1734. Decode XORed Permutation
Description
There is an integer array perm
that is a permutation of the first n
positive integers, where n
is always odd.
It was encoded into another integer array encoded
of length n - 1
, such that encoded[i] = perm[i] XOR perm[i + 1]
. For example, if perm = [1,3,2]
, then encoded = [2,1]
.
Given the encoded
array, return the original array perm
. It is guaranteed that the answer exists and is unique.
Example 1:
Input: encoded = [3,1] Output: [1,2,3] Explanation: If perm = [1,2,3], then encoded = [1 XOR 2,2 XOR 3] = [3,1]
Example 2:
Input: encoded = [6,5,4,6] Output: [2,4,1,5,3]
Constraints:
3 <= n < 105
n
is odd.encoded.length == n - 1
Solutions
Solution 1: Bitwise Operation
We notice that the array $perm$ is a permutation of the first $n$ positive integers, so the XOR of all elements in $perm$ is $1 \oplus 2 \oplus \cdots \oplus n$, denoted as $a$. And $encode[i]=perm[i] \oplus perm[i+1]$, if we denote the XOR of all elements $encode[0],encode[2],\cdots,encode[n-3]$ as $b$, then $perm[n-1]=a \oplus b$. Knowing the last element of $perm$, we can find all elements of $perm$ by traversing the array $encode$ in reverse order.
The time complexity is $O(n)$, where $n$ is the length of the array $perm$. Ignoring the space consumption of the answer, the space complexity is $O(1)$.
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