# 2935. Maximum Strong Pair XOR II

## Description

You are given a 0-indexed integer array nums. A pair of integers x and y is called a strong pair if it satisfies the condition:

• |x - y| <= min(x, y)

You need to select two integers from nums such that they form a strong pair and their bitwise XOR is the maximum among all strong pairs in the array.

Return the maximum XOR value out of all possible strong pairs in the array nums.

Note that you can pick the same integer twice to form a pair.

Example 1:

Input: nums = [1,2,3,4,5]
Output: 7
Explanation: There are 11 strong pairs in the array nums: (1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (3, 5), (4, 4), (4, 5) and (5, 5).
The maximum XOR possible from these pairs is 3 XOR 4 = 7.


Example 2:

Input: nums = [10,100]
Output: 0
Explanation: There are 2 strong pairs in the array nums: (10, 10) and (100, 100).
The maximum XOR possible from these pairs is 10 XOR 10 = 0 since the pair (100, 100) also gives 100 XOR 100 = 0.


Example 3:

Input: nums = [500,520,2500,3000]
Output: 1020
Explanation: There are 6 strong pairs in the array nums: (500, 500), (500, 520), (520, 520), (2500, 2500), (2500, 3000) and (3000, 3000).
The maximum XOR possible from these pairs is 500 XOR 520 = 1020 since the only other non-zero XOR value is 2500 XOR 3000 = 636.


Constraints:

• 1 <= nums.length <= 5 * 104
• 1 <= nums[i] <= 220 - 1

## Solutions

Solution 1: Sorting + Binary Trie

 Observing the inequality $x - y \leq \min(x, y)$, which involves absolute value and minimum value, we can assume $x \leq y$, then we have $y - x \leq x$, that is, $y \leq 2x$. We can enumerate $y$ from small to large, then $x$ must satisfy the inequality $y \leq 2x$.

Therefore, we sort the array $nums$, and then enumerate $y$ from small to large. We use two pointers to maintain a window so that the elements $x$ in the window satisfy the inequality $y \leq 2x$. We can use a binary trie to maintain the elements in the window, so we can find the maximum XOR value in the window in $O(1)$ time. Each time we add $y$ to the trie, and remove the elements at the left end of the window that do not satisfy the inequality, this can ensure that the elements in the window satisfy the inequality $y \leq 2x$. Then query the maximum XOR value from the trie and update the answer.

The time complexity is $O(n \times \log M)$, and the space complexity is $O(n \times \log M)$. Here, $n$ is the length of the array $nums$, and $M$ is the maximum value in the array $nums$. In this problem, $M = 2^{20}$.

• class Trie {
private Trie[] children = new Trie[2];
private int cnt = 0;

public Trie() {
}

public void insert(int x) {
Trie node = this;
for (int i = 20; i >= 0; --i) {
int v = x >> i & 1;
if (node.children[v] == null) {
node.children[v] = new Trie();
}
node = node.children[v];
++node.cnt;
}
}

public int search(int x) {
Trie node = this;
int ans = 0;
for (int i = 20; i >= 0; --i) {
int v = x >> i & 1;
if (node.children[v ^ 1] != null && node.children[v ^ 1].cnt > 0) {
ans |= 1 << i;
node = node.children[v ^ 1];
} else {
node = node.children[v];
}
}
return ans;
}

public void remove(int x) {
Trie node = this;
for (int i = 20; i >= 0; --i) {
int v = x >> i & 1;
node = node.children[v];
--node.cnt;
}
}
}

class Solution {
public int maximumStrongPairXor(int[] nums) {
Arrays.sort(nums);
Trie tree = new Trie();
int ans = 0, i = 0;
for (int y : nums) {
tree.insert(y);
while (y > nums[i] * 2) {
tree.remove(nums[i++]);
}
ans = Math.max(ans, tree.search(y));
}
return ans;
}
}

• class Trie {
public:
Trie* children[2];
int cnt;

Trie()
: cnt(0) {
children[0] = nullptr;
children[1] = nullptr;
}

void insert(int x) {
Trie* node = this;
for (int i = 20; ~i; --i) {
int v = (x >> i) & 1;
if (node->children[v] == nullptr) {
node->children[v] = new Trie();
}
node = node->children[v];
++node->cnt;
}
}

int search(int x) {
Trie* node = this;
int ans = 0;
for (int i = 20; ~i; --i) {
int v = (x >> i) & 1;
if (node->children[v ^ 1] != nullptr && node->children[v ^ 1]->cnt > 0) {
ans |= 1 << i;
node = node->children[v ^ 1];
} else {
node = node->children[v];
}
}
return ans;
}

void remove(int x) {
Trie* node = this;
for (int i = 20; ~i; --i) {
int v = (x >> i) & 1;
node = node->children[v];
--node->cnt;
}
}
};

class Solution {
public:
int maximumStrongPairXor(vector<int>& nums) {
sort(nums.begin(), nums.end());
Trie* tree = new Trie();
int ans = 0, i = 0;
for (int y : nums) {
tree->insert(y);
while (y > nums[i] * 2) {
tree->remove(nums[i++]);
}
ans = max(ans, tree->search(y));
}
return ans;
}
};

• class Trie:
__slots__ = ("children", "cnt")

def __init__(self):
self.children: List[Trie | None] = [None, None]
self.cnt = 0

def insert(self, x: int):
node = self
for i in range(20, -1, -1):
v = x >> i & 1
if node.children[v] is None:
node.children[v] = Trie()
node = node.children[v]
node.cnt += 1

def search(self, x: int) -> int:
node = self
ans = 0
for i in range(20, -1, -1):
v = x >> i & 1
if node.children[v ^ 1] and node.children[v ^ 1].cnt:
ans |= 1 << i
node = node.children[v ^ 1]
else:
node = node.children[v]
return ans

def remove(self, x: int):
node = self
for i in range(20, -1, -1):
v = x >> i & 1
node = node.children[v]
node.cnt -= 1

class Solution:
def maximumStrongPairXor(self, nums: List[int]) -> int:
nums.sort()
tree = Trie()
ans = i = 0
for y in nums:
tree.insert(y)
while y > nums[i] * 2:
tree.remove(nums[i])
i += 1
ans = max(ans, tree.search(y))
return ans


• type Trie struct {
children [2]*Trie
cnt      int
}

func newTrie() *Trie {
return &Trie{}
}

func (t *Trie) insert(x int) {
node := t
for i := 20; i >= 0; i-- {
v := (x >> uint(i)) & 1
if node.children[v] == nil {
node.children[v] = newTrie()
}
node = node.children[v]
node.cnt++
}
}

func (t *Trie) search(x int) int {
node := t
ans := 0
for i := 20; i >= 0; i-- {
v := (x >> uint(i)) & 1
if node.children[v^1] != nil && node.children[v^1].cnt > 0 {
ans |= 1 << uint(i)
node = node.children[v^1]
} else {
node = node.children[v]
}
}
return ans
}

func (t *Trie) remove(x int) {
node := t
for i := 20; i >= 0; i-- {
v := (x >> uint(i)) & 1
node = node.children[v]
node.cnt--
}
}

func maximumStrongPairXor(nums []int) (ans int) {
sort.Ints(nums)
tree := newTrie()
i := 0
for _, y := range nums {
tree.insert(y)
for ; y > nums[i]*2; i++ {
tree.remove(nums[i])
}
ans = max(ans, tree.search(y))
}
return ans
}

• class Trie {
children: (Trie | null)[];
cnt: number;

constructor() {
this.children = [null, null];
this.cnt = 0;
}

insert(x: number): void {
let node: Trie | null = this;
for (let i = 20; i >= 0; i--) {
const v = (x >> i) & 1;
if (node.children[v] === null) {
node.children[v] = new Trie();
}
node = node.children[v] as Trie;
node.cnt++;
}
}

search(x: number): number {
let node: Trie | null = this;
let ans = 0;
for (let i = 20; i >= 0; i--) {
const v = (x >> i) & 1;
if (node.children[v ^ 1] !== null && (node.children[v ^ 1] as Trie).cnt > 0) {
ans |= 1 << i;
node = node.children[v ^ 1] as Trie;
} else {
node = node.children[v] as Trie;
}
}
return ans;
}

remove(x: number): void {
let node: Trie | null = this;
for (let i = 20; i >= 0; i--) {
const v = (x >> i) & 1;
node = node.children[v] as Trie;
node.cnt--;
}
}
}

function maximumStrongPairXor(nums: number[]): number {
nums.sort((a, b) => a - b);
const tree = new Trie();
let ans = 0;
let i = 0;

for (const y of nums) {
tree.insert(y);

while (y > nums[i] * 2) {
tree.remove(nums[i++]);
}

ans = Math.max(ans, tree.search(y));
}

return ans;
}