LeetCode #2935 — HARD

Maximum Strong Pair XOR II

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

Solve on LeetCode
The Problem

Problem Statement

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
Patterns Used
⚡Bit Manipulation→🔤Trie→🪟Sliding Window→

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: 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.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Hash Map · Bit Manipulation · Trie · Sliding Window

Example 1

[1,2,3,4,5]

Example 2

[10,100]

Example 3

[500,520,2500,3000]

Related Problems

  • Maximum XOR of Two Numbers in an Array (maximum-xor-of-two-numbers-in-an-array)
  • Maximum XOR With an Element From Array (maximum-xor-with-an-element-from-array)
Step 02

Core Insight

What unlocks the optimal approach

  • Sort the array, now let <code>x <= y</code> which means <code>|x - y| <= min(x, y)</code> can now be written as <code>y - x <= x</code> or in other words, <code>y <= 2 * x</code>.
  • If <code>x</code> and <code>y</code> have the same number of bits, try making<code>y</code>’s bits different from x if possible for each bit starting from the second most significant bit.
  • If <code>y</code> has 1 more bit than <code>x</code> and <code>y <= 2 * x</code> use the idea about Digit DP to make <code>y</code>’s prefix smaller than <code>2 * x + 1</code> as well as trying to make each bit different from <code>x</code> using a Hashmap.
  • Alternatively, use Trie data structure to find the pair with maximum <code>XOR</code>.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. Continue until all input is consumed.
Use the first example testcase as your mental trace to verify each transition.
Step 04

Edge Cases

Minimum Input
Single element / shortest valid input
Validate boundary behavior before entering the main loop or recursion.
Duplicates & Repeats
Repeated values / repeated states
Decide whether duplicates should be merged, skipped, or counted explicitly.
Extreme Constraints
Largest constraint values
Re-check complexity target against constraints to avoid time-limit issues.
Invalid / Corner Shape
Empty collections, zeros, or disconnected structures
Handle special-case structure before the core algorithm path.
Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2935: Maximum Strong Pair XOR II
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;
    }
}
Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n × log M)
Space
O(n × log M)

Approach Breakdown

SORT + SCAN
O(n log n) time
O(n) space

Sort the array in O(n log n), then scan for the missing or unique element by comparing adjacent pairs. Sorting requires O(n) auxiliary space (or O(1) with in-place sort but O(n log n) time remains). The sort step dominates.

BIT MANIPULATION
O(n) time
O(1) space

Bitwise operations (AND, OR, XOR, shifts) are O(1) per operation on fixed-width integers. A single pass through the input with bit operations gives O(n) time. The key insight: XOR of a number with itself is 0, which eliminates duplicates without extra space.

Shortcut: Bit operations are O(1). XOR cancels duplicates. Single pass → O(n) time, O(1) space.
Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.

Shrinking the window only once

Wrong move: Using `if` instead of `while` leaves the window invalid for multiple iterations.

Usually fails on: Over-limit windows stay invalid and produce wrong lengths/counts.

Fix: Shrink in a `while` loop until the invariant is valid again.

Related Problems
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