Question

1829. Maximum XOR for Each Query

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Description

You are given a sorted array nums of n non-negative integers and an integer maximumBit. You want to perform the following query n times:

1. Find a non-negative integer k < 2maximumBit such that nums[0] XOR nums[1] XOR ... XOR nums[nums.length-1] XOR k is maximized. k is the answer to the ith query.
2. Remove the last element from the current array nums.

Return _an array_ answer_, where _answer[i]_ is the answer to the _ith_ query_.

 

Example 1:

Input: nums = [0,1,1,3], maximumBit = 2
Output: [0,3,2,3]
Explanation: The queries are answered as follows:
1st query: nums = [0,1,1,3], k = 0 since 0 XOR 1 XOR 1 XOR 3 XOR 0 = 3.
2nd query: nums = [0,1,1], k = 3 since 0 XOR 1 XOR 1 XOR 3 = 3.
3rd query: nums = [0,1], k = 2 since 0 XOR 1 XOR 2 = 3.
4th query: nums = [0], k = 3 since 0 XOR 3 = 3.

Example 2:

Input: nums = [2,3,4,7], maximumBit = 3
Output: [5,2,6,5]
Explanation: The queries are answered as follows:
1st query: nums = [2,3,4,7], k = 5 since 2 XOR 3 XOR 4 XOR 7 XOR 5 = 7.
2nd query: nums = [2,3,4], k = 2 since 2 XOR 3 XOR 4 XOR 2 = 7.
3rd query: nums = [2,3], k = 6 since 2 XOR 3 XOR 6 = 7.
4th query: nums = [2], k = 5 since 2 XOR 5 = 7.

Example 3:

Input: nums = [0,1,2,2,5,7], maximumBit = 3
Output: [4,3,6,4,6,7]

 

Constraints:

• nums.length == n
• 1 <= n <= 105
• 1 <= maximumBit <= 20
• 0 <= nums[i] < 2maximumBit
• nums​​​ is sorted in ascending order.

Solutions

Solution 1: Bitwise Operation + Enumeration

First, we preprocess the XOR sum $xs$ of the array nums, i.e., $xs=nums[0] \oplus nums[1] \oplus \cdots \oplus nums[n-1]$.

Next, we enumerate each element $x$ in the array nums from back to front. The current XOR sum is $xs$. We need to find a number $k$ such that the value of $xs \oplus k$ is as large as possible, and $k \lt 2^{maximumBit}$.

That is to say, we start from the $maximumBit - 1$ bit of $xs$ and enumerate to the lower bit. If a bit of $xs$ is $0$, then we set the corresponding bit of $k$ to $1$. Otherwise, we set the corresponding bit of $k$ to $0$. In this way, the final $k$ is the answer to each query. Then, we update $xs$ to $xs \oplus x$ and continue to enumerate the next element.

The time complexity is $O(n \times m)$, where $n$ and $m$ are the values of the array nums and maximumBit respectively. Ignoring the space consumption of the answer, the space complexity is $O(1)$.

class Solution:
  def getMaximumXor(self, nums: List[int], maximumBit: int) -> List[int]:
    ans = []
    xs = reduce(xor, nums)
    for x in nums[::-1]:
      k = 0
      for i in range(maximumBit - 1, -1, -1):
        if (xs >> i & 1) == 0:
          k |= 1 << i
      ans.append(k)
      xs ^= x
    return ans

class Solution {
  public int[] getMaximumXor(int[] nums, int maximumBit) {
    int n = nums.length;
    int xs = 0;
    for (int x : nums) {
      xs ^= x;
    }
    int[] ans = new int[n];
    for (int i = 0; i < n; ++i) {
      int x = nums[n - i - 1];
      int k = 0;
      for (int j = maximumBit - 1; j >= 0; --j) {
        if (((xs >> j) & 1) == 0) {
          k |= 1 << j;
        }
      }
      ans[i] = k;
      xs ^= x;
    }
    return ans;
  }
}

class Solution {
public:
  vector<int> getMaximumXor(vector<int>& nums, int maximumBit) {
    int xs = 0;
    for (int& x : nums) {
      xs ^= x;
    }
    int n = nums.size();
    vector<int> ans(n);
    for (int i = 0; i < n; ++i) {
      int x = nums[n - i - 1];
      int k = 0;
      for (int j = maximumBit - 1; ~j; --j) {
        if ((xs >> j & 1) == 0) {
          k |= 1 << j;
        }
      }
      ans[i] = k;
      xs ^= x;
    }
    return ans;
  }
};

func getMaximumXor(nums []int, maximumBit int) (ans []int) {
  xs := 0
  for _, x := range nums {
    xs ^= x
  }
  for i := range nums {
    x := nums[len(nums)-i-1]
    k := 0
    for j := maximumBit - 1; j >= 0; j-- {
      if xs>>j&1 == 0 {
        k |= 1 << j
      }
    }
    ans = append(ans, k)
    xs ^= x
  }
  return
}

function getMaximumXor(nums: number[], maximumBit: number): number[] {
  let xs = 0;
  for (const x of nums) {
    xs ^= x;
  }
  const n = nums.length;
  const ans = new Array(n);
  for (let i = 0; i < n; ++i) {
    const x = nums[n - i - 1];
    let k = 0;
    for (let j = maximumBit - 1; j >= 0; --j) {
      if (((xs >> j) & 1) == 0) {
        k |= 1 << j;
      }
    }
    ans[i] = k;
    xs ^= x;
  }
  return ans;
}

/**
 * @param {number[]} nums
 * @param {number} maximumBit
 * @return {number[]}
 */
var getMaximumXor = function (nums, maximumBit) {
  let xs = 0;
  for (const x of nums) {
    xs ^= x;
  }
  const n = nums.length;
  const ans = new Array(n);
  for (let i = 0; i < n; ++i) {
    const x = nums[n - i - 1];
    let k = 0;
    for (let j = maximumBit - 1; j >= 0; --j) {
      if (((xs >> j) & 1) == 0) {
        k |= 1 << j;
      }
    }
    ans[i] = k;
    xs ^= x;
  }
  return ans;
};

public class Solution {
  public int[] GetMaximumXor(int[] nums, int maximumBit) {
    int xs = 0;
    foreach (int x in nums) {
      xs ^= x;
    }
    int n = nums.Length;
    int[] ans = new int[n];
    for (int i = 0; i < n; ++i) {
      int x = nums[n - i - 1];
      int k = 0;
      for (int j = maximumBit - 1; j >= 0; --j) {
        if ((xs >> j & 1) == 0) {
          k |= 1 << j;
        }
      }
      ans[i] = k;
      xs ^= x;
    }
    return ans;
  }
}

Solution 2: Enumeration Optimization

Similar to Solution 1, we first preprocess the XOR sum $xs$ of the array nums, i.e., $xs=nums[0] \oplus nums[1] \oplus \cdots \oplus nums[n-1]$.

Next, we calculate $2^{maximumBit} - 1$, which is $2^{maximumBit}$ minus $1$, denoted as $mask$. Then, we enumerate each element $x$ in the array nums from back to front. The current XOR sum is $xs$, then $k=xs \oplus mask$ is the answer to each query. Then, we update $xs$ to $xs \oplus x$ and continue to enumerate the next element.

The time complexity is $O(n)$, where $n$ is the length of the array nums. Ignoring the space consumption of the answer, the space complexity is $O(1)$.

class Solution:
  def getMaximumXor(self, nums: List[int], maximumBit: int) -> List[int]:
    ans = []
    xs = reduce(xor, nums)
    mask = (1 << maximumBit) - 1
    for x in nums[::-1]:
      k = xs ^ mask
      ans.append(k)
      xs ^= x
    return ans

class Solution {
  public int[] getMaximumXor(int[] nums, int maximumBit) {
    int xs = 0;
    for (int x : nums) {
      xs ^= x;
    }
    int mask = (1 << maximumBit) - 1;
    int n = nums.length;
    int[] ans = new int[n];
    for (int i = 0; i < n; ++i) {
      int x = nums[n - i - 1];
      int k = xs ^ mask;
      ans[i] = k;
      xs ^= x;
    }
    return ans;
  }
}

class Solution {
public:
  vector<int> getMaximumXor(vector<int>& nums, int maximumBit) {
    int xs = 0;
    for (int& x : nums) {
      xs ^= x;
    }
    int mask = (1 << maximumBit) - 1;
    int n = nums.size();
    vector<int> ans(n);
    for (int i = 0; i < n; ++i) {
      int x = nums[n - i - 1];
      int k = xs ^ mask;
      ans[i] = k;
      xs ^= x;
    }
    return ans;
  }
};

func getMaximumXor(nums []int, maximumBit int) (ans []int) {
  xs := 0
  for _, x := range nums {
    xs ^= x
  }
  mask := (1 << maximumBit) - 1
  for i := range nums {
    x := nums[len(nums)-i-1]
    k := xs ^ mask
    ans = append(ans, k)
    xs ^= x
  }
  return
}

function getMaximumXor(nums: number[], maximumBit: number): number[] {
  let xs = 0;
  for (const x of nums) {
    xs ^= x;
  }
  const mask = (1 << maximumBit) - 1;
  const n = nums.length;
  const ans = new Array(n);
  for (let i = 0; i < n; ++i) {
    const x = nums[n - i - 1];
    let k = xs ^ mask;
    ans[i] = k;
    xs ^= x;
  }
  return ans;
}

/**
 * @param {number[]} nums
 * @param {number} maximumBit
 * @return {number[]}
 */
var getMaximumXor = function (nums, maximumBit) {
  let xs = 0;
  for (const x of nums) {
    xs ^= x;
  }
  const mask = (1 << maximumBit) - 1;
  const n = nums.length;
  const ans = new Array(n);
  for (let i = 0; i < n; ++i) {
    const x = nums[n - i - 1];
    let k = xs ^ mask;
    ans[i] = k;
    xs ^= x;
  }
  return ans;
};

public class Solution {
  public int[] GetMaximumXor(int[] nums, int maximumBit) {
    int xs = 0;
    foreach (int x in nums) {
      xs ^= x;
    }
    int mask = (1 << maximumBit) - 1;
    int n = nums.Length;
    int[] ans = new int[n];
    for (int i = 0; i < n; ++i) {
      int x = nums[n - i - 1];
      int k = xs ^ mask;
      ans[i] = k;
      xs ^= x;
    }
    return ans;
  }
}