#1819

Hard

LeetCode Problem Workspace

Number of Different Subsequences GCDs

Given an array of positive integers, find the number of different subsequences' GCDs.

Problem Statement

You are given an array of positive integers. Your task is to find how many different GCD values can be formed from all possible subsequences of the array.

A subsequence is derived by removing some elements (possibly none) of the array while maintaining the order of the remaining elements. You need to return the number of distinct GCD values that can be formed from the subsequences.

Examples

Example 1

Input: nums = [6,10,3]

Output: 5

The figure shows all the non-empty subsequences and their GCDs. The different GCDs are 6, 10, 3, 2, and 1.

Example 2

Input: nums = [5,15,40,5,6]

Output: 7

Example details omitted.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 2 * 105

Solution Approach

Mathematical Approach to Counting Subsequences

To solve this problem, focus on the mathematical properties of the GCD. For each potential GCD value, check whether there exists a subsequence whose GCD equals that value. This requires iterating over the array and considering each number's contribution to possible subsequences.

Efficient Subsequence Construction

Instead of checking every possible subsequence (which is computationally expensive), efficiently track the subsequences' GCDs by using dynamic programming or similar approaches. Use a set or array to store the results and update it progressively as you go through the array.

Optimizing with Divisibility and Modular Arithmetic

Take advantage of divisibility rules and modular arithmetic to minimize redundant calculations. For each element, determine its contribution to subsequences of a certain GCD. This helps you avoid recalculating GCDs unnecessarily and accelerates the solution process.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the final approach, but it generally involves iterating over the array and updating a set of possible GCD values, resulting in a time complexity proportional to the size of the array and the possible GCD values. Space complexity is also influenced by how you store the GCDs during calculations, often requiring O(N) space.

What Interviewers Usually Probe

Ability to optimize solutions using number theory principles.
Efficiency in handling large input sizes and array lengths.
Understanding of dynamic programming or optimization strategies.

Common Pitfalls or Variants

Common pitfalls

Attempting to brute-force through all subsequences without leveraging mathematical properties.
Overlooking optimizations related to divisibility and GCD calculation.
Not handling large input sizes efficiently, leading to time limit exceeded errors.

Follow-up variants

How does the problem change if we are given an array of size 10^6?
What if the array only contains powers of two?
How does the problem behave with very large numbers up to 2 * 10^5?

FAQ

How can I calculate the number of different GCDs from subsequences efficiently?

Focus on using dynamic programming or optimized set-based calculations that avoid brute-force enumeration of all subsequences.

What is the main challenge in solving the "Number of Different Subsequences GCDs" problem?

The main challenge lies in efficiently calculating subsequences' GCDs without overcomplicating the solution or exceeding time limits.

What should I consider when approaching this problem's optimization?

Consider utilizing properties of divisibility and GCD to minimize redundant checks and iterating efficiently over the array.

How does number theory apply to this problem?

Number theory helps by leveraging divisibility rules and GCD properties to efficiently determine the number of distinct GCD values.

Can GhostInterview assist with understanding GCD properties for subsequences?

Yes, GhostInterview provides insights into the mathematical approach for optimizing GCD calculations, ensuring an efficient solution.

terminal

Solution

Solution 1: Enumeration + Mathematics

For all sub-sequences of the array $nums$, their greatest common divisor (GCD) will not exceed the maximum value $mx$ in the array.

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class Solution:
    def countDifferentSubsequenceGCDs(self, nums: List[int]) -> int:
        mx = max(nums)
        vis = set(nums)
        ans = 0
        for x in range(1, mx + 1):
            g = 0
            for y in range(x, mx + 1, x):
                if y in vis:
                    g = gcd(g, y)
                    if g == x:
                        ans += 1
                        break
        return ans

Continue Practicing

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