Leetcode #1994: The Number of Good Subsets
In this guide, we solve Leetcode #1994 The Number of Good Subsets in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
You are given an integer array nums. We call a subset of nums good if its product can be represented as a product of one or more distinct prime numbers.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Bit Manipulation, Array, Hash Table, Math, Dynamic Programming, Bitmask, Counting, Number Theory
Intuition
Fast membership checks and value lookups are the heart of this problem, which makes a hash map the natural choice.
By storing what we have already seen (or counts/indexes), we can answer the question in one pass without backtracking.
Approach
Scan the input once, using the map to detect when the condition is satisfied and to update state as you go.
This keeps the solution linear while remaining easy to explain in an interview setting.
Steps:
- Initialize a hash map for seen items or counts.
- Iterate through the input, querying/updating the map.
- Return the first valid result or the final computed value.
Example
Input: nums = [1,2,3,4]
Output: 6
Explanation: The good subsets are:
- [1,2]: product is 2, which is the product of distinct prime 2.
- [1,2,3]: product is 6, which is the product of distinct primes 2 and 3.
- [1,3]: product is 3, which is the product of distinct prime 3.
- [2]: product is 2, which is the product of distinct prime 2.
- [2,3]: product is 6, which is the product of distinct primes 2 and 3.
- [3]: product is 3, which is the product of distinct prime 3.
Python Solution
class Solution:
def numberOfGoodSubsets(self, nums: List[int]) -> int:
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
cnt = Counter(nums)
mod = 10**9 + 7
n = len(primes)
f = [0] * (1 << n)
f[0] = pow(2, cnt[1])
for x in range(2, 31):
if cnt[x] == 0 or x % 4 == 0 or x % 9 == 0 or x % 25 == 0:
continue
mask = 0
for i, p in enumerate(primes):
if x % p == 0:
mask |= 1 << i
for state in range((1 << n) - 1, 0, -1):
if state & mask == mask:
f[state] = (f[state] + cnt[x] * f[state ^ mask]) % mod
return sum(f[i] for i in range(1, 1 << n)) % mod
Complexity
The time complexity is O(n). The space complexity is O(n).
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.