Count Substrings Divisible By Last Digit

This problem challenges you to count the number of valid substrings where each substring is divisible by its last non-zero digit. Use dynamic programming to transition through states, efficiently solving it for large strings. This approach breaks down the problem into manageable subproblems by analyzing divisibility based on substring properties.

Problem Statement

You are given a string s consisting of digits. Your task is to return the number of substrings of s that are divisible by their non-zero last digit.

Note that a substring may contain leading zeros, but it must be divisible by its last non-zero digit to count as valid. For example, in the string 12936, substrings like 129 and 293 are invalid as they are not divisible by their last digit.

Examples

Example 1

Input: s = "12936"

Output: 11

Substrings "29" , "129" , "293" and "2936" are not divisible by their last digit. There are 15 substrings in total, so the answer is 15 - 4 = 11 .

Example 2

Input: s = "5701283"

Output: 18

Substrings "01" , "12" , "701" , "012" , "128" , "5701" , "7012" , "0128" , "57012" , "70128" , "570128" , and "701283" are all divisible by their last digit. Additionally, all substrings that are just 1 non-zero digit are divisible by themselves. Since there are 6 such digits, the answer is 12 + 6 = 18 .

Example 3

Input: s = "1010101010"

Output: 25

Only substrings that end with digit '1' are divisible by their last digit. There are 25 such substrings.

Constraints

• 1 <= s.length <= 105
• s consists of digits only.

Solution Approach

State Transition Dynamic Programming

Use dynamic programming to model the problem by considering each index and possible modulus states. Let dp[index][i][j] represent the number of subarrays s[start...index] such that s[start...index] % i == j. This helps to track divisibility conditions for every possible substring efficiently.

Efficiently Handle Leading Zeros

Substrings with leading zeros need careful consideration. You must ensure they are valid only if divisible by their last non-zero digit. Dynamic programming helps efficiently exclude these cases while still counting valid substrings.

Optimize Time Complexity

The approach must be optimized for time complexity since the input string can be as large as 100,000 characters. Ensure that the solution takes advantage of memoization or efficient state transitions to minimize the number of recalculations.

Complexity Analysis

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

The time and space complexity of this problem depend on the approach used. A naive solution might lead to high time complexity due to the number of substrings, but a dynamic programming solution reduces redundant calculations, potentially improving performance.

What Interviewers Usually Probe

• The candidate demonstrates an understanding of dynamic programming patterns and state transitions.
• The solution should address how leading zeros are handled properly in the divisibility check.
• The candidate should efficiently optimize the solution for large inputs, considering both time and space complexity.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to handle substrings with leading zeros correctly.
• Not considering edge cases where substrings end with '1' or other digits that are trivially divisible.
• Inefficient solutions that do not optimize for large inputs may lead to timeouts or excessive space usage.

Follow-up variants

• Extend the problem to count substrings divisible by any arbitrary digit, not just the last non-zero digit.
• Modify the problem to check divisibility for substrings of a specified length, instead of all substrings.
• Implement a solution that also checks if substrings can be divisible by any digit in their range, adding another layer of complexity.

FAQ

What is the main pattern used in solving the Count Substrings Divisible By Last Digit problem?

The main pattern is state transition dynamic programming, where you track the divisibility of substrings using a modular approach.

How do I handle leading zeros in this problem?

Leading zeros are handled by ensuring that only substrings divisible by their non-zero last digit are counted, avoiding invalid results.

How can I optimize the solution for large inputs in the Count Substrings Divisible By Last Digit problem?

Optimizing the solution involves using dynamic programming with memoization to avoid recalculating results for the same subproblems, reducing time complexity.

What are the common mistakes when solving this problem?

Common mistakes include not handling leading zeros properly and failing to optimize the solution for large inputs.

Can I apply the same dynamic programming approach to similar problems?

Yes, the state transition dynamic programming approach can be applied to other problems involving divisibility, string manipulation, and modulus calculations.