1707. Maximum XOR With an Element From Array

Description

You are given an array nums consisting of non-negative integers. You are also given a queries array, where queries[i] = [xi, mi].

The answer to the ith query is the maximum bitwise XOR value of xi and any element of nums that does not exceed mi. In other words, the answer is max(nums[j] XOR xi) for all j such that nums[j] <= mi. If all elements in nums are larger than mi, then the answer is -1.

Return an integer array answer where answer.length == queries.length and answer[i] is the answer to the ith query.

 

Example 1:

Input: nums = [0,1,2,3,4], queries = [[3,1],[1,3],[5,6]]
Output: [3,3,7]
Explanation:
1) 0 and 1 are the only two integers not greater than 1. 0 XOR 3 = 3 and 1 XOR 3 = 2. The larger of the two is 3.
2) 1 XOR 2 = 3.
3) 5 XOR 2 = 7.

Example 2:

Input: nums = [5,2,4,6,6,3], queries = [[12,4],[8,1],[6,3]]
Output: [15,-1,5]

 

Constraints:

• 1 <= nums.length, queries.length <= 105
• queries[i].length == 2
• 0 <= nums[j], xi, mi <= 109

 

Solutions

Solution: Trie + Bit Manipulation

• Time complexity: O(mlogm+nlogn+(m+n)*log(Max(nums[i])))
• Space complexity: O(m*log(Max(nums[i]))+n)

 

JavaScript

/**
 * @param {number[]} nums
 * @param {number[][]} queries
 * @return {number[]}
 */
const maximizeXor = function (nums, queries) {
  const m = nums.length;
  const n = queries.length;
  const indexedQueries = queries.map((query, index) => {
    const [xor, max] = query;

    return { xor, max, index };
  });

  nums.sort((a, b) => a - b);
  indexedQueries.sort((a, b) => a.max - b.max);

  const maxNum = Math.max(nums[m - 1], indexedQueries[n - 1].max);
  const trie = new BitTrie(maxNum);
  const result = Array.from({ length: n }, () => -1);
  let index = 0;

  for (const query of indexedQueries) {
    while (index < m && nums[index] <= query.max) {
      trie.insert(nums[index]);
      index += 1;
    }

    if (index) {
      result[query.index] = trie.getMaxXor(query.xor);
    }
  }

  return result;
};

class TrieNode {
  zero = null;
  one = null;
}

class BitTrie {
  root = new TrieNode();

  constructor(maxNum) {
    this.maxBit = Math.ceil(Math.log2(maxNum));
  }

  insert(num) {
    let node = this.root;

    for (let index = this.maxBit; index >= 0; index--) {
      if ((num >> index) & 1) {
        node.one = node.one ?? new TrieNode();
        node = node.one;
      } else {
        node.zero = node.zero ?? new TrieNode();
        node = node.zero;
      }
    }
  }

  getMaxXor(num) {
    let node = this.root;
    let xor = 0;

    for (let index = this.maxBit; index >= 0; index--) {
      if ((num >> index) & 1) {
        if (node.zero) {
          xor |= 1 << index;
          node = node.zero;
        } else {
          node = node.one;
        }
      } else {
        if (node.one) {
          xor |= 1 << index;
          node = node.one;
        } else {
          node = node.zero;
        }
      }
    }

    return xor;
  }
}