#2478

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Number of Beautiful Partitions

The problem involves finding the number of beautiful partitions in a string with dynamic programming and state transitions.

string dynamic programming

Problem Statement

You are given a string s consisting of the digits '1' to '9', an integer k, and an integer minLength. You need to partition the string into exactly k parts such that each part has a length of at least minLength and no part has leading zeros. A partition is considered beautiful if these conditions are met.

Your task is to determine how many different ways you can partition s into exactly k beautiful parts. Since the result may be very large, return the answer modulo 10^9 + 7.

Examples

Example 1

Input: s = "23542185131", k = 3, minLength = 2

Output: 3

There exists three ways to create a beautiful partition: "2354 | 218 | 5131" "2354 | 21851 | 31" "2354218 | 51 | 31"

Example 2

Input: s = "23542185131", k = 3, minLength = 3

Output: 1

There exists one way to create a beautiful partition: "2354 | 218 | 5131".

Example 3

Input: s = "3312958", k = 3, minLength = 1

Output: 1

There exists one way to create a beautiful partition: "331 | 29 | 58".

Constraints

1 <= k, minLength <= s.length <= 1000
s consists of the digits '1' to '9'.

Solution Approach

Dynamic Programming with State Transitions

The solution involves using dynamic programming to calculate the number of ways to partition the string. At each step, consider every possible valid partition from the left of the string while keeping track of the number of valid partitions for each state.

Greedy Choices for Partitioning

To optimize the process, try using a greedy approach where you take as many digits as possible from the left for each partition. This simplifies the problem and works well when combined with dynamic programming.

Modulo Operation for Large Results

Since the number of partitions can be large, take the result modulo 10^9 + 7 at each step to ensure that the output fits within the required range.

Complexity Analysis

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

The time complexity of this solution depends on the implementation of the dynamic programming algorithm, typically O(n * k), where n is the length of the string. The space complexity also depends on the approach used, but it is typically O(n * k).

What Interviewers Usually Probe

Look for the candidate's understanding of dynamic programming with state transitions.
Check if the candidate suggests a greedy approach in combination with dynamic programming.
Evaluate if the candidate considers edge cases such as large strings and the modulo operation.

Common Pitfalls or Variants

Common pitfalls

Failing to handle partitions with leading zeros.
Overcomplicating the problem by not applying the greedy approach effectively.
Ignoring the modulo operation and producing numbers too large to handle.

Follow-up variants

Consider a variation where the number of partitions k is fixed, but the minimum length minLength varies.
Explore a scenario where the string contains only a single digit repeated, requiring careful handling of partitions.
Try solving the problem with a more restrictive set of partition conditions, such as ensuring each partition forms a valid number.

FAQ

What is the primary algorithm used in the "Number of Beautiful Partitions" problem?

The problem is solved using dynamic programming with state transitions.

How can a greedy approach help in this problem?

A greedy approach helps by taking as many digits as possible from the left for each partition, simplifying the decision-making process.

What should be considered when working with large results in this problem?

You should take the result modulo 10^9 + 7 to ensure that the numbers stay within the acceptable range.

How do you handle partitions with leading zeros in this problem?

You must ensure that no partition has a leading zero, as this would invalidate the partition.

What is the time complexity of the "Number of Beautiful Partitions" problem?

The time complexity is typically O(n * k), where n is the length of the string and k is the number of partitions.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i][j]$ as the number of schemes for dividing the first $i$ characters into $j$ sections. Initialize $f[0][0] = 1$, and the rest $f[i][j] = 0$.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

class Solution:
    def beautifulPartitions(self, s: str, k: int, minLength: int) -> int:
        primes = '2357'
        if s[0] not in primes or s[-1] in primes:
            return 0
        mod = 10**9 + 7
        n = len(s)
        f = [[0] * (k + 1) for _ in range(n + 1)]
        g = [[0] * (k + 1) for _ in range(n + 1)]
        f[0][0] = g[0][0] = 1
        for i, c in enumerate(s, 1):
            if i >= minLength and c not in primes and (i == n or s[i] in primes):
                for j in range(1, k + 1):
                    f[i][j] = g[i - minLength][j - 1]
            for j in range(k + 1):
                g[i][j] = (g[i - 1][j] + f[i][j]) % mod
        return f[n][k]

Continue Practicing

#2472 maximum number of non overlapping palindrome substrings #2484 count palindromic subsequences #2522 partition string into substrings with values at most k