Number of Ways to Separate Numbers - Video Solutions

Problem Overview: You receive a numeric string and must count how many ways it can be split into a sequence of integers where each number is non‑decreasing compared to the previous one. Leading zeros are not allowed. The challenge is comparing many substrings efficiently while exploring valid partitions.

Approach 1: Recursive Approach with Memoization (Exponential → Pruned with DP)

This approach treats the problem as recursive partitioning of the string. At each index, choose a substring representing the next number and verify two rules: it cannot start with '0', and it must be greater than or equal to the previous number. The recursion continues from the next index. Memoization stores results based on the current index and previous substring length so repeated states are not recomputed. Even with caching, substring comparisons can be expensive, and the number of candidate splits grows quickly. Time complexity is roughly O(n^3) in the worst case due to substring comparisons, and space complexity is O(n^2) for memoization states and recursion stack. This method helps build intuition for how the partitioning works.

Approach 2: Dynamic Programming with Fast String Comparison (Optimal)

The optimized approach uses dynamic programming combined with precomputed substring comparison. Let dp[i][len] represent the number of valid ways to form a sequence ending at index i where the last number has length len. For each candidate number ending at i, you compare it with the previous number of the same or smaller length to ensure the non‑decreasing condition. Direct string comparison would make the solution too slow, so an LCP (longest common prefix) table is precomputed. This idea is closely related to structures used in suffix arrays and enables constant‑time substring comparison.

With LCP values, you can determine whether two substrings are lexicographically ordered without scanning characters repeatedly. Prefix sums over DP states allow fast transitions when the previous number length is smaller. Each position processes possible lengths while enforcing the no‑leading‑zero rule. The final answer is the sum of valid sequences that consume the entire string.

This reduces the overall complexity to O(n^2) time and O(n^2) space. The algorithm relies heavily on string processing and dynamic programming state transitions, which makes it the expected solution for this hard problem.

Recommended for interviews: Start by describing the recursive partition idea to show you understand the constraints and ordering rule. Then move to the dynamic programming solution with LCP‑based substring comparison. Interviewers typically expect the optimized O(n^2) DP approach because it demonstrates strong control over string comparison, state design, and performance optimization.