201563: [AtCoder]ARC156 D - Xor Sum 5

Memory Limit:1024 MB Time Limit:2 S
Judge Style:Text Compare Creator:
Submit:0 Solved:0

Description

Score : $700$ points

Problem Statement

You are given a sequence of $N$ non-negative integers $A=(A_1,A_2,\dots,A_N)$ and a positive integer $K$.

Find the bitwise $\mathrm{XOR}$ of $\displaystyle \sum_{i=1}^{K} A_{X_i}$ over all $N^K$ sequences of $K$ positive integer sequences $X=(X_1,X_2,\dots,X_K)$ such that $1 \leq X_i \leq N\ (1\leq i \leq K)$.

What is bitwise $\mathrm{XOR}$?

The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows:

  • When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of the digits in that place of $A$ and $B$ is $1$, and $0$ otherwise.
For example, we have $3 \oplus 5 = 6$ (in base two: $011 \oplus 101 = 110$).
Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$.

Constraints

  • $1 \leq N \leq 1000$
  • $1 \leq K \leq 10^{12}$
  • $0 \leq A_i \leq 1000$
  • All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $K$
$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answer.


Sample Input 1

2 2
10 30

Sample Output 1

40

There are four sequences to consider: $(X_1,X_2)=(1,1),(1,2),(2,1),(2,2)$, for which $A_{X_1}+A_{X_2}$ is $20,40,40,60$, respectively. Thus, the answer is $20 \oplus 40 \oplus 40 \oplus 60=40$.


Sample Input 2

4 10
0 0 0 0

Sample Output 2

0

Sample Input 3

11 998244353
314 159 265 358 979 323 846 264 338 327 950

Sample Output 3

236500026047

Input

题意翻译

给定 $n$ 个数的数列 $a$ 和一个整数 $k$。 对于所有长度为 $k$,值域为 $[1,n]$ 的数列 $p$,求出 $\sum _{i=1}^{k} a_{p_i}$ 的异或和。

加入题单

上一题 下一题 算法标签: