XOR Cross Over

AtCoder

IDagc071_a

Time2000ms

Memory256MB

Difficulty—

_(In this problem statement, all sequences consist of non-negative integers.)_ There is a blackboard on which sequences are written. Initially, the blackboard only has one length-$N$ sequence $A=(A_1,A_2,\dots,A_N)$. We repeat the following operation until all sequences on the blackboard are of length $1$: * Among the sequences on the blackboard, choose one that has length at least $2$, and remove it from the blackboard. Let the chosen sequence be $B=(B_1,B_2,\dots,B_n)$. Then, choose an integer $k$ satisfying $1 \leq k < n$, and define $X = B_1 \oplus B_2 \oplus \dots \oplus B_k$ and $Y=B_{k+1} \oplus B_{k+2} \oplus \dots \oplus B_n$. Finally, write two sequences $(B_1 \oplus Y, B_2 \oplus Y,\dots,B_k \oplus Y)$ and $(B_{k+1} \oplus X,\ B_{k+2} \oplus X,\dots,B_n \oplus X)$ on the blackboard. Here, $\oplus$ denotes the bitwise $\mathrm{XOR}$ operation. After this series of operations, let $(C_1),(C_2),\dots,(C_N)$ be the $N$ sequences of length $1$ on the blackboard. Find the minimum possible value of $\sum_{i=1}^{N} C_i$. Notes on the bitwise $\mathrm{XOR}$ operationFor two non-negative integers $A, B$, the bitwise $\mathrm{XOR}$ operation $A \oplus B$ is defined as follows. * When $A \oplus B$ is written in binary, for each $2^k$ place ($k \geq 0$), the bit is 1 if and only if exactly one of the bits in the $2^k$ place of $A$ and $B$ (in binary) is 1, and 0 otherwise. For example, $3 \oplus 5 = 6$ (in binary notation: $011 \oplus 101 = 110$). In general, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$, and it can be proved that this value does not depend on the order of $p_1, p_2, \dots, p_k$. ## Constraints * $1 \leq N \leq 500$ * $0 \leq A_i < 2^{40}$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

4
1 3 4 5

Output #1

6

Initially, the blackboard has $(1,3,4,5)$ written on it. From here, we perform operations in the following steps.

1.  Remove $(1,3,4,5)$ from the blackboard and set $k=2$. Then, $X=1 \oplus 3=2$ and $Y=4 \oplus 5=1$, and two new sequences $(1 \oplus 1, 3 \oplus 1)=(0,2)$ and $(4 \oplus 2, 5 \oplus 2)=(6,7)$ are written on the blackboard.
2.  Remove $(0,2)$ from the blackboard and set $k=1$. Two new sequences $(2),(2)$ are written on the blackboard.
3.  Remove $(6,7)$ from the blackboard and set $k=1$. Two new sequences $(1),(1)$ are written on the blackboard.

After these steps, the four sequences on the blackboard are $(2),(2),(1),(1)$, each of length $1$. Their sum is $2+2+1+1=6$.

Input #2

7
1 5 14 23 20 18 20

Output #2

Input #3

20
246260893965 450834729933 1091503137264 661761201979 238822689279 375606126051 183045127603 1004516515418 976478741401 957665143474 451659136716 828528157302 204109014940 184065081345 122138832666 130646707415 144391522538 87966805947 381909891703 343575641318

Output #3

3597854777632

API Response (JSON)

{
  "problem": {
    "name": "XOR Cross Over",
    "description": {
      "content": "_(In this problem statement, all sequences consist of non-negative integers.)_ There is a blackboard on which sequences are written. Initially, the blackboard only has one length-$N$ sequence $A=(A_1,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc071_a"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "_(In this problem statement, all sequences consist of non-negative integers.)_\nThere is a blackboard on which sequences are written. Initially, the blackboard only has one length-$N$ sequence $A=(A_1,...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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