903. Valid Permutations for DI Sequence
Description
You are given a string s
of length n
where s[i]
is either:
'D'
means decreasing, or'I'
means increasing.
A permutation perm
of n + 1
integers of all the integers in the range [0, n]
is called a valid permutation if for all valid i
:
- If
s[i] == 'D'
, thenperm[i] > perm[i + 1]
, and - If
s[i] == 'I'
, thenperm[i] < perm[i + 1]
.
Return the number of valid permutations perm
. Since the answer may be large, return it modulo 109 + 7
.
Example 1:
Input: s = "DID" Output: 5 Explanation: The 5 valid permutations of (0, 1, 2, 3) are: (1, 0, 3, 2) (2, 0, 3, 1) (2, 1, 3, 0) (3, 0, 2, 1) (3, 1, 2, 0)
Example 2:
Input: s = "D" Output: 1
Constraints:
n == s.length
1 <= n <= 200
s[i]
is either'I'
or'D'
.
Solutions
Solution 1: Dynamic Programming
We define $f[i][j]$ as the number of permutations that satisfy the problem's requirements with the first $i$ characters of the string ending with the number $j$. Initially, $f[0][0]=1$, and the rest $f[0][j]=0$. The answer is $\sum_{j=0}^n f[n][j]$.
Consider $f[i][j]$, where $j \in [0, i]$.
If the $i$th character $s[i-1]$ is 'D'
, then $f[i][j]$ can be transferred from $f[i-1][k]$, where $k \in [j+1, i]$. Since $k-1$ can only be up to $i-1$, we move $k$ one place to the left, so $k \in [j, i-1]$. Therefore, we have $f[i][j] = \sum_{k=j}^{i-1} f[i-1][k]$.
If the $i$th character $s[i-1]$ is 'I'
, then $f[i][j]$ can be transferred from $f[i-1][k]$, where $k \in [0, j-1]$. Therefore, we have $f[i][j] = \sum_{k=0}^{j-1} f[i-1][k]$.
The final answer is $\sum_{j=0}^n f[n][j]$.
The time complexity is $O(n^3)$, and the space complexity is $O(n^2)$. Here, $n$ is the length of the string.
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We can optimize the time complexity to $O(n^2)$ using prefix sums.
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Additionally, we can optimize the space complexity to $O(n)$ using a rolling array.
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