## Saturday, November 12, 2016

### LeetCode OJ - Maximum XOR of Two Numbers in an Array

Problem:

Please find the problem here.

Analysis:

I started with a greedy thinking. Suppose I can turn the MSB on, there is no reason for me not to do so.

Therefore we partitioned the numbers into two sets, one with the MSB turned on, one with the MSB turned off. We must pick one from one partition and another from another.

The second bit is critical, we know they want to be different, but otherwise we do not know which one should be 0, which one should be 1, or it is also possible that it is impossible to make them differ. It looks like if we could make them differ, we should always do that, but otherwise, make them the same and move on to the third bit.

The algorithm goes on until we reach the last bit, and we are done. If we were to implement this idea brute force, we would have to partition the sets 32 times, and figure out someway to pass the sets around, a complicated business.

Solution:

The key observation here is that we can build a trie. Building the trie takes linear time, and a trie node perfectly denote a set of the shape we wanted, so we can simply use the trie node to represent the sets. Whether or not we can do the same bit can be queried in constant time, making it a perfect candidate to use.

Performance wise - notice we walk a node at most twice, together with the trie node building time, the overall algorithm is linear time!

Code:

// LEET_MAXIMUM_XOR_OF_TWO_NUMBERS_IN_AN_ARRAY
#include "stdafx.h"

// https://leetcode.com/problems/maximum-xor-of-two-numbers-in-an-array/

#include "LEET_MAXIMUM_XOR_OF_TWO_NUMBERS_IN_AN_ARRAY.h"
#include <stack>
#include <iostream>
#include <algorithm>
#include <vector>
#include <string>

using namespace std;

namespace _LEET_MAXIMUM_XOR_OF_TWO_NUMBERS_IN_AN_ARRAY
{
struct MyTreeNode
{
MyTreeNode* left;
MyTreeNode* right;
};
class Solution
{
public:
int findMaximumXOR(MyTreeNode* p1, MyTreeNode* p2, int level)
{
int result = 0;
if (level == 0)
{
return 0;
}
if ((p1->left != nullptr && p2->right != nullptr) || (p1->right != nullptr && p2->left != nullptr))
{
if (p1->left != nullptr && p2->right != nullptr)
{
result = findMaximumXOR(p1->left, p2->right, level - 1);
}
if (p1->right != nullptr && p2->left != nullptr)
{
result = max(result, findMaximumXOR(p1->right, p2->left, level - 1));
}
result += 1 << (level - 1);
}
else
{
if (p1->left != nullptr && p2->left != nullptr)
{
result = findMaximumXOR(p1->left, p2->left, level - 1);
}
if (p1->right != nullptr && p2->right != nullptr)
{
result = max(result, findMaximumXOR(p1->right, p2->right, level - 1));
}
}
return result;
}
int findMaximumXOR(vector<int>& nums)
{
// Step 1: Build a trie out of the binary digits of the numbers
int n = nums.size();
MyTreeNode* root = new MyTreeNode();
int current_level = -1;
for (int i = 0; i < n; i++)
{
int d = nums[i];
stack<int> digits;
while (d > 0)
{
digits.push(d % 2);
d /= 2;
}
if (current_level == -1)
{
current_level= digits.size();
}
else
{
while (digits.size() < current_level)
{
digits.push(0);
}
while (current_level < digits.size())
{
MyTreeNode* newRoot = new MyTreeNode();
newRoot->left = root;
root = newRoot;
current_level++;
}
}
MyTreeNode* cur = root;
while (digits.size() > 0)
{
int d = digits.top();
digits.pop();
if (d == 0)
{
if (cur->left == nullptr)
{
cur->left = new MyTreeNode();
}
cur = cur->left;
}
else
{
if (cur->right == nullptr)
{
cur->right = new MyTreeNode();
}
cur = cur->right;
}
}
}
// Walk the trie with two pointers to find the answer
return findMaximumXOR(root, root, current_level);
}
};
};

using namespace _LEET_MAXIMUM_XOR_OF_TWO_NUMBERS_IN_AN_ARRAY;

int LEET_MAXIMUM_XOR_OF_TWO_NUMBERS_IN_AN_ARRAY()
{
Solution solution;
int input_array[] = { 3, 10, 5, 25, 2, 8 };
vector<int> input(input_array, input_array + _countof(input_array));
cout << solution.findMaximumXOR(input);
return 0;
}