For each query, we identify the GCD at a specific index after generating all possible pairs and sorting them. By using a hash table to track counts of each GCD and applying efficient array scanning, the solution avoids direct pair generation, making it scalable. This method handles large input arrays by combining number theory insights with counting techniques for rapid query resolution.
Problem Statement
You are given an integer array nums of length n and a queries array. Compute all possible GCDs for each pair (nums[i], nums[j]) where 0 <= i < j < n, then sort these GCDs in ascending order.
For each index in queries, return the GCD at that position from the sorted list. Ensure the solution efficiently handles large arrays without explicitly generating every pair, leveraging counting and hash-based strategies.
Examples
Example 1
Input: nums = [2,3,4], queries = [0,2,2]
Output: [1,2,2]
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]
gcdPairs sorted in ascending order is [1, 1, 1, 2, 2, 4] .
Example 3
Input: nums = [2,2], queries = [0,0]
Output: [2,2]
gcdPairs = [2] .
Constraints
- 2 <= n == nums.length <= 105
- 1 <= nums[i] <= 5 * 104
- 1 <= queries.length <= 105
- 0 <= queries[i] < n * (n - 1) / 2
Solution Approach
Count GCD Frequencies Using Hash Map
Iterate through nums and count occurrences of each number. For each possible GCD g, calculate how many pairs produce g using combinatorial formulas and store results in a hash table for quick access.
Generate Sorted GCD Prefix Sum
Convert the hash map of GCD counts into a prefix sum array representing the cumulative number of pairs up to each GCD value. This allows fast lookup for the query indices without sorting all pairs explicitly.
Answer Queries with Binary Search
For each query index, perform a binary search on the prefix sum array to locate the GCD corresponding to that index. This efficiently resolves all queries in O(log M) per query, where M is the number of distinct GCDs.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity depends on the number of distinct GCDs and the prefix sum processing, typically O(N log N + Q log M). Space complexity is O(M) for storing GCD counts and prefix sums.
What Interviewers Usually Probe
- The candidate considers counting pairs instead of brute-force pair generation.
- Mentions using hash maps to store GCD frequencies for fast queries.
- Uses prefix sums or cumulative counts to answer multiple queries efficiently.
Common Pitfalls or Variants
Common pitfalls
- Attempting to generate all pairs explicitly, which leads to memory overflow on large arrays.
- Ignoring duplicates in nums, which affects the correct pair count for GCDs.
- Not using cumulative counts, resulting in O(N^2) query lookups.
Follow-up variants
- Return the k-th largest GCD instead of the k-th smallest.
- Handle dynamic updates to nums and answer queries in real time.
- Count pairs with GCD divisible by a given integer instead of exact GCD values.
How GhostInterview Helps
- Automatically constructs GCD frequency tables to accelerate query resolution.
- Generates prefix sum arrays to map indices to GCD values without manual sorting.
- Highlights edge cases like duplicate numbers or large input sizes to prevent performance pitfalls.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the primary pattern in Sorted GCD Pair Queries?
The pattern is array scanning plus hash lookup to count GCDs efficiently without enumerating all pairs.
Can I solve this problem by generating all pairs?
Generating all pairs is feasible only for small arrays; the recommended method uses counting and hash maps to scale.
How does prefix sum help answer queries quickly?
Prefix sums accumulate pair counts for each GCD, enabling direct index-based lookup via binary search instead of scanning all pairs.
Do duplicate numbers in nums affect results?
Yes, duplicates increase the number of pairs with the same GCD, which must be correctly counted to answer queries accurately.
What is an efficient approach to large query arrays?
Use the hash table to store GCD counts and prefix sums to resolve each query in O(log M) time, avoiding explicit pair iteration.
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