Create Maximum Number

To solve this problem, we’ll have to find the maximum number of length k that can be formed from the digits of the two numbers. We’ll do this in a few steps:

Find the Maximum Number from a Single Array: Write a helper function to find the maximum number that can be formed from a single array with given length k.
Merge Two Arrays: Write another helper function to merge two arrays into the largest possible number.
Iterate through Lengths and Merge: Iterate through possible lengths for the first array and find the maximum merged array using the two functions from above.
Return the Result: Return the maximum merged array found in step 3.

Here’s the code implementation:

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class Solution:
    def maxNumber(self, nums1: List[int], nums2: List[int], k: int) -> List[int]:
        def max_single_number(nums, k):
            stack = []
            to_pop = len(nums) - k
            for num in nums:
                while to_pop > 0 and stack and stack[-1] < num:
                    stack.pop()
                    to_pop -= 1
                stack.append(num)
            return stack[:k]

        def merge(nums1, nums2):
            return [max(nums1, nums2).pop(0) for _ in range(len(nums1) + len(nums2))]

        result = []
        for i in range(max(0, k - len(nums2)), min(k, len(nums1)) + 1):
            current = merge(max_single_number(nums1, i), max_single_number(nums2, k - i))
            result = max(result, current)
        return result

Here’s a breakdown of the code:

The max_single_number function is used to get the maximum single number of length k from a given array.
The merge function takes two lists and merges them into a single maximum number.
Finally, we iterate through the possible lengths of the numbers taken from nums1, and use the helper functions to find the maximum number.

The time complexity of this code is O(k * (m + n)), where m and n are the lengths of nums1 and nums2, respectively, and the space complexity is O(m + n).

The problem is about creating the maximum possible number of length k from the given two integer arrays nums1 and nums2. The relative order of the digits from the same array must be preserved.

Python solution:

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from collections import deque

class Solution:
    def maxNumber(self, nums1: List[int], nums2: List[int], k: int) -> List[int]:
        # Function to find the maximum number of length n from a single list of numbers
        def max_single_number(nums, n):
            stack = []
            to_pop = len(nums) - n
            for num in nums:
                while to_pop > 0 and stack and stack[-1] < num:
                    stack.pop()
                    to_pop -= 1
                stack.append(num)
            return stack[:n]

        # Function to merge two lists of numbers to maintain the relative order
        def merge(nums1, nums2):
            return [max(nums1, nums2).pop(0) for _ in range(len(nums1) + len(nums2))]

        max_result = []
        # Iterate for different lengths of numbers taken from nums1 and nums2
        for i in range(max(0, k - len(nums2)), min(k, len(nums1)) + 1):
            current = merge(max_single_number(nums1, i), max_single_number(nums2, k - i))
            max_result = max(max_result, current)
        return max_result

The code defines a class Solution with a method maxNumber that takes in two integer arrays and an integer k. It returns the maximum number of length k by using the numbers from both arrays while preserving their relative order.

max_single_number function is used to get the maximum number of a certain length from a single list.
merge function is used to combine two lists of numbers while maintaining their relative order.
The main part of the code iterates through different combinations of lengths for numbers taken from nums1 and nums2, and finds the maximum possible combination.

It should return the result for the given examples and meet the constraints.

Identifying Problem Isomorphism

“Create Maximum Number” has a simpler version “Largest Number”.

In “Create Maximum Number”, you’re given two arrays of digits and are tasked with forming the maximum possible number of a certain length, using elements from both arrays. The relative order of the digits from each individual array must be maintained.

“Largest Number” also involves creating the largest possible number, but from a single array of non-negative integers. The task here is to rearrange the integers in such a way that when they are concatenated together, they form the largest possible number.

In both problems, the main objective is to arrange numbers in a way that results in the largest possible number. However, “Create Maximum Number” adds an extra level of complexity as you have to deal with two arrays and maintain the relative order of the digits from each array.

Thus, if you understand the simpler problem “Largest Number”, it can be a stepping stone to solving the more complex problem “Create Maximum Number”.

10 Prerequisite LeetCode Problems

The problem “321. Create Maximum Number” involves understanding of array manipulations and greedy algorithms. Here are 10 problems to prepare for this problem:

These cover how to handle different array manipulations and use greedy algorithms to solve complex problems.

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class Solution:
    def maxNumber(self, nums1: List[int], nums2: List[int], k: int) -> List[int]:        
        def merge(n1, n2):
            res = []
            while (n1 or n2) :
                if n1>n2:
                    res.append(n1[0])
                    n1 = n1[1:]
                else:
                    res.append(n2[0])
                    n2 = n2[1:]
            return res

        def findmax(nums, length):
            l = []
            maxpop = len(nums)-length
            for i in range(len(nums)):
                while maxpop>0 and len(l) and nums[i]>l[-1]:
                    l.pop()
                    maxpop -= 1
                l.append(nums[i])
            return l[:length]

        n1 = len(nums1)
        n2 = len(nums2)
        res = [0]*k
        for i in range(k+1):
            j = k-i
            if i>n1 or j>n2:    continue
            l1 = findmax(nums1, i)
            l2 = findmax(nums2, j)
            res = max(res, merge(l1,l2))
        return res

Problem Classification

Language Agnostic Coding Drills

The Python code uses a greedy strategy to solve the problem of finding the maximum number that can be formed from two input arrays. Let’s break down the main concepts used in this code. Each concept will be followed by a coding drill you can implement separately before combining them into the final solution.

Creating custom functions for reusable operations: The merge and findmax functions are implemented to perform certain tasks multiple times in the main function. You need to understand how to define and use functions in your chosen programming language.
Array Manipulation: The code involves accessing and manipulating arrays, so it’s important to understand the basics of arrays/lists in your chosen programming language. This includes accessing elements, array slicing, and modifying elements.
Loops and conditionals: Understanding the usage of for-loops, while-loops, and if-statements is crucial here. The findmax function uses a while-loop inside a for-loop, and the main function uses a for-loop with if-statements inside it.
Greedy Strategy: The problem is solved using a greedy approach, which involves making the locally optimal choice at each step with the hope that these local choices lead to a global optimum. The findmax function is a practical implementation of this strategy.
Comparisons and Sorting: Comparing lists and choosing the maximum one is a central part of the problem. It’s crucial to understand how comparisons work in your chosen programming language, as well as how to sort data.

Problem-solving approach:

Given two input arrays and a number ‘k’, the task is to create the largest number possible of length ‘k’ using digits from the input arrays.

The problem is approached by dividing it into smaller tasks - finding the maximum number from a single array (using the findmax function), and then merging these maximum numbers in a way that results in the largest number possible (using the merge function).
The findmax function uses a greedy strategy to keep the maximum number at every step while ensuring that there are enough remaining elements to reach the desired length.
For every possible division of ‘k’ between the two arrays, it finds the maximum number for each array and then merges them.
The solution keeps track of the maximum result seen so far, and finally, it returns this maximum number.

The final solution involves combining all these drills and using the learned concepts to implement the merge and findmax functions. Once these functions are properly implemented, you can easily solve the main problem using a for-loop and if-statements, as shown in the provided code.

Targeted Drills in Python

The key concepts and operations involved in this problem include array manipulation, function definition and invocation, loops and conditional statements, greedy algorithm, and comparisons.

Drill 1 - Creating custom functions for reusable operations: Define a function called merge that takes two lists as input and merges them into a single list. It should compare the input lists at every step and append the larger element to the output list.

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def merge(n1, n2):
    res = []
    while n1 or n2:
        if n1 > n2:
            res.append(n1[0])
            n1 = n1[1:]
        else:
            res.append(n2[0])
            n2 = n2[1:]
    return res

Drill 2 - Function to find the maximum sublist of a specific length: Define a function called findmax that takes a list and an integer as input, and returns the maximum sublist of the given length.

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def findmax(nums, length):
    l = []
    maxpop = len(nums) - length
    for i in range(len(nums)):
        while maxpop > 0 and len(l) and nums[i] > l[-1]:
            l.pop()
            maxpop -= 1
        l.append(nums[i])
    return l[:length]

Drill 3 - Implement a for-loop and use if-statements to determine how to divide k: Using a for-loop, iterate over the range k+1 and store k-i in j. Then use if-statements to check whether i or j exceed n1 or n2 respectively.

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n1 = len(nums1)
n2 = len(nums2)
res = [0]*k
for i in range(k+1):
    j = k-i
    if i>n1 or j>n2:    continue

Drill 4 - Call findmax and merge functions and update res accordingly: For each i and j from the previous drill, call findmax on nums1 and nums2, then call merge on the two results, and update res if the result of merge is larger.

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l1 = findmax(nums1, i)
l2 = findmax(nums2, j)
res = max(res, merge(l1, l2))

After practicing and understanding these drills, you can combine them into the final solution. Make sure to implement them in the correct order and with the right parameters.

Unit Tests

The following are the test cases that will test various conditions of the problem.

Test Case 1: Test with smallest possible inputs.

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def test_case_1():
    solution = Solution()
    assert solution.maxNumber([1], [2], 1) == [2]

Test Case 2: Test where numbers in both arrays are the same.

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def test_case_2():
    solution = Solution()
    assert solution.maxNumber([1, 2, 3], [1, 2, 3], 3) == [3, 2, 3]

Test Case 3: Test where the maximum possible number can be formed by picking numbers from both arrays.

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def test_case_3():
    solution = Solution()
    assert solution.maxNumber([6, 7], [6, 0, 4], 5) == [6, 7, 6, 0, 4]

Test Case 4: Test with larger lists of numbers.

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def test_case_4():
    solution = Solution()
    assert solution.maxNumber([3, 4, 6, 5], [9, 1, 2, 5, 8, 3], 5) == [9, 8, 6, 5, 3]

Test Case 5: Test with both arrays containing single digit numbers in random order.

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def test_case_5():
    solution = Solution()
    assert solution.maxNumber([6, 0, 7, 6, 5, 7, 6, 2, 0, 1], [4, 8, 1, 9, 2, 8, 8, 2], 15) == [7, 6, 2, 0, 1, 4, 8, 1, 9, 2, 8, 8, 2, 0, 1]

Test Case 6: Test with one array being empty.

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def test_case_6():
    solution = Solution()
    assert solution.maxNumber([], [6, 0, 4], 2) == [6, 0]

To run these tests, you would use a testing framework like pytest. First, you would need to import the Solution class at the top of your test file (from solution import Solution if your solution is in a file named solution.py). Then you can define each of these functions. To run the tests, you would use the command pytest test_file.py, replacing test_file.py with the name of your test file.

Acceptance tests are typically higher-level tests that cover a broad scope of the application and are meant to evaluate the system’s compliance with the business requirements and the needs of the stakeholders. In this context, they should test the overall behavior of the maxNumber function, typically by using more complex and real-world scenarios.

For this function, acceptance test can be like:

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def test_acceptance():
    solution = Solution()
    assert solution.maxNumber([3, 4, 6, 5], [9, 1, 2, 5, 8, 3], 5) == [9, 8, 6, 5, 3]
    assert solution.maxNumber([6, 7], [6, 0, 4], 5) == [6, 7, 6, 0, 4]
    assert solution.maxNumber([6, 0, 7, 6, 5, 7, 6, 2, 0, 1], [4, 8, 1, 9, 2, 8, 8, 2], 15) == [7, 6, 2, 0, 1, 4, 8, 1, 9, 2, 8, 8, 2, 0, 1]

These acceptance test cases are checking if the function can handle arrays of different lengths and structures, and return the correct maximum number possible. They are designed to test the functionality of the solution in a way that is close to how it would be used in a real-world scenario.

Do note that it’s important to have a wide range of test cases - testing not only expected and normal operation, but also edge cases and potential points of failure. Acceptance tests are often supplemented with unit tests that test specific aspects of the functionality in more detail.

10 Prerequisite LeetCode Problems

The problem “321. Create Maximum Number” is related to Greedy Algorithms, Dynamic Programming, and understanding of Monotonic Stack concept. Here are 10 simpler problems to prepare:

455. Assign Cookies: This problem requires a basic understanding of greedy algorithms.
392. Is Subsequence: This problem can help you understand the concept of subsequences.
406. Queue Reconstruction by Height: A problem involving ordering and placement based on multiple factors.
334. Increasing Triplet Subsequence: This problem is about identifying a subsequence of a certain property, which is a skill useful for this problem.
316. Remove Duplicate Letters: This problem requires greedy selection and maintaining an optimal substructure while removing characters, which is a good practice for the problem in question.
402. Remove K Digits: A problem with a similar flavor to the main problem, where you need to remove elements to achieve a certain objective.
376. Wiggle Subsequence: This problem helps in understanding how to choose elements from an array to form a sequence with maximum length.
714. Best Time to Buy and Sell Stock with Transaction Fee: This problem combines dynamic programming with a greedy approach.
907. Sum of Subarray Minimums: A problem that requires understanding of stack data structure and a flavor of dynamic programming.
739. Daily Temperatures: This problem requires understanding of Monotonic Stack concept.

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Problem Statement:You are given two integer arrays nums1 and nums2 of lengths m and n respectively. nums1 and nums2 represent the digits of two numbers. You are also given an integer k.

Create the maximum number of length k <= m + n from digits of the two numbers. The relative order of the digits from the same array must be preserved.

Return an array of the k digits representing the answer.

Example 1:

Input: nums1 = [3,4,6,5], nums2 = [9,1,2,5,8,3], k = 5 Output: [9,8,6,5,3] Example 2:

Input: nums1 = [6,7], nums2 = [6,0,4], k = 5 Output: [6,7,6,0,4] Example 3:

Input: nums1 = [3,9], nums2 = [8,9], k = 3 Output: [9,8,9]

Constraints:

m == nums1.length n == nums2.length 1 <= m, n <= 500 0 <= nums1[i], nums2[i] <= 9 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.

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

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.

Provide names by categorizing these cases

What are the edge cases?

What are the key insights from analyzing the different cases?

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?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

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

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

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

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

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

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

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.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

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?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

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.

Q&A

What are the reasons for making these mistakes in the given code?

Create Maximum Number

Identifying Problem Isomorphism

10 Prerequisite LeetCode Problems

Problem Classification

Language Agnostic Coding Drills

Targeted Drills in Python

Unit Tests

10 Prerequisite LeetCode Problems

Problem Analysis and Key Insights

Problem Boundary

Problem Classification

Distilling the Problem to Its Core Elements

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Identify Invariant

Identify Loop Invariant

Thought Process

Establishing Preconditions and Postconditions

Problem Decomposition

From Brute Force to Optimal Solution

Code Explanation and Design Decisions

Coding Constructs

Language Agnostic Coding Drills

Targeted Drills in Python

Q&A

Similar Problems