $A_i$ written on the front and an integer $B_i$ written on the back. Consider choosing one or more cards so that the exclusive logical sum of the integers written on the front of the chosen cards is at most $K$. Find the maximum possible exclusive logical sum of the integers written on the back of the chosen cards. What is the exclusive logical sum? The exclusive logical sum $a \oplus b$ of two integers $a$ and $b$ is defined as follows. * The $2^k$'s place ($k \geq 0$) in the binary notation of $a \oplus b$ is $1$ if exactly one of the $2^k$'s places in the binary notation of $a$ and $b$ is $1$; otherwise, it is $0$. For example, $3 \oplus 5 = 6$ (In binary notation: $011 \oplus 101 = 110$). In general, the exclusive logical sum of $k$ integers $p_1, \dots, p_k$ is defined as $(\cdots ((p_1 \oplus p_2) \oplus p_3) \oplus \cdots \oplus p_k)$. We can prove that it is independent of the order of $p_1, \dots, p_k$. ## Constraints * $1 \leq N \leq 1000$ * $0 \leq K \lt 2^{30}$ * $0 \leq A_i, B_i \lt 2^{30} \, (1 \leq i \leq N)$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ $A_1$ $B_1$ $\vdots$ $A_N$ $B_N$ [samples]

Samples

Input #1

Output #1

3

By choosing Cards $1$ and $2$, the exclusive logical sum of the integers written on the front of them is $2$, and that on the back of them is $3$, which is the maximum.

Input #2

1 2
3 4

Output #2

\-1

It is impossible to choose cards so that the condition is satisfied.

Input #3

10 326872757
487274679 568989827
267359104 968688210
669234369 189421955
1044049637 253386228
202278801 233212012
436646715 769734012
478066962 376960084
491389944 1033137442
214977048 1051768288
803550682 1053605300

Output #3

1064164329

API Response (JSON)

{
  "problem": {
    "name": "Xor Cards",
    "description": {
      "content": "There are $N$ cards numbered $1, \\dots, N$. Card $i \\, (1 \\leq i \\leq N)$ has an integer $A_i$ written on the front and an integer $B_i$ written on the back. Consider choosing one or more cards so tha",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc249_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ cards numbered $1, \\dots, N$. Card $i \\, (1 \\leq i \\leq N)$ has an integer $A_i$ written on the front and an integer $B_i$ written on the back.\nConsider choosing one or more cards so tha...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments