LeetCode #3312 — HARD

Sorted GCD Pair Queries

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 an integer array nums of length n and an integer array queries.

Let gcdPairs denote an array obtained by calculating the GCD of all possible pairs (nums[i], nums[j]), where 0 <= i < j < n, and then sorting these values in ascending order.

For each query queries[i], you need to find the element at index queries[i] in gcdPairs.

Return an integer array answer, where answer[i] is the value at gcdPairs[queries[i]] for each query.

The term gcd(a, b) denotes the greatest common divisor of a and b.

Example 1:

Input: nums = [2,3,4], queries = [0,2,2]

Output: [1,2,2]

Explanation:

gcdPairs = [gcd(nums[0], nums[1]), gcd(nums[0], nums[2]), gcd(nums[1], nums[2])] = [1, 2, 1].

After sorting in ascending order, gcdPairs = [1, 1, 2].

So, the answer is [gcdPairs[queries[0]], gcdPairs[queries[1]], gcdPairs[queries[2]]] = [1, 2, 2].

Example 2:

Input: nums = [4,4,2,1], queries = [5,3,1,0]

Output: [4,2,1,1]

Explanation:

gcdPairs sorted in ascending order is [1, 1, 1, 2, 2, 4].

Example 3:

Input: nums = [2,2], queries = [0,0]

Output: [2,2]

Explanation:

gcdPairs = [2].

Constraints:

  • 2 <= n == nums.length <= 105
  • 1 <= nums[i] <= 5 * 104
  • 1 <= queries.length <= 105
  • 0 <= queries[i] < n * (n - 1) / 2
Patterns Used
🔍Modified Binary Search→

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 an integer array nums of length n and an integer array queries. Let gcdPairs denote an array obtained by calculating the GCD of all possible pairs (nums[i], nums[j]), where 0 <= i < j < n, and then sorting these values in ascending order. For each query queries[i], you need to find the element at index queries[i] in gcdPairs. Return an integer array answer, where answer[i] is the value at gcdPairs[queries[i]] for each query. The term gcd(a, b) denotes the greatest common divisor of a and b.

Baseline thinking

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

Pattern signal: Array · Hash Map · Math · Binary Search

Example 1

[2,3,4]
[0,2,2]

Example 2

[4,4,2,1]
[5,3,1,0]

Example 3

[2,2]
[0,0]
Step 02

Core Insight

What unlocks the optimal approach

  • Try counting the number of pairs that have a GCD of <code>g</code.
  • Use inclusion-exclusion.
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 #3312: Sorted GCD Pair Queries
class Solution {
    public int[] gcdValues(int[] nums, long[] queries) {
        int mx = Arrays.stream(nums).max().getAsInt();
        int[] cnt = new int[mx + 1];
        long[] cntG = new long[mx + 1];
        for (int x : nums) {
            ++cnt[x];
        }
        for (int i = mx; i > 0; --i) {
            int v = 0;
            for (int j = i; j <= mx; j += i) {
                v += cnt[j];
                cntG[i] -= cntG[j];
            }
            cntG[i] += 1L * v * (v - 1) / 2;
        }
        for (int i = 2; i <= mx; ++i) {
            cntG[i] += cntG[i - 1];
        }
        int m = queries.length;
        int[] ans = new int[m];
        for (int i = 0; i < m; ++i) {
            ans[i] = search(cntG, queries[i]);
        }
        return ans;
    }

    private int search(long[] nums, long x) {
        int n = nums.length;
        int l = 0, r = n;
        while (l < r) {
            int mid = l + r >> 1;
            if (nums[mid] > x) {
                r = mid;
            } else {
                l = mid + 1;
            }
        }
        return l;
    }
}
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 + (M + q)
Space
O(M)

Approach Breakdown

LINEAR SCAN
O(n) time
O(1) space

Check every element from left to right until we find the target or exhaust the array. Each comparison is O(1), and we may visit all n elements, giving O(n). No extra space needed.

BINARY SEARCH
O(log n) time
O(1) space

Each comparison eliminates half the remaining search space. After k comparisons, the space is n/2ᵏ. We stop when the space is 1, so k = log₂ n. No extra memory needed — just two pointers (lo, hi).

Shortcut: Halving the input each step → O(log n). Works on any monotonic condition, not just sorted arrays.
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.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

Boundary update without `+1` / `-1`

Wrong move: Setting `lo = mid` or `hi = mid` can stall and create an infinite loop.

Usually fails on: Two-element ranges never converge.

Fix: Use `lo = mid + 1` or `hi = mid - 1` where appropriate.

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