Codeforces Round 812 (Div. 2) |
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Finished |
A long time ago, Spyofgame invented the famous array $$$a$$$ ($$$1$$$-indexed) of length $$$n$$$ that contains information about the world and life. After that, he decided to convert it into the matrix $$$b$$$ ($$$0$$$-indexed) of size $$$(n + 1) \times (n + 1)$$$ which contains information about the world, life and beyond.
Spyofgame converted $$$a$$$ into $$$b$$$ with the following rules.
Here $$$\oplus$$$ denotes the bitwise XOR operation.
Today, archaeologists have discovered the famous matrix $$$b$$$. However, many elements of the matrix has been lost. They only know the values of $$$b_{i,n}$$$ for $$$1 \leq i \leq n$$$ (note that these are some elements of the last column, not the last row).
The archaeologists want to know what a possible array of $$$a$$$ is. Can you help them reconstruct any array that could be $$$a$$$?
The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 5 \cdot 10^5$$$).
The second line contains $$$n$$$ integers $$$b_{1,n}, b_{2,n}, \ldots, b_{n,n}$$$ ($$$0 \leq b_{i,n} < 2^{30}$$$).
If some array $$$a$$$ is consistent with the information, print a line containing $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$. If there are multiple solutions, output any.
If such an array does not exist, output $$$-1$$$ instead.
3 0 2 1
1 2 3
1 199633
199633
10 346484077 532933626 858787727 369947090 299437981 416813461 865836801 141384800 157794568 691345607
725081944 922153789 481174947 427448285 516570428 509717938 855104873 280317429 281091129 1050390365
If we let $$$a = [1,2,3]$$$, then $$$b$$$ will be:
$$$\bf{0}$$$ | $$$\bf{1}$$$ | $$$\bf{2}$$$ | $$$\bf{3}$$$ |
$$$\bf{0}$$$ | $$$1$$$ | $$$3$$$ | $$$0$$$ |
$$$\bf{0}$$$ | $$$1$$$ | $$$2$$$ | $$$2$$$ |
$$$\bf{0}$$$ | $$$1$$$ | $$$3$$$ | $$$1$$$ |
The values of $$$b_{1,n}, b_{2,n}, \ldots, b_{n,n}$$$ generated are $$$[0,2,1]$$$ which is consistent with what the archaeologists have discovered.
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