diff --git a/solution/1900-1999/1955.Count Number of Special Subsequences/README.md b/solution/1900-1999/1955.Count Number of Special Subsequences/README.md index b96be4d54015b..d03e72a664b8d 100644 --- a/solution/1900-1999/1955.Count Number of Special Subsequences/README.md +++ b/solution/1900-1999/1955.Count Number of Special Subsequences/README.md @@ -86,17 +86,17 @@ tags: 如果 $nums[i] = 0$:如果我们不选择 $nums[i]$,则 $f[i][0] = f[i-1][0]$;如果我们选择 $nums[i]$,那么 $f[i][0]=f[i-1][0]+1$,因为我们可以在任何一个以 $0$ 结尾的特殊子序列后面加上一个 $0$ 得到一个新的特殊子序列,也可以将 $nums[i]$ 单独作为一个特殊子序列。因此 $f[i][0] = 2 \times f[i - 1][0] + 1$。其余的 $f[i][j]$ 与 $f[i-1][j]$ 相等。 -如果 $nums[i] = 1$:如果我们不选择 $nums[i]$,则 $f[i][1] = f[i-1][1]$;如果我们选择 $nums[i]$,那么 $f[i][1]=f[i-1][1]+f[i-1][0]$,因为我们可以在任何一个以 $0$ 或 $1$ 结尾的特殊子序列后面加上一个 $1$ 得到一个新的特殊子序列。因此 $f[i][1] = f[i-1][1] + 2 \times f[i - 1][0]$。其余的 $f[i][j]$ 与 $f[i-1][j]$ 相等。 +如果 $nums[i] = 1$:如果我们不选择 $nums[i]$,则 $f[i][1] = f[i-1][1]$;如果我们选择 $nums[i]$,那么 $f[i][1]=f[i-1][1]+f[i-1][0]$,因为我们可以在任何一个以 $0$ 或 $1$ 结尾的特殊子序列后面加上一个 $1$ 得到一个新的特殊子序列。因此 $f[i][1] = f[i-1][0] + 2 \times f[i - 1][1]$。其余的 $f[i][j]$ 与 $f[i-1][j]$ 相等。 -如果 $nums[i] = 2$:如果我们不选择 $nums[i]$,则 $f[i][2] = f[i-1][2]$;如果我们选择 $nums[i]$,那么 $f[i][2]=f[i-1][2]+f[i-1][1]$,因为我们可以在任何一个以 $1$ 或 $2$ 结尾的特殊子序列后面加上一个 $2$ 得到一个新的特殊子序列。因此 $f[i][2] = f[i-1][2] + 2 \times f[i - 1][1]$。其余的 $f[i][j]$ 与 $f[i-1][j]$ 相等。 +如果 $nums[i] = 2$:如果我们不选择 $nums[i]$,则 $f[i][2] = f[i-1][2]$;如果我们选择 $nums[i]$,那么 $f[i][2]=f[i-1][2]+f[i-1][1]$,因为我们可以在任何一个以 $1$ 或 $2$ 结尾的特殊子序列后面加上一个 $2$ 得到一个新的特殊子序列。因此 $f[i][2] = f[i-1][1] + 2 \times f[i - 1][2]$。其余的 $f[i][j]$ 与 $f[i-1][j]$ 相等。 综上,我们可以得到如下的状态转移方程: $$ \begin{aligned} f[i][0] &= 2 \times f[i - 1][0] + 1, \quad nums[i] = 0 \\ -f[i][1] &= f[i-1][1] + 2 \times f[i - 1][0], \quad nums[i] = 1 \\ -f[i][2] &= f[i-1][2] + 2 \times f[i - 1][1], \quad nums[i] = 2 \\ +f[i][1] &= f[i-1][0] + 2 \times f[i - 1][1], \quad nums[i] = 1 \\ +f[i][2] &= f[i-1][1] + 2 \times f[i - 1][2], \quad nums[i] = 2 \\ f[i][j] &= f[i-1][j], \quad nums[i] \neq j \end{aligned} $$ @@ -105,8 +105,6 @@ $$ 时间复杂度 $O(n)$,空间复杂度 $O(n)$。其中 $n$ 是数组 $nums$ 的长度。 -我们注意到,上述的状态转移方程中,$f[i][j]$ 的值仅与 $f[i-1][j]$ 有关,因此我们可以去掉第一维,将空间复杂度优化到 $O(1)$。 - #### Python3 @@ -258,7 +256,9 @@ function countSpecialSubsequences(nums: number[]): number { -### 方法二 +### 方法二:动态规划(空间优化) + +我们注意到,上述的状态转移方程中,$f[i][j]$ 的值仅与 $f[i-1][j]$ 有关,因此我们可以去掉第一维,将空间复杂度优化到 $O(1)$。 diff --git a/solution/1900-1999/1955.Count Number of Special Subsequences/README_EN.md b/solution/1900-1999/1955.Count Number of Special Subsequences/README_EN.md index 3d29305a5c3be..27c41bb546d5b 100644 --- a/solution/1900-1999/1955.Count Number of Special Subsequences/README_EN.md +++ b/solution/1900-1999/1955.Count Number of Special Subsequences/README_EN.md @@ -76,7 +76,34 @@ tags: -### Solution 1 +### Solution 1: Dynamic Programming + +We define $f[i][j]$ to represent the number of special subsequences ending with $j$ among the first $i+1$ elements. Initially, $f[i][j]=0$, and if $nums[0]=0$, then $f[0][0]=1$. + +For $i \gt 0$, we consider the value of $nums[i]$: + +If $nums[i] = 0$: If we do not choose $nums[i]$, then $f[i][0] = f[i-1][0]$; if we choose $nums[i]$, then $f[i][0]=f[i-1][0]+1$, because we can add a $0$ to the end of any special subsequence ending with $0$ to get a new special subsequence, or we can use $nums[i]$ alone as a special subsequence. Therefore, $f[i][0] = 2 \times f[i - 1][0] + 1$. The rest of $f[i][j]$ is equal to $f[i-1][j]$. + +If $nums[i] = 1$: If we do not choose $nums[i]$, then $f[i][1] = f[i-1][1]$; if we choose $nums[i]$, then $f[i][1]=f[i-1][1]+f[i-1][0]$, because we can add a $1$ to the end of any special subsequence ending with $0$ or $1$ to get a new special subsequence. Therefore, $f[i][1] = f[i-1][0] + 2 \times f[i - 1][1]$. The rest of $f[i][j]$ is equal to $f[i-1][j]$. + +If $nums[i] = 2$: If we do not choose $nums[i]$, then $f[i][2] = f[i-1][2]$; if we choose $nums[i]$, then $f[i][2]=f[i-1][2]+f[i-1][1]$, because we can add a $2$ to the end of any special subsequence ending with $1$ or $2$ to get a new special subsequence. Therefore, $f[i][2] = f[i-1][1] + 2 \times f[i - 1][2]$. The rest of $f[i][j]$ is equal to $f[i-1][j]$. + +In summary, we can get the following state transition equations: + +$$ +\begin{aligned} +f[i][0] &= 2 \times f[i - 1][0] + 1, \quad nums[i] = 0 \\ +f[i][1] &= f[i-1][0] + 2 \times f[i - 1][1], \quad nums[i] = 1 \\ +f[i][2] &= f[i-1][1] + 2 \times f[i - 1][2], \quad nums[i] = 2 \\ +f[i][j] &= f[i-1][j], \quad nums[i] \neq j +\end{aligned} +$$ + +The final answer is $f[n-1][2]$. + +The time complexity is $O(n)$, and the space complexity is $O(n)$. Where $n$ is the length of the array $nums$. + +Similar code found with 1 license type @@ -229,7 +256,13 @@ function countSpecialSubsequences(nums: number[]): number { -### Solution 2 +### Solution 2: Dynamic Programming (Space Optimization) + +We notice that in the above state transition equations, the value of $f[i][j]$ is only related to $f[i-1][j]$. Therefore, we can remove the first dimension and optimize the space complexity to $O(1)$. + +We can use an array $f$ of length 3 to represent the number of special subsequences ending with 0, 1, and 2, respectively. For each element in the array, we update the array $f$ according to the value of the current element. + +The time complexity is $O(n)$, and the space complexity is $O(1)$. Where $n$ is the length of the array $nums$.