Score : $500$ points

问题描述

商店 supaisu-ya 销售 $2^N-1$ 种香料：香料 $1$、香料 $2$、……、香料 $2^N-1$。每种香料库存一包。对于每个 $i = 1, 2, \ldots, 2^N-1$，香料 $i$ 的价格为 $c_i$ 日元。Takahashi 可以购买这些香料中的任意一种。

他计划在回家后选择并混合所购买的一种或多种香料来制作咖喱。
如果混合了 $k$ 种香料，即香料 $A_1$、香料 $A_2$、……、香料 $A_k$，那么得到的咖喱辣度为 $A_1 \oplus A_2 \oplus \cdots \oplus A_k$，其中 $\oplus$ 表示按位异或运算。

根据回家后的感受，Takahashi 想决定咖喱的辣度。现在，他将购买一组香料，使得他能制作出从 $1$ 到 $2^N-1$ 的任何辣度的咖喱。请输出 Takahashi 最少需要支付的金额。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

The shop supaisu-ya sells $2^N-1$ kinds of spices: Spice $1$, Spice $2$, $\ldots$, Spice $2^N-1$. One pack of each spice is in stock. For each $i = 1, 2, \ldots, 2^N-1$, Spice $i$ costs $c_i$ yen. Takahashi can buy any of these spices.

He plans to make curry after getting home by choosing one or more of the bought spices and mixing them.
If $k$ spices, Spice $A_1$, Spice $A_2$, $\ldots$, Spice $A_k$, are mixed, the hotness of the resulting curry is $A_1 \oplus A_2 \oplus \cdots \oplus A_k$, where $\oplus$ denotes the bitwise XOR.

Takahashi wants to decide the hotness of the curry based on his feeling after getting home. For now, he will buy a set of spices that allows him to make curry of any hotness from $1$ through $2^N-1$. Print the minimum possible amount of money Takahashi pays.

Constraints

• $2 \leq N \leq 16$
• $1 \leq c_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$c_1$ $c_2$ $\ldots$ $c_{2^N-1}$

Output

Print the minimum possible amount of money Takahashi pays.

Sample Input 1

2
4 5 3

Sample Output 1

7

If Takahashi buys Spice $1$ and $3$, he can make curry of any hotness from $1$ through $3$, as follows.

• To make curry of hotness $1$, use just Spice $1$.
• To make curry of hotness $2$, mix Spice $1$ and Spice $3$.
• To make curry of hotness $3$, use just Spice $3$.

Here, Takahashi pays $c_1 + c_3 = 4 + 3 = 7$ yen, which is the minimum possible amount he pays.

Sample Input 2

4
9 7 9 7 10 4 3 9 4 8 10 5 6 3 8

Sample Output 2

15

update @ 2024/3/10 10:14:23