Kth Smallest Number in Multiplication Table

To find the kth smallest element in the m x n multiplication table, we can use a binary search method. Here’s the code with explanation:

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class Solution:
    def findKthNumber(self, m: int, n: int, k: int) -> int:
        # The binary search will run on the range of values in the multiplication table.
        # Since it's 1-indexed, the minimum value is 1 and the maximum value is m * n.
        left, right = 1, m * n

        # Function to count how many numbers are less than or equal to x in the multiplication table.
        def count_less_equal(x: int) -> int:
            count = 0
            for i in range(1, m + 1):
                # For each row i, there are min(x // i, n) numbers less than or equal to x.
                count += min(x // i, n)
            return count

        # Binary search to find the kth smallest number.
        while left < right:
            mid = (left + right) // 2
            if count_less_equal(mid) < k:
                left = mid + 1
            else:
                right = mid

        return left

Explanation:

  • The count_less_equal function counts how many numbers in the m x n multiplication table are less than or equal to a given number x.
  • We perform binary search over the range [1, m * n], calculating how many numbers are less than or equal to the midpoint mid.
  • If the count is less than k, we know that the kth smallest number must be greater than mid, so we set left to mid + 1.
  • If the count is greater or equal to k, we know that the kth smallest number must be less than or equal to mid, so we set right to mid.
  • Finally, left will be the value of the kth smallest number in the multiplication table.

Identifying Problem Isomorphism

“668. Kth Smallest Number in Multiplication Table” involves identifying the Kth smallest number in a multiplication table of m * n, where every row and column is in ascending order. The problem utilizes a binary search method on the set of possible values in the multiplication table.

An analogous problem is “378. Kth Smallest Element in a Sorted Matrix” on LeetCode. In this problem, the task is to find the Kth smallest element in a row and column-wise sorted matrix. The problem also applies a binary search technique on the range of numbers found within the matrix.

Both center around the usage of binary search on a range of potential values to find a particular element, instead of applying it directly to a data structure. The difference is in the specifics of what comprises the ‘matrix’ (a multiplication table versus a literal matrix). Therefore, this is an approximate mapping.

Understanding these problems can provide valuable insights into the application of binary search in a non-traditional manner, beneficial for tackling similar problems.

10 Prerequisite LeetCode Problems

“668. Kth Smallest Number in Multiplication Table” involves binary search, sorting, and understanding properties of a multiplication table. Here are some problems to understand these concepts better:

  1. LeetCode 704. Binary Search

    • This problem will help you master the basics of binary search.

  2. LeetCode 33. Search in Rotated Sorted Array

    • This problem is a bit more advanced application of binary search, which will give you more confidence with this method.

  3. LeetCode 74. Search a 2D Matrix

    • This problem will give you practice using binary search in a 2D matrix, which is similar to the multiplication table in the main problem.

  4. LeetCode 378. Kth Smallest Element in a Sorted Matrix

    • This problem is a close analogue to the main problem, where instead of a multiplication table, you have a sorted matrix.

  5. LeetCode 153. Find Minimum in Rotated Sorted Array

    • This problem helps to understand the concept of modified binary search in which the properties of the array elements are used.

  6. LeetCode 162. Find Peak Element

    • This problem is about binary search on an array with unique properties, helping you prepare for the main problem.

  7. LeetCode 34. Find First and Last Position of Element in Sorted Array

    • This problem involves binary search and deals with finding specific elements in a sorted array, which will be helpful in the main problem.

  8. LeetCode 230. Kth Smallest Element in a BST

    • This problem helps to understand the concept of finding the Kth smallest element in a unique structure.

  9. LeetCode 540. Single Element in a Sorted Array

    • This problem further explores the idea of binary search where the search space has certain unique properties.

  10. LeetCode 35. Search Insert Position

    • This problem will help you understand how to perform a binary search when the target may not exist in the array.

By working through these problems, you should get a solid understanding of binary search and working with sorted structures, which are important concepts for the problem “668. Kth Smallest Number in Multiplication Table”.

Problem Classification

Problem Statement:Nearly everyone has used the Multiplication Table. The multiplication table of size m x n is an integer matrix mat where mat[i][j] == i * j (1-indexed).

Given three integers m, n, and k, return the kth smallest element in the m x n multiplication table.

Example 1:

Input: m = 3, n = 3, k = 5 Output: 3 Explanation: The 5th smallest number is 3.

Example 2:

Input: m = 2, n = 3, k = 6 Output: 6 Explanation: The 6th smallest number is 6.

Constraints:

1 <= m, n <= 3 * 104 1 <= k <= m * n

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 ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

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.

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.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. 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.

  1. 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.

  2. 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.

  3. 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.

  1. 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.

  2. 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.

  3. 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.

Q&A

Similar Problems

Given the problem , identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.

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class Solution:
    def findKthNumber(self, m: int, n: int, k: int) -> int:
        lo, hi = 1, m*n
        while lo < hi:
            mid, count = (lo+hi)//2, 0

            for i in range(1, m+1):
                count += n if n<mid//i else mid//i
            if count>=k:
                hi = mid
            else:
                lo = mid+1
        return lo

Problem Classification

Domain Categorization: This problem falls under the “Mathematics” and “Searching” domain in Computer Science.

Problem Components Analysis:

  1. ‘What’ Components:
    • The inputs to the problem are three integers: m, n, and k.
    • m and n are used to create an m x n multiplication table where the value at each cell is the product of its row and column indices (1-indexed).
    • k is the ranking of the element in the multiplication table that we need to return. If we were to flatten the table and sort it in ascending order, k would be the index we are interested in (1-indexed).

Classification of the Problem: This problem can be further classified as a “Binary Search” problem. Although the problem description doesn’t directly imply a binary search solution, the nature of the task (finding the kth smallest element in a sorted structure) is a common scenario where binary search can be applied efficiently. This is especially the case considering that generating the entire multiplication table might not be feasible for large m and n, and thus a more efficient approach is required.

The problem statement is about finding the Kth smallest number in a multiplication table of two numbers M and N. Based on this, we can classify this problem as follows:

  1. Discrete Mathematics/Number Theory: The problem deals with integers and their properties. It involves understanding of multiplication tables and ordering of numbers.

  2. Order Statistics: It requires finding the Kth smallest number in a specific order, which is a common theme in order statistics.

  3. Binary Search: Although this is slightly veering towards implementation, the problem inherently requires an efficient way to narrow down the potential answers, which falls under the technique of binary search in algorithmic problem solving.

  4. Array/Matrix Processing: Since we are dealing with a multiplication table which can be viewed as a 2D grid or matrix, the problem also belongs to this category.

These classifications are based on the problem domain and do not refer to any specific implementation details.

Language Agnostic Coding Drills

Concept 1: Binary Search The algorithm uses a binary search to find the Kth smallest number in the multiplication table. Binary search is an efficient algorithm for finding an item from a sorted list of items. It works by repeatedly dividing in half the portion of the list that could contain the item until you’ve narrowed down the possible locations to just one.

Concept 2: Iteration For each mid-point in the binary search, the algorithm runs a loop from 1 to m to count how many numbers in the multiplication table are less than or equal to mid. This is done by comparing n and mid//i and choosing the smaller one. If the number of elements less than or equal to mid is greater than or equal to k, then the mid-point is a potential answer.

Concept 3: Mathematical Operations The algorithm uses a couple of mathematical operations like floor division (to get the mid-point and to calculate count), multiplication (in calculating the high end of binary search), and comparison (in calculating count and updating low or high).

The algorithm starts by setting the low end of the binary search to 1 and the high end to m*n, the largest possible number in the table. Then it starts the binary search loop. For each mid-point, it calculates the number of elements that are less than or equal to mid. If the count is greater than or equal to k, it means the Kth smallest number is less than or equal to mid, so it updates the high end to mid. If the count is less than k, it means the Kth smallest number is greater than mid, so it updates the low end to mid+1. The loop continues until the low end is equal to the high end, at which point it returns the low end (or high end) as the answer.

Targeted Drills in Python

Drill 1: Binary Search The binary search algorithm is used to efficiently find a target value within a sorted array. In this problem, the array is not explicitly given, but the binary search is used to narrow down the search range for the Kth smallest number in the multiplication table.

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def binary_search_example():
    data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
    target = 7

    low = 0
    high = len(data) - 1

    while low <= high:
        mid = (low + high) // 2
        if data[mid] == target:
            return mid
        elif data[mid] < target:
            low = mid + 1
        else:
            high = mid - 1
    return -1

print(binary_search_example())

Drill 2: Counting in a loop This is an essential part of the problem, as for each mid-point in the binary search, a loop runs from 1 to m to count how many numbers in the multiplication table are less than or equal to mid.

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def count_less_equal_mid(m, n, mid):
    count = 0
    for i in range(1, m+1):
        count += min(n, mid//i)
    return count

print(count_less_equal_mid(3, 3, 6))  # returns: 8

Drill 3: Finding the kth smallest element With the help of the binary search and counting in a loop, we can now write a function that returns the kth smallest number in the m*n multiplication table.

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def find_kth_number(m, n, k):
    lo, hi = 1, m * n
    while lo < hi:
        mid = (lo + hi) // 2
        count = 0
        for i in range(1, m + 1):
            count += min(n, mid // i)
        if count < k:
            lo = mid + 1
        else:
            hi = mid
    return lo

print(find_kth_number(3, 3, 5))  # returns: 3

All of the drills build up to the final solution, where the function find_kth_number utilizes the concept of binary search (Drill 1) and counting within a loop (Drill 2) to find the kth smallest element in the multiplication table (Drill 3).