Xor Sum 5

AtCoder

IDarc156_d

Time2000ms

Memory256MB

Difficulty—

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: $N$ $K$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

2 2
10 30

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 $20 \oplus 40 \oplus 40 \oplus 60=40$.

Input #2

4 10
0 0 0 0

Output #2

Input #3

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

Output #3

236500026047

API Response (JSON)

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    "name": "Xor Sum 5",
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      "content": "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 $N^K$ seque",
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    "platform": "AtCoder",
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      "statement_type": "Markdown",
      "content": "You are given a sequence of $N$ non-negative integers $A=(A_1,A_2,\\dots,A_N)$ and a positive integer $K$.\nFind the bitwise $\\mathrm{XOR}$ of $\\displaystyle \\sum_{i=1}^{K} A_{X_i}$ over all $N^K$ seque...",
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