#903

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Valid Permutations for DI Sequence

The problem asks to find the number of valid permutations for a given DI sequence string using dynamic programming and state transitions.

string dynamic programming prefix sum

Problem Statement

You are given a string s of length n, where each character is either 'I' or 'D'. The string represents a sequence of constraints for a permutation of integers 0 to n. 'I' means the number at that index must be greater than the one before it, and 'D' means the number at that index must be smaller.

Return the number of valid permutations of integers from 0 to n that satisfy the conditions represented by the string s. Since the result may be large, return it modulo 10^9 + 7.

Examples

Example 1

Input: s = "DID"

Output: 5

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

Example details omitted.

Constraints

n == s.length
1 <= n <= 200
s[i] is either 'I' or 'D'.

Solution Approach

Dynamic Programming Approach

This problem can be solved efficiently using dynamic programming. Create a dp array where dp[i][j] represents the number of valid permutations of length i with j values still available for use. The transitions depend on the current state of the sequence and whether the character in the string is 'I' or 'D'.

State Transition Modeling

State transitions are central to solving this problem. When a 'D' is encountered, consider the possible valid permutations by considering the decrementing constraints. Similarly, for an 'I', consider the valid ways to increment the numbers. This approach ensures that the constraints are maintained at every step of the transition.

Modulo Operations

Given the potential size of the answer, performing calculations modulo 10^9 + 7 is essential to avoid overflow and meet problem constraints. Ensure that each transition step and the final result respects the modulo operation.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this solution is O(n^2), where n is the length of the string s, as we must compute the transitions for every possible pair of elements. The space complexity is O(n^2) as well due to the dp table used to store the state of each step in the process.

What Interviewers Usually Probe

Candidate understands dynamic programming concepts and is able to apply them to a string-based problem.
The candidate can model state transitions based on string patterns like 'I' and 'D'.
The candidate is capable of handling large numbers with modulo arithmetic.

Common Pitfalls or Variants

Common pitfalls

Failing to properly handle the modulo operation, leading to overflow errors.
Not accounting for all valid transitions when switching between 'I' and 'D' sequences.
Overcomplicating the solution without recognizing that dynamic programming and state transitions can simplify the problem.

Follow-up variants

Modify the problem to work with permutations of a different set of integers, such as from 0 to n-1.
Introduce additional constraints, like allowing both 'I' and 'D' at the same index in the sequence.
Consider a similar problem where the goal is to find the number of valid permutations that satisfy multiple sequences of 'I' and 'D'.

FAQ

What is the approach to solving Valid Permutations for DI Sequence?

The problem is solved using dynamic programming with state transitions, ensuring that the conditions set by the DI sequence are met.

How can I optimize the solution for Valid Permutations for DI Sequence?

Focus on reducing unnecessary recalculations by properly utilizing dynamic programming and ensuring each transition step is calculated only once.

What is the time complexity of solving Valid Permutations for DI Sequence?

The time complexity is O(n^2), where n is the length of the sequence, due to the need to evaluate all possible transitions.

What is the modulo value used in Valid Permutations for DI Sequence?

The modulo value used in this problem is 10^9 + 7 to handle large numbers and prevent overflow.

What are some common mistakes when solving Valid Permutations for DI Sequence?

Common mistakes include mishandling the modulo operation, incorrect state transitions, and overcomplicating the dynamic programming setup.

terminal

Solution

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 $\sum_{j=0}^n f[n][j]$.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

class Solution:
    def numPermsDISequence(self, s: str) -> int:
        mod = 10**9 + 7
        n = len(s)
        f = [[0] * (n + 1) for _ in range(n + 1)]
        f[0][0] = 1
        for i, c in enumerate(s, 1):
            if c == "D":
                for j in range(i + 1):
                    for k in range(j, i):
                        f[i][j] = (f[i][j] + f[i - 1][k]) % mod
            else:
                for j in range(i + 1):
                    for k in range(j):
                        f[i][j] = (f[i][j] + f[i - 1][k]) % mod
        return sum(f[n][j] for j in range(n + 1)) % mod

Continue Practicing

#1871 jump game vii #902 numbers at most n given digit set #926 flip string to monotone increasing