XOR Matching

AtCoder

IDabc126_f

Time2000ms

Memory256MB

Difficulty—

Construct a sequence $a$ = {$a_1,\ a_2,\ ...,\ a_{2^{M + 1}}$} of length $2^{M + 1}$ that satisfies the following conditions, if such a sequence exists. * Each integer between $0$ and $2^M - 1$ (inclusive) occurs twice in $a$. * For any $i$ and $j$ $(i < j)$ such that $a_i = a_j$, the formula $a_i \ xor \ a_{i + 1} \ xor \ ... \ xor \ a_j = K$ holds. What is xor (bitwise exclusive or)? The xor of integers $c_1, c_2, ..., c_n$ is defined as follows: * When $c_1 \ xor \ c_2 \ xor \ ... \ xor \ c_n$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if the number of integers among $c_1, c_2, ...c_m$ whose binary representations have $1$ in the $2^k$'s place is odd, and $0$ if that count is even. For example, $3 \ xor \ 5 = 6$. (If we write it in base two: `011` $xor$ `101` $=$ `110`.) ## Constraints * All values in input are integers. * $0 \leq M \leq 17$ * $0 \leq K \leq 10^9$ ## Input Input is given from Standard Input in the following format: $M$ $K$ [samples]

Samples

Input #1

1 0

Output #1

0 0 1 1

For this case, there are multiple sequences that satisfy the condition.
For example, when $a$ = {$0, 0, 1, 1$}, there are two pairs $(i,\ j)\ (i < j)$ such that $a_i = a_j$: $(1, 2)$ and $(3, 4)$. Since $a_1 \ xor \ a_2 = 0$ and $a_3 \ xor \ a_4 = 0$, this sequence $a$ satisfies the condition.

Input #2

1 1

Output #2

\-1

No sequence satisfies the condition.

Input #3

5 58

Output #3

\-1

No sequence satisfies the condition.

API Response (JSON)

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    "name": "XOR Matching",
    "description": {
      "content": "Construct a sequence $a$ = {$a_1,\\ a_2,\\ ...,\\ a_{2^{M + 1}}$} of length $2^{M + 1}$ that satisfies the following conditions, if such a sequence exists. *   Each integer between $0$ and $2^M - 1$ (in",
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc126_f"
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  "statements": [
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      "statement_type": "Markdown",
      "content": "Construct a sequence $a$ = {$a_1,\\ a_2,\\ ...,\\ a_{2^{M + 1}}$} of length $2^{M + 1}$ that satisfies the following conditions, if such a sequence exists.\n\n*   Each integer between $0$ and $2^M - 1$ (in...",
      "is_translate": false,
      "language": "English"
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}

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