How do you solve Sum of Beautiful Subsequences on LeetCode?

Sum of Beautiful Subsequences (LeetCode #3671) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n log n) time complexity and O(n) space complexity. Key insight: Understanding GCD properties helps in filtering elements.

What is the time complexity of Sum of Beautiful Subsequences?

The optimal solution for Sum of Beautiful Subsequences runs in O(n log n) time and uses O(n) space. Filtering and counting subsequences using a Fenwick Tree allows efficient updates and queries, leading to O(n log n) overall.

What is the brute force solution for Sum of Beautiful Subsequences?

The brute force approach for Sum of Beautiful Subsequences is Brute Force, with O(n²) time complexity and O(1) space complexity. Generate all possible strictly increasing subsequences and calculate their GCDs. Count the occurrences of each GCD and compute the beauty values. The optimal Optimal Solution approach improves this to O(n log n).

What algorithm or data structure is used to solve Sum of Beautiful Subsequences?

Sum of Beautiful Subsequences uses the following concepts: Array, Math, Binary Indexed Tree, Number Theory. The recommended patterns to study are: Hash Map, Array.

What are the interview tips for Sum of Beautiful Subsequences?

Explain your thought process clearly. Use examples to illustrate your approach. Be prepared to discuss time and space complexities.

What are common mistakes when solving Sum of Beautiful Subsequences?

Overlooking subsequences that might not be strictly increasing. Not considering all possible GCDs up to the maximum element.

#3671

Sum of Beautiful Subsequences

Hard

ArrayMathBinary Indexed TreeNumber TheoryHash MapArray

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

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

💡

Intuition

Time O(n log n)Space O(n)

Use a candidate GCD and filter elements divisible by it. Count increasing subsequences using a Fenwick Tree for efficiency.

⚙️

Algorithm

3 steps

1Step 1: For each candidate GCD g, filter nums to only include elements divisible by g and scale them down.
2Step 2: Count strictly increasing subsequences using coordinate compression and a Fenwick Tree.
3Step 3: Calculate beauty values for each GCD and sum them.

solution.py16 lines

1def sum_of_beautiful_subsequences(nums):
2    from collections import defaultdict
3    from bisect import bisect_left
4    MOD = 10**9 + 7
5    max_num = max(nums)
6    total_beauty = 0
7    for g in range(1, max_num + 1):
8        filtered = [x // g for x in nums if x % g == 0]
9        if not filtered: continue
10        dp = [0] * (max(filtered) + 1)
11        for x in filtered:
12            dp[x] = (dp[x] + 1) % MOD
13            for j in range(x + 1, len(dp)):
14                dp[j] = (dp[j] + dp[x]) % MOD
15        total_beauty = (total_beauty + g * sum(dp)) % MOD
16    return total_beauty

ℹ

Complexity note: Filtering and counting subsequences using a Fenwick Tree allows efficient updates and queries, leading to O(n log n) overall.

1Understanding GCD properties helps in filtering elements.
2Using data structures like Fenwick Tree optimizes counting subsequences.

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