Problem Description

In this problem, we have a string num which consists of concatenated positive integers. We're told that the original list of integers was written in a non-decreasing order and none of the integers had leading zeros. The challenge is to figure out how many different lists of positive integers could have been combined to form the initial string num.

Since the answer could be very large, we need to return it modulo 10^9 + 7.

To illustrate with an example, if the input string num is "327", there are two possible lists of non-decreasing integers: [3, 27] and [327]. Hence the function would return 2.

Intuition

Approaching this problem requires that we solve it piece by piece. We need to determine under what conditions we can split the string num into different integers such that they maintain a non-decreasing order. We can start from the first character and continue appending the following characters until we get a valid number, then process the remainder of the string in a similar manner.

Here's how we might think about the solution:

1. Dynamic Programming (DP): We can consider this problem as one that can be solved using dynamic programming. We try to build up the number of ways to form the string num from smaller subproblems based on the length of the last number used.

2. Longest Common Prefix (LCP): To maintain the non-decreasing order of numbers, we need a way to compare the numbers in num efficiently. The LCP (Longest Common Prefix) array can be used to store the length of the longest common prefix between two substrings of num. This helps us compare numbers quickly without generating all substrings, which would be inefficient.

3. Construction of DP Table: We construct a 2D DP table with dp[i][j] meaning the number of ways to form the first i digits of num where the last number used has j digits.

4. Transition: We consider two cases for the transition. Either the last two numbers used are the same length - in which case we can form a new list of integers by using the same length of the number only if the two numbers are in non-decreasing order. Or, the last number used is shorter than the previous - in which case you can always append it because it won’t violate the non-decrease constraint.

5. Base Case: We initiate dp[0][0] as 1 because there is one way to form an empty string with an empty list.

This solution iteratively builds upon smaller subproblems and combines their results to find the total number of combinations for larger lengths of the string num. At each step, it ensures that numbers are added in a non-decreasing order and without leading zeros.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution utilizes a nested function cmp to compare substrings of num, and a Dynamic Programming (DP) table to keep track of the number of valid combinations that can be formed up to certain points in the string num.

Let's walk through the components of the implementation:

1. Longest Common Prefix (LCP) Calculation: Before the main DP logic, the function computes the LCP array using a nested loop. For each pair of indices i and j in the string num, lcp[i][j] is calculated to store the length of the longest common prefix between substrings starting from i and j. This pre-computation is essential for the cmp function.

2. Comparison function (cmp): The function cmp(i, j, k) is used to compare two numbers within the string num of length k starting from i and j respectively. This function will indirectly utilize the precomputed LCP values to efficiently determine if the first number starting at index i is greater than or equal to the second number starting at index j.

3. Dynamic Programming: The 2D DP table dp is created to keep track of valid combinations. dp[i][j] represents the number of ways to partition num[0:i] such that the last number used has j digits.

4. DP Initialization: The dp table is initialized such that dp[0][0] is 1, since there is exactly one way to create an empty partition.

5. DP Iteration: The code iterates through each end index i of the substring and considers all possible lengths j for the last number. Inside this nested loop, the algorithm decides whether the substring of length j ending at i can form a valid number (it cannot start with '0') and if it maintains the non-decreasing property compared to the previous number.

6. DP Transition: The crucial part of this solution is how the DP values transition from one state to another. For each i and j, the implementation considers the number of combinations if the last two numbers are of equal length (and the second is greater or equal) or if the second number may length less.

   • If the last two numbers are of equal length and the comparison via cmp is true, it takes the value from dp[i-j][j] (which represents the count up to the first number's end index and the subsequent number also ending at i-j and also has length j).
   • Otherwise, it uses the value from dp[i-j][min(j-1,i-j)] to represent scenarios wherein the most recent number is of smaller size and doesn't threaten the non-decreasing order.

7. DP State Updates: After ascertaining which previous DP state can be transitioned from, dp[i][j] is updated with the count of combinations by considering the previous number lengths into account. It sums the count from the previously calculated number of ways and the new count, keeping in mind that we must perform this addition modulo 10^9 + 7.

8. Result Extraction: Lastly, after populating the DP table, the result is the total combinations possible which is dp[n][n], meaning the number of combinations to form the entire string with all possible last number lengths considered.

The solution is a well-crafted example of combining DP with string comparisons optimized by LCP to ensure polynomial time complexity and manage the large answer space creatively with modular arithmetic.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Suppose our input string num is "1123". We want to find out in how many different ways we can split this string into a list of non-decreasing positive integers.

Step 1: Longest Common Prefix (LCP) Calculation: We calculate the LCP for every pair of starting positions in the string num. For simplicity, let's assume we have a function that does this efficiently.

Step 2: Comparison function (cmp): This function will compare two numbers within the string num based on their starting indices and length using the LCP information.

Step 3: Dynamic Programming (DP): We initialize a DP table dp where dp[i][j] will represent the number of ways to split the substring num[0:i] such that the last number has length j.

Step 4: DP Initialization: We set dp[0][0] to 1 because there's one way to form an empty string which is an empty list of integers.

Step 5: DP Iteration:

• For i = 1, the substring "1" can form one list, [1].
• For i = 2, the substring "11" can form two lists, [1, 1] or [11].
• For i = 3, the substring "112" can form three lists, [1, 1, 2], [1, 12], or [11, 2].
• For i = 4, the substring "1123" can form the following lists: [1, 1, 2, 3], [1, 1, 23], [1, 12, 3], [1, 123], [11, 2, 3], [11, 23]. That's six valid lists.

Step 6: DP Transition:

• We check if we can append a number to the existing one to make sure they are in non-decreasing order.
• For instance, at i = 4, j = 2, we compare "23" with "12" (previous number of length 2). Since "23" >= "12", we can consider the count from dp[2][2].

Step 7: DP State Updates: We update the dp table by adding the number of ways to the current count, ensuring we consider each possible length of the last number.

Step 8: Result Extraction: We find the total count of combinations to form the string "1123", which will be dp[4][4] + dp[4][3] + dp[4][2] + ... + dp[4][1] after iterating through all possible lengths for the last number.

For this example, by walking through the DP table, we find that there are six valid lists that can be formed from "1123". This approach ensures we explore all possibilities iteratively while keeping the problem manageable using the principles of dynamic programming.

Solution Implementation

Time and Space Complexity

The time complexity of the provided algorithm is primarily determined by two nested loops that iterate over the length of the input string num and the computation of the longest common prefix (LCP) for all substrings. Calculating the LCP takes O(n^2) time since it requires two nested loops, each running for n iterations, where n is the length of the string num. Afterward, the main part of the algorithm uses dynamic programming where there are two nested loops that also run for n iterations each, thus also taking O(n^2) time. In the innermost part of these loops, the cmp function is called, which takes constant time due to the precomputed LCP values. Therefore, the dominant part of the time complexity is O(n^2).

The space complexity of the algorithm comes from the storage of the dynamic programming table dp, and the longest common prefix table lcp, each of size n x n. Therefore, the space complexity is O(n^2) to store these two tables.

Learn more about how to find time and space complexity quickly using problem constraints.