How do you solve Find the Number of Subsequences With Equal GCD on LeetCode?

Find the Number of Subsequences With Equal GCD (LeetCode #3336) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n * max_num) time complexity and O(max_num) space complexity. Key insight: The GCD of a set of numbers can be computed incrementally, allowing us to build upon previous results.

What is the time complexity of Find the Number of Subsequences With Equal GCD?

The optimal solution for Find the Number of Subsequences With Equal GCD runs in O(n * max_num) time and uses O(max_num) space. The time complexity is O(n * max_num) because we iterate through each number and update the GCD counts for all possible GCDs, where max_num is the maximum value in nums.

What is the brute force solution for Find the Number of Subsequences With Equal GCD?

The brute force approach for Find the Number of Subsequences With Equal GCD is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves generating all possible pairs of subsequences and checking their GCDs. This is straightforward but inefficient due to the exponential number of subsequences. The optimal Optimal Solution approach improves this to O(n * max_num).

What algorithm or data structure is used to solve Find the Number of Subsequences With Equal GCD?

Find the Number of Subsequences With Equal GCD uses the following concepts: Array, Math, Dynamic Programming, Number Theory. The recommended patterns to study are: Dynamic Programming, GCD Computation.

What are the interview tips for Find the Number of Subsequences With Equal GCD?

Always discuss your thought process and approach before jumping into coding. Consider edge cases, such as arrays with all identical elements or the minimum and maximum constraints. Practice explaining your solution clearly, as communication is key in interviews.

What are common mistakes when solving Find the Number of Subsequences With Equal GCD?

Failing to account for the modulo operation when counting pairs. Not considering the efficiency of generating subsequences, leading to timeouts.

#3336

Find the Number of Subsequences With Equal GCD

Hard

ArrayMathDynamic ProgrammingNumber TheoryDynamic ProgrammingGCD Computation

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n * max_num)
Space	O(1)	O(max_num)

💡

Intuition

Time O(n * max_num)Space O(max_num)

The optimal solution uses dynamic programming to count subsequences with the same GCD efficiently. Instead of generating all subsequences, we keep track of how many subsequences yield each possible GCD.

⚙️

Algorithm

3 steps

1Step 1: Initialize a frequency array to count subsequences for each GCD from 1 to max(nums).
2Step 2: For each number in nums, update the frequency array for all GCDs that can be formed with this number.
3Step 3: Calculate the total pairs of subsequences with equal GCD using the frequency array.

solution.py22 lines

1# Full working Python code
2from math import gcd
3from collections import defaultdict
4
5MOD = 10**9 + 7
6
7def countSubsequences(nums):
8    max_num = max(nums)
9    freq = [0] * (max_num + 1)
10    total_pairs = 0
11
12    for num in nums:
13        for g in range(max_num, 0, -1):
14            if freq[g] > 0:
15                freq[gcd(g, num)] = (freq[gcd(g, num)] + freq[g]) % MOD
16        freq[num] = (freq[num] + 1) % MOD
17
18    for g in range(1, max_num + 1):
19        if freq[g] > 1:
20            total_pairs = (total_pairs + freq[g] * (freq[g] - 1) // 2) % MOD
21
22    return total_pairs

ℹ

Complexity note: The time complexity is O(n * max_num) because we iterate through each number and update the GCD counts for all possible GCDs, where max_num is the maximum value in nums.

1The GCD of a set of numbers can be computed incrementally, allowing us to build upon previous results.
2Counting subsequences with the same GCD can be efficiently managed using a frequency array.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.