Count of Integers

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class Solution:
    def __init__(self):
        self.dp = [[[-1 for _ in range(2)] for _ in range(9 * 22)] for _ in range(23)]
        self.mod = 1000000007

    def dfs(self, i, sum, min_sum, max_sum, num, lim, equal):
        if i == 0:
            for x in range(len(num)):
                for y in range(9 * 22):
                    for z in range(2):
                        self.dp[x][y][z] = -1
        if i == len(num) or sum > max_sum:
            return sum >= min_sum and sum <= max_sum and (equal or not lim)
        if self.dp[i][sum][lim] == -1:
            for n in range(ord('0'), ord((num[i] if lim else '9')) + 1):
                char_n = chr(n)
                self.dp[i][sum][lim] = (max(0, self.dp[i][sum][lim]) + 
                    self.dfs(i + 1, sum + ord(char_n) - ord('0'), min_sum, max_sum, num, lim and char_n == num[i], equal)) % self.mod
        return self.dp[i][sum][lim]

    def count(self, num1, num2, min_sum, max_sum):
        return (self.mod + self.dfs(0, 0, min_sum, max_sum, num2, True, True) -
                self.dfs(0, 0, min_sum, max_sum, num1, True, False)) % self.mod

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

“Count of Integers” can be approximately mapped to “Find All Numbers Disappeared in an Array”.

Reasoning:

In both problems, you’re required to examine a collection of integers and derive some kind of count based on them.

In the “Count of Integers”, you might be asked to count specific integers, count the occurrence of certain digits, or perform some other operation that requires you to analyze the integers in some way.

In “Find All Numbers Disappeared in an Array”, you are examining an array of integers and counting the integers that are not present.

While the specific tasks are not exactly the same, both problems involve counting elements in a collection of integers, making them approximately isomorphic.

“Find All Numbers Disappeared in an Array” is simpler, because it doesn’t require any manipulation of the integers themselves - only an understanding of which integers are missing from the array. The complexity of “Count of Integers” can vary based on the specific conditions provided in the problem.

10 Prerequisite LeetCode Problems

This problem can be solved using dynamic programming and digit DP concept. The main idea is to construct the number from the most significant digit to the least significant digit, at the same time keeping track of the sum of the digits. If the sum is out of the range [min_sum, max_sum], you can stop early. The state of the dynamic programming can be the index of the current digit, the current sum, and two boolean variables indicating whether the current number is less than num1 and greater than num2.

Here are 10 problems for preparing to solve this problem:

902. Numbers At Most N Given Digit Set: This problem requires similar techniques to calculate the number of integers less than or equal to a given number.
1015. Smallest Integer Divisible by K: In this problem, you need to find the smallest number that is divisible by K and only contains the digit 1.
935. Knight Dialer: This is a dynamic programming problem where you count the number of different phone numbers of a certain length.
639. Decode Ways II: This is a dynamic programming problem that involves counting the number of ways to decode a string.
673. Number of Longest Increasing Subsequence: This problem involves counting the number of longest increasing subsequences.
368. Largest Divisible Subset: This problem involves dynamic programming and counting, where you find the largest subset such that every pair of elements is divisible.
62. Unique Paths: This is a dynamic programming problem where you count the number of unique paths in a grid.
494. Target Sum: This problem involves finding the number of ways to assign symbols to make the sum of numbers equal to target.
139. Word Break: This problem involves dynamic programming and counting, where you determine if the string can be segmented into a space-separated sequence of one or more dictionary words.
377. Combination Sum IV: In this problem, you’re counting the number of possible combinations that add up to a certain number.

Here are 9 problems that can help you develop the needed concepts:

91. Decode Ways: This problem helps with understanding dynamic programming and counting, with constraints based on the content of the digits.
70. Climbing Stairs: A simple problem for understanding basic counting in dynamic programming.
509. Fibonacci Number: Basic problem to understand dynamic programming.
322. Coin Change: This problem helps with understanding dynamic programming and constraints.
139. Word Break: This problem helps with understanding how to approach a problem using dynamic programming where previous solutions can affect future ones.
279. Perfect Squares: This problem introduces the concept of a number being represented as the sum of perfect square numbers.
152. Maximum Product Subarray: This problem helps with dynamic programming, especially dealing with arrays.
238. Product of Array Except Self: This problem involves manipulation of numbers in an array, similar to dealing with the digit sum in the “Count of Integers” problem.
560. Subarray Sum Equals K: This problem helps understand how to deal with sums in an array, which is somewhat similar to dealing with the sum of digits.

Problem Classification

Problem Statement:You are given two numeric strings num1 and num2 and two integers max_sum and min_sum. We denote an integer x to be good if:

num1 <= x <= num2 min_sum <= digit_sum(x) <= max_sum. Return the number of good integers. Since the answer may be large, return it modulo 109 + 7.

Note that digit_sum(x) denotes the sum of the digits of x.

Example 1:

Input: num1 = “1”, num2 = “12”, min_sum = 1, max_sum = 8 Output: 11 Explanation: 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 Explanation: 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

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.