Problem Description
You wrote down many positive integers in a string called num. However, you realized that you forgot to add commas to separate 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 10^9 + 7.
Key Insights
- The numbers must be positive integers.
- Numbers cannot have leading zeros unless they are the number '0' itself.
- The integers must be in non-decreasing order.
- The problem can be approached using dynamic programming to count valid partitions of the string.
Space and Time Complexity
Time Complexity: O(n^2), where n is the length of the string num, due to nested iterations for checking valid partitions.
Space Complexity: O(n), for storing the dynamic programming results.
Solution
The solution involves using a dynamic programming approach. We maintain a DP array where dp[i] represents the number of ways to partition the substring num[0:i]. We iterate through each end index i of the substring and check all possible starting indices j. For each valid partition num[j:i], we ensure that it does not have leading zeros and that it is greater than or equal to the last number added to the partition. We update our DP array accordingly.
The algorithm checks for valid partitions by comparing current segments of the string with the last added number in the partition to maintain the non-decreasing order.