#2719

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Count of Integers

Count of Integers challenges you to find the number of integers with a digit sum between a given range using dynamic programming.

math string dynamic programming

Problem Statement

You are given two numeric strings, num1 and num2, along with two integers, min_sum and max_sum. You need to determine how many integers between num1 and num2 (inclusive) have a sum of digits that falls between min_sum and max_sum, both inclusive. An integer x is considered good if its digit sum lies within this range.

To solve this, you must calculate the count of valid integers by checking all possibilities in the range. As the result could be large, return the count modulo 10^9 + 7. The challenge lies in efficiently evaluating all possible integers without directly iterating over every possibility.

Examples

Example 1

Input: num1 = "1", num2 = "12", min_sum = 1, max_sum = 8

Output: 11

There are 11 integers whose sum of digits lies between 1 and 8 are 1,2,3,4,5,6,7,8,10,11, and 12. Thus, we return 11.

Example 2

Input: num1 = "1", num2 = "5", min_sum = 1, max_sum = 5

Output: 5

The 5 integers whose sum of digits lies between 1 and 5 are 1,2,3,4, and 5. Thus, we return 5.

Constraints

1 <= num1 <= num2 <= 1022
1 <= min_sum <= max_sum <= 400

Solution Approach

Dynamic Programming Setup

Define a function f(n, l, r) which represents the count of numbers between 1 and n with a digit sum between l and r. This approach avoids directly iterating over each number by breaking down the problem into manageable subproblems using dynamic programming.

Digit Sum Transitions

Use state transition to process each digit of the numbers. Starting from the most significant digit, compute the possible transitions based on the digit sum constraints. This allows us to compute the count of numbers efficiently while ensuring the sum of digits remains within the required bounds.

Modulo Optimization

Since the result may be large, use modulo 10^9 + 7 to prevent overflow and ensure the answer fits within standard integer limits. This step is crucial in both the dynamic programming approach and when returning the final answer.

Complexity Analysis

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

The time complexity is dependent on the approach used for dynamic programming. It generally involves processing each digit of the numbers in the range, leading to a time complexity of O(d * range), where d is the number of digits and range is the difference between the bounds. Space complexity is similarly affected by the state space of the dynamic programming table, typically O(d * range).

What Interviewers Usually Probe

Candidate's ability to define and implement dynamic programming states is crucial.
Look for a strong understanding of digit-based state transitions.
Ensure the candidate handles large numbers correctly using modulo operations.

Common Pitfalls or Variants

Common pitfalls

Not optimizing the solution using dynamic programming, leading to a time complexity that is too high.
Incorrectly calculating the sum of digits or failing to properly track the transition states between digits.
Failing to apply modulo 10^9 + 7 correctly, potentially causing overflow or incorrect results.

Follow-up variants

Variant where the sum of digits can only be calculated for specific subranges of num1 and num2.
Variant where num1 and num2 are very large strings, requiring optimizations beyond basic dynamic programming.
Variant with additional constraints on the maximum sum of digits, which would require more advanced state transitions.

FAQ

What is the key dynamic programming pattern used in the Count of Integers problem?

The problem leverages state transition dynamic programming to count numbers within a given range that satisfy the digit sum constraints.

How do I optimize the Count of Integers solution to handle large inputs?

By applying dynamic programming with state transitions over the digits of num1 and num2, and using modulo 10^9 + 7 to manage large results.

What is the time complexity of the Count of Integers problem?

The time complexity is generally O(d * range), where d is the number of digits in the largest number and range is the difference between num1 and num2.

How does GhostInterview help with the Count of Integers problem?

GhostInterview provides targeted assistance with the state transition approach and helps ensure correctness with dynamic programming techniques for counting valid numbers.

What are some common pitfalls when solving the Count of Integers problem?

Common pitfalls include failing to optimize with dynamic programming, incorrect digit sum calculations, and neglecting to apply modulo 10^9 + 7 correctly.

terminal

Solution

Solution 1: Digit DP

The problem is actually asking for the number of integers in the range $[num1,..num2]$ whose digit sum is in the range $[min\_sum,..max\_sum]$. For this kind of range $[l,..r]$ problem, we can consider transforming it into finding the answers for $[1,..r]$ and $[1,..l-1]$, and then subtracting the latter from the former.

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class Solution:
    def count(self, num1: str, num2: str, min_sum: int, max_sum: int) -> int:
        @cache
        def dfs(pos: int, s: int, limit: bool) -> int:
            if pos >= len(num):
                return int(min_sum <= s <= max_sum)
            up = int(num[pos]) if limit else 9
            return (
                sum(dfs(pos + 1, s + i, limit and i == up) for i in range(up + 1)) % mod
            )

        mod = 10**9 + 7
        num = num2
        a = dfs(0, 0, True)
        dfs.cache_clear()
        num = str(int(num1) - 1)
        b = dfs(0, 0, True)
        return (a - b) % mod

Continue Practicing

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