3129. Find All Possible Stable Binary Arrays I

Description

You are given 3 positive integers zero, one, and limit.

A binary array arr is called stable if:

The number of occurrences of 0 in arr is exactly zero.
The number of occurrences of 1 in arr is exactly one.
Each subarray of arr with a size greater than limit must contain both 0 and 1.

Return the total number of stable binary arrays.

Since the answer may be very large, return it modulo 10⁹ + 7.

Example 1:

Input: zero = 1, one = 1, limit = 2

Output: 2

Explanation:

The two possible stable binary arrays are [1,0] and [0,1], as both arrays have a single 0 and a single 1, and no subarray has a length greater than 2.

Example 2:

Input: zero = 1, one = 2, limit = 1

Output: 1

Explanation:

The only possible stable binary array is [1,0,1].

Note that the binary arrays [1,1,0] and [0,1,1] have subarrays of length 2 with identical elements, hence, they are not stable.

Example 3:

Input: zero = 3, one = 3, limit = 2

Output: 14

Explanation:

All the possible stable binary arrays are [0,0,1,0,1,1], [0,0,1,1,0,1], [0,1,0,0,1,1], [0,1,0,1,0,1], [0,1,0,1,1,0], [0,1,1,0,0,1], [0,1,1,0,1,0], [1,0,0,1,0,1], [1,0,0,1,1,0], [1,0,1,0,0,1], [1,0,1,0,1,0], [1,0,1,1,0,0], [1,1,0,0,1,0], and [1,1,0,1,0,0].

Constraints:

1 <= zero, one, limit <= 200

Solutions

Solution 1: Memoization Search

We design a function $dfs(i, j, k)$ to represent the number of stable binary arrays that satisfy the problem conditions when there are $i$ $0$s and $j$ $1$s left, and the next number to be filled is $k$. The answer is $dfs(zero, one, 0) + dfs(zero, one, 1)$.

The calculation process of the function $dfs(i, j, k)$ is as follows:

If $i < 0$ or $j < 0$, return $0$.
If $i = 0$, return $1$ when $k = 1$ and $j \leq \textit{limit}$, otherwise return $0$.
If $j = 0$, return $1$ when $k = 0$ and $i \leq \textit{limit}$, otherwise return $0$.
If $k = 0$, we consider the case where the previous number is $0$, $dfs(i - 1, j, 0)$, and the case where the previous number is $1$, $dfs(i - 1, j, 1)$. If the previous number is $0$, it may cause more than $\textit{limit}$ $0$s in the subarray, i.e., the situation where the $\textit{limit} + 1$

Python3JavaC++GoTypeScript

class Solution:
    def numberOfStableArrays(self, zero: int, one: int, limit: int) -> int:
        @cache
        def dfs(i: int, j: int, k: int) -> int:
            if i == 0:
                return int(k == 1 and j <= limit)
            if j == 0:
                return int(k == 0 and i <= limit)
            if k == 0:
                return (
                    dfs(i - 1, j, 0)
                    + dfs(i - 1, j, 1)
                    - (0 if i - limit - 1 < 0 else dfs(i - limit - 1, j, 1))
                )
            return (
                dfs(i, j - 1, 0)
                + dfs(i, j - 1, 1)
                - (0 if j - limit - 1 < 0 else dfs(i, j - limit - 1, 0))
            )

        mod = 10**9 + 7
        ans = (dfs(zero, one, 0) + dfs(zero, one, 1)) % mod
        dfs.cache_clear()
        return ans

class Solution {
    private final int mod = (int) 1e9 + 7;
    private Long[][][] f;
    private int limit;

    public int numberOfStableArrays(int zero, int one, int limit) {
        f = new Long[zero + 1][one + 1][2];
        this.limit = limit;
        return (int) ((dfs(zero, one, 0) + dfs(zero, one, 1)) % mod);
    }

    private long dfs(int i, int j, int k) {
        if (i < 0 || j < 0) {
            return 0;
        }
        if (i == 0) {
            return k == 1 && j <= limit ? 1 : 0;
        }
        if (j == 0) {
            return k == 0 && i <= limit ? 1 : 0;
        }
        if (f[i][j][k] != null) {
            return f[i][j][k];
        }
        if (k == 0) {
            f[i][j][k] = (dfs(i - 1, j, 0) + dfs(i - 1, j, 1) - dfs(i - limit - 1, j, 1) + mod) % mod;
        } else {
            f[i][j][k] = (dfs(i, j - 1, 0) + dfs(i, j - 1, 1) - dfs(i, j - limit - 1, 0) + mod) % mod;
        }
        return f[i][j][k];
    }
}

class