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Number of Ways to Separate Numbers

You wrote down many positive integers in a string called num. However, you realized that you forgot to add commas to seperate the different numbers. You remember that the list of integers was non-decreasing and that no integer had leading zeros.

Return the number of possible lists of integers that you could have written down to get the string num. Since the answer may be large, return it modulo 109 + 7.

 

Example 1:

Input: num = "327"
Output: 2
Explanation: You could have written down the numbers:
3, 27
327

Example 2:

Input: num = "094"
Output: 0
Explanation: No numbers can have leading zeros and all numbers must be positive.

Example 3:

Input: num = "0"
Output: 0
Explanation: No numbers can have leading zeros and all numbers must be positive.

 

Constraints:

  • 1 <= num.length <= 3500
  • num consists of digits '0' through '9'.

Solution Explanation

This problem asks to find the number of ways to separate a given string num into a non-decreasing sequence of positive integers. The solution uses dynamic programming along with a Longest Common Prefix (LCP) array to efficiently check for the non-decreasing condition.

1. Longest Common Prefix (LCP) Array:

The lcp array is a crucial component. lcp[i][j] stores the length of the longest common prefix between the suffixes of num starting at indices i and j. This array is pre-computed using a nested loop iterating from the end of the string. This preprocessing step allows for efficient comparison of potential integers later.

2. Dynamic Programming:

The core of the solution is a dynamic programming approach. dp[i][j] represents the number of ways to separate the first i digits of num such that the last integer in the separation has length j.

  • Base Case: dp[0][0] = 1 (empty string has one way to be separated).

  • Iteration: The nested loops iterate through possible lengths j of the last integer and ending positions i in the string.

  • Transition: If the current digit num[i-j] is not '0' (to avoid leading zeros), it checks if adding a new integer of length j maintains the non-decreasing order. This is done using the cmp function which leverages the pre-computed lcp array. cmp(i - j, i - j - j, j) compares the newly formed integer with the previous integer. If the order is maintained, dp[i][j] is updated by adding dp[i-j][j] (the number of ways to separate the preceding part plus the new integer). If not, it falls back to dp[i-j][min(j-1, i-j)], considering the possibility that the previous integer was shorter.

3. Result:

Finally, dp[n][n] (where n is the length of num) stores the total number of ways to separate the entire string.

Time Complexity Analysis:

  • LCP Array Computation: O(n²) - nested loops iterating through all pairs of indices.
  • DP: O(n²) - nested loops iterating through all possible ending positions and integer lengths.
  • Total: O(n²) - dominated by the nested loops.

Space Complexity Analysis:

  • LCP Array: O(n²) - to store the LCP array.
  • DP Array: O(n²) - to store the DP table.
  • Total: O(n²)

Code Examples (Python, Java, C++, Go):

The code examples provided implement the described algorithm in different programming languages. They demonstrate the LCP array calculation, the dynamic programming approach, and the handling of modular arithmetic to avoid integer overflow. The key parts (like the cmp function and the DP transition) directly correspond to the explanation above.