Count the Number of Ideal Arrays

Count the Number of Ideal Arrays

What is this problem about?

The Count the Number of Ideal Arrays interview question defines an "ideal array" of length $n$ where each element is between 1 and $m$ inclusive, and every element is a multiple of its predecessor (i.e., arr[i] % arr[i-1] == 0).

You need to find the total number of such ideal arrays.

Why is this asked in interviews?

This "Hard" problem is a favorite at Microsoft and Amazon because it blends dynamic programming interview pattern with combinatorics and number theory. It tests the ability to recognize that since $arr[i] \ge arr[i-1] imes 2$ (unless they are equal), the number of distinct values in an ideal array is very small ($\le \log_2(m)$). This observation is the key to an efficient solution.

Algorithmic pattern used

The problem is solved using DP and Stars and Bars.

1. DP for Sequences: Let dp[len][val] be the number of strictly increasing sequences of length len ending with value val, where each element divides the next. Max len is $\sim 14$ because $2^{14} > 10^4$.
2. Combinatorics: For a strictly increasing sequence of length $L$, how many ways can we expand it to length $n$ by repeating elements? This is a "stars and bars" problem: choosing $L$ positions out of $n$ to change values, which is $\binom{n-1}{L-1}$.
3. Combine: Total = $\sum_{L=1}^{14} (\sum_{v=1}^{m} dp[L][v]) imes \binom{n-1}{L-1}$.

Example explanation

$n=2, m=5$

• Strictly increasing sequences:
  • Length 1: [1], [2], [3], [4], [5]. (5 total)
  • Length 2: [1,2], [1,3], [1,4], [1,5], [2,4]. (4 total)
• Expanding to $n=2$:
  • Length 1 sequences: $\binom{2-1}{1-1} = 1$ way. $5 imes 1 = 5$.
  • Length 2 sequences: $\binom{2-1}{2-1} = 1$ way. $4 imes 1 = 4$. Total = 9.

Common mistakes candidates make

• Full DP: Attempting $O(N imes M)$ DP, which fails when $n=10^9$.
• Divisor calculation: Inefficiently finding divisors for the DP transition. Precomputing a sieve is better.
• Combinations: Forgetting to handle large $n$ in the $\binom{n-1}{L-1}$ calculation using modular inverse.

Interview preparation tip

Remember: If a sequence grows exponentially (like multiples), its length is bounded by $\log(Max)$. This "logarithmic bound" trick is common in number theory problems to reduce one dimension of a DP table.