2024-12-07：找出所有稳定的二进制数组 Ⅰ。用go语言，给定三个正整数 zero、one 和 limit，定义一个稳定的二进制数组需要满足以下条件：
数组中 0 的数量为 zero，1 的数量为 one，且每个长度超过 limit 的子数组都必须同时包含 0 和 1。
求出满足条件的稳定二进制数组的总数，结果需对 1000000007 取模后返回。
输入：zero = 1, one = 1, limit = 2。
输出：2。
解释：
两个稳定的二进制数组为 [1,0] 和 [0,1] ，两个数组都有一个 0 和一个 1 ，且没有子数组长度大于 2 。
答案2024-12-07：
chatgpt
题目来自leetcode3129。
大体步骤如下：
1.初始化变量：
- 初始化动态规划数组
dp，它是一个三维数组，dp[i][j][k]表示包含i个 0 和j个 1 的子数组中，最后一个数字是k的所有稳定二进制数组的数量。 - 初始化模数
mod为 1e9 + 7，用于取模操作。
2.动态规划填表：
- 遍历填充
dp数组，根据限制条件计算每个子问题的解。 - 内层循环处理
0和1的数量，更新dp[i][j][0]和dp[i][j][1]的值，考虑超过 limit 的限制情况。
3.返回结果：
- 最后返回
(dp[zero][one][0] + dp[zero][one][1]) % mod，即得到满足条件的稳定二进制数组的总数。
总的时间复杂度：
- 填表部分需要二重循环遍历
zero和one，时间复杂度为 O(zero * one)，内部还有限制条件的判断，整体时间复杂度为 O(zero * one)。 - 计算结果的取模操作时间复杂度为 O(1)。
总的额外空间复杂度：
- 动态规划数组
dp的空间复杂度为 O(zero * one)，因此总的额外空间复杂度为 O(zero * one)。
Go完整代码如下：
package main
import (
"fmt"
)
func numberOfStableArrays(zero int, one int, limit int) int {
dp := make([][][2]int, zero+1)
mod := int(1e9 + 7)
for i := 0; i <= zero; i++ {
dp[i] = make([][2]int, one+1)
}
for i := 0; i <= min(zero, limit); i++ {
dp[i][0][0] = 1
}
for j := 0; j <= min(one, limit); j++ {
dp[0][j][1] = 1
}
for i := 1; i <= zero; i++ {
for j := 1; j <= one; j++ {
if i > limit {
dp[i][j][0] = dp[i-1][j][0] + dp[i-1][j][1] - dp[i-limit-1][j][1]
} else {
dp[i][j][0] = dp[i-1][j][0] + dp[i-1][j][1]
}
dp[i][j][0] = (dp[i][j][0]%mod + mod) % mod
if j > limit {
dp[i][j][1] = dp[i][j-1][1] + dp[i][j-1][0] - dp[i][j-limit-1][0]
} else {
dp[i][j][1] = dp[i][j-1][1] + dp[i][j-1][0]
}
dp[i][j][1] = (dp[i][j][1]%mod + mod) % mod
}
}
return (dp[zero][one][0] + dp[zero][one][1]) % mod
}
func main() {
zero := 1
one := 1
limit := 2
fmt.Println(numberOfStableArrays(zero, one, limit))
}
Rust完整代码如下：
const MOD: i64 = 1000000007;
fn min(x: i32, y: i32) -> i32 {
if x < y {
x
} else {
y
}
}
fn number_of_stable_arrays(zero: i32, one: i32, limit: i32) -> i64 {
let mut dp = vec![vec![[0, 0]; one as usize + 1]; zero as usize + 1];
for i in 0..=zero {
dp[i as usize][0][0] = 1;
}
for j in 0..=one {
dp[0][j as usize][1] = 1;
}
for i in 1..=zero {
for j in 1..=one {
if i > limit {
dp[i as usize][j as usize][0] = (dp[i as usize - 1][j as usize][0]
+ dp[i as usize - 1][j as usize][1]
- dp[(i - limit - 1) as usize][j as usize][1])
% MOD;
} else {
dp[i as usize][j as usize][0] =
(dp[i as usize - 1][j as usize][0] + dp[i as usize - 1][j as usize][1]) % MOD;
}
if dp[i as usize][j as usize][0] < 0 {
dp[i as usize][j as usize][0] += MOD;
}
if j > limit {
dp[i as usize][j as usize][1] = (dp[i as usize][j as usize - 1][1]
+ dp[i as usize][j as usize - 1][0]
- dp[i as usize][j as usize - limit as usize - 1][0])
% MOD;
} else {
dp[i as usize][j as usize][1] =
(dp[i as usize][j as usize - 1][1] + dp[i as usize][j as usize - 1][0]) % MOD;
}
if dp[i as usize][j as usize][1] < 0 {
dp[i as usize][j as usize][1] += MOD;
}
}
}
(dp[zero as usize][one as usize][0] + dp[zero as usize][one as usize][1]) % MOD
}
fn main() {
let zero = 1;
let one = 1;
let limit = 2;
println!("{}", number_of_stable_arrays(zero, one, limit));
}
