Count Numbers with Non-Decreasing Digits

Use digit dynamic programming to traverse each digit from most to least significant, tracking non-decreasing sequences. Handle bounds l and r carefully, applying modulo arithmetic to prevent overflow. This approach leverages state transitions efficiently for base b digits, ensuring correct counts without enumerating every number.

Problem Statement

Given two strings l and r representing integers and an integer b, compute the number of integers in the inclusive range [l, r] whose digits, when expressed in base b, never decrease from left to right. Return the result modulo 10^9 + 7.

An integer's digits are non-decreasing if each digit is greater than or equal to the previous digit reading from most significant to least significant. The input strings l and r contain no leading zeros, and l represents a value less than or equal to r.

Examples

Example 1

Input: l = "23", r = "28", b = 8

Output: 3

Example 2

Input: l = "2", r = "7", b = 2

Output: 2

Constraints

• 1 <= l.length <= r.length <= 100
• 2 <= b <= 10
• l and r consist only of digits.
• The value represented by l is less than or equal to the value represented by r.
• l and r do not contain leading zeros.

Solution Approach

Digit Dynamic Programming Setup

Define a DP function dp(pos, prevDigit, tight) representing the number of valid numbers from current position with previous digit prevDigit, respecting the upper bound if tight is true. Iterate through digits 0 to b-1 while maintaining non-decreasing property.

Handling Bounds and Modulo

Use separate calls for l and r to count numbers up to r and subtract numbers up to l-1. Apply modulo 10^9 + 7 at each step to prevent overflow and handle large ranges efficiently.

Optimization and State Pruning

Memoize states by position, previous digit, and tight flag to avoid redundant calculations. Prune impossible transitions where next digit is less than prevDigit to reduce computation.

Complexity Analysis

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

Time complexity is O(length * b * 2) due to position, previous digit, and tight states in DP. Space complexity is O(length * b * 2) for memoization table.

What Interviewers Usually Probe

• Watch if candidate recognizes digit DP and tight bounds for non-decreasing sequences.
• Check handling of base b conversions and modulo arithmetic.
• Look for memoization to avoid recomputation and reduce time complexity.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to respect the tight upper bound in digit DP leading to overcounting.
• Not applying modulo at each step causing integer overflow for large ranges.
• Incorrectly handling prevDigit for non-decreasing constraint, allowing invalid sequences.

Follow-up variants

• Count numbers with strictly increasing digits instead of non-decreasing.
• Find numbers with non-increasing digits in a given base.
• Count numbers with additional constraints like divisible by k while non-decreasing.

FAQ

What does 'non-decreasing digits' mean in this problem?

Digits never decrease from left to right; each digit is greater than or equal to the previous when read in base b.

How does digit dynamic programming apply here?

Digit DP tracks counts of valid sequences position by position, considering previous digit and bounds to ensure correctness.

Why do we need modulo 10^9 + 7?

Because the number of valid integers can be extremely large, modulo prevents overflow and ensures consistent output.

Can this approach handle different bases?

Yes, the DP iterates digits from 0 to b-1, making it adaptable to any base between 2 and 10.

What is the main challenge in Count Numbers with Non-Decreasing Digits?

The challenge is efficiently counting without enumerating all numbers while respecting the non-decreasing property and bounds using state transition DP.