[GDCPC 2023] Swapping Operation

Luogu
IDLGP9696
Time1000ms
Memory1024MB
DifficultyP6
2023广东O2优化前缀和省赛/邀请赛
Given a non-negative integer sequence $A = a_1, a_2, \dots, a_n$ of length $n$, define $$ F(A)=\max\limits_{1\leq k<n} ((a_1 \,\&\, a_2 \,\&\, \cdots \,\&\, a_k)+(a_{k+1} \,\&\, a_{k+2} \,\&\, \cdots \,\&\, a_n)) $$ where $\&$ is the bitwise-and operator. You can perform the swapping operation at most once: choose two indices $i$ and $j$ such that $1\leq i < j\leq n$ and then swap the values of $a_i$ and $a_j$. Calculate the maximum possible value of $F(A)$ after performing at most one swapping operation. ## Input There are multiple test cases. The first line of the input contains an integer $T$ indicating the number of test cases. For each test case: The first line contains an integer $n$ ($2\leq n\leq 10^5$) indicating the length of sequence $A$. The second line contains $n$ integers $a_1, a_2, \cdots, a_n$ ($0\leq a_i\leq 10^9$) indicating the given sequence $A$. It's guaranteed that the sum of $n$ of all test cases will not exceed $10^5$. ## Output For each test case output one line containing one integer indicating the maximum possible value of $F(A)$ after performing at most one swapping operation. [samples] ## Note For the first sample test case, we can swap $a_4$ and $a_6$ so the sequence becomes $\{6, 5, 4, 6, 5, 3\}$. We can then choose $k = 5$ so that $F(A) = (6 \,\&\, 5 \,\&\, 4 \,\&\, 6 \,\&\, 5) + (3) = 7$. For the second sample test case, we can swap $a_2$ and $a_4$ so the sequence becomes $\{1, 1, 1, 2, 2, 2\}$. We can then choose $k = 3$ so that $F(A) = (1 \,\&\, 1 \,\&\, 1) + (2 \,\&\, 2 \,\&\, 2) = 3$. For the third sample test case we do not perform the swapping operation. We can then choose $k = 2$ so that $F(A) = (1 \,\&\, 1) + (2 \,\&\, 2 \,\&\, 2) = 3$.
Samples
Input #1
3
6
6 5 4 3 5 6
6
1 2 1 1 2 2
5
1 1 2 2 2
Output #1
7
3
3
API Response (JSON)
{
  "problem": {
    "name": "[GDCPC 2023] Swapping Operation",
    "description": {
      "content": "Given a non-negative integer sequence $A = a_1, a_2, \\dots, a_n$ of length $n$, define $$ F(A)=\\max\\limits_{1\\leq k<n} ((a_1 \\,\\&\\, a_2 \\,\\&\\, \\cdots \\,\\&\\, a_k)+(a_{k+1} \\,\\&\\, a_{k+2} \\,\\&\\, \\cdots",
      "description_type": "Markdown"
    },
    "platform": "Luogu",
    "limit": {
      "time_limit": 1000,
      "memory_limit": 1048576
    },
    "difficulty": {
      "LuoguStyle": "P6"
    },
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "LGP9696"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given a non-negative integer sequence $A = a_1, a_2, \\dots, a_n$ of length $n$, define\n\n$$\nF(A)=\\max\\limits_{1\\leq k<n} ((a_1 \\,\\&\\, a_2 \\,\\&\\, \\cdots \\,\\&\\, a_k)+(a_{k+1} \\,\\&\\, a_{k+2} \\,\\&\\, \\cdots...",
      "is_translate": false,
      "language": "English"
    }
  ]
}
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