Leetcode #2338: Count the Number of Ideal Arrays
In this guide, we solve Leetcode #2338 Count the Number of Ideal Arrays 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 two integers n and maxValue, which are used to describe an ideal array. A 0-indexed integer array arr of length n is considered ideal if the following conditions hold: Every arr[i] is a value from 1 to maxValue, for 0 <= i < n.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Math, Dynamic Programming, Combinatorics, Number Theory
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: n = 2, maxValue = 5
Output: 10
Explanation: The following are the possible ideal arrays:
- Arrays starting with the value 1 (5 arrays): [1,1], [1,2], [1,3], [1,4], [1,5]
- Arrays starting with the value 2 (2 arrays): [2,2], [2,4]
- Arrays starting with the value 3 (1 array): [3,3]
- Arrays starting with the value 4 (1 array): [4,4]
- Arrays starting with the value 5 (1 array): [5,5]
There are a total of 5 + 2 + 1 + 1 + 1 = 10 distinct ideal arrays.
Python Solution
class Solution:
def idealArrays(self, n: int, maxValue: int) -> int:
def dfs(i, cnt):
res = c[-1][cnt - 1]
if cnt < n:
k = 2
while k * i <= maxValue:
res = (res + dfs(k * i, cnt + 1)) % mod
k += 1
return res
c = [[0] * 16 for _ in range(n)]
mod = 10**9 + 7
for i in range(n):
for j in range(min(16, i + 1)):
c[i][j] = 1 if j == 0 else (c[i - 1][j] + c[i - 1][j - 1]) % mod
ans = 0
for i in range(1, maxValue + 1):
ans = (ans + dfs(i, 1)) % mod
return ans
Complexity
The time complexity is O(n·m) (typical). The space complexity is O(n·m) or optimized.
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.