OR Sum

AtCoder

IDabc291_g

Time8000ms

Memory256MB

Difficulty—

There are length-$N$ sequences $A=(A_0,A_1,\ldots,A_{N-1})$ and $B=(B_0,B_1,\ldots,B_{N-1})$. Takahashi may perform the following operation on $A$ any number of times (possibly zero): * apply a left cyclic shift to the sequence $A$. In other words, replace $A$ with $A'$ defined by $A'_i=A_{(i+1)\% N}$, where $x\% N$ denotes the remainder when $x$ is divided by $N$. Takahashi's objective is to maximize $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)$, where $x|y$ denotes the bitwise logical sum (bitwise OR) of $x$ and $y$. Find the maximum possible $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)$. What is the bitwise logical sum (bitwise OR)? The **logical sum** (or the OR operation) is an operation on two one-bit integers ($0$ or $1$) defined by the table below. The **bitwise logical sum (bitwise OR)** is an operation of applying the logical sum bitwise. $x$ $y$ $x|y$ $0$ $0$ $0$ $0$ $1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ The logical sum yields $1$ if at least one of the bits $x$ and $y$ is $1$. Conversely, it yields $0$ only if both of them are $0$. ##### Example 0110 | 0101 = 0111 ## Constraints * $2 \leq N \leq 5\times 10^5$ * $0\leq A_i,B_i \leq 31$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_0$ $A_1$ $\ldots$ $A_{N-1}$ $B_0$ $B_1$ $\ldots$ $B_{N-1}$ [samples]

Samples

Input #1

3
0 1 3
0 2 3

Output #1

8

If Takahashi does not perform the operation, $A$ remains $(0,1,3)$, and we have $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)=(0|0)+(1|2)+(3|3)=0+3+3=6$;  
if he performs the operation once, making $A=(1,3,0)$, we have $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)=(1|0)+(3|2)+(0|3)=1+3+3=7$; and  
if he performs the operation twice, making $A=(3,0,1)$, we have $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)=(3|0)+(0|2)+(1|3)=3+2+3=8$.  
If he performs the operation three or more times, $A$ becomes one of the sequences above, so the maximum possible $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)$ is $8$, which should be printed.

Input #2

5
1 6 1 4 3
0 6 4 0 1

Output #2

23

The value is maximized if he performs the operation three times, making $A=(4,3,1,6,1)$,  
where $\displaystyle\sum_{i=0}^{N-1} (A_i|B_i)=(4|0)+(3|6)+(1|4)+(6|0)+(1|1)=4+7+5+6+1=23$.

API Response (JSON)

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  "problem": {
    "name": "OR Sum",
    "description": {
      "content": "There are length-$N$ sequences $A=(A_0,A_1,\\ldots,A_{N-1})$ and $B=(B_0,B_1,\\ldots,B_{N-1})$.   Takahashi may perform the following operation on $A$ any number of times (possibly zero): *   apply a l",
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    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
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    "sync_url": null,
    "sign": "abc291_g"
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    {
      "statement_type": "Markdown",
      "content": "There are length-$N$ sequences $A=(A_0,A_1,\\ldots,A_{N-1})$ and $B=(B_0,B_1,\\ldots,B_{N-1})$.  \nTakahashi may perform the following operation on $A$ any number of times (possibly zero):\n\n*   apply a l...",
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