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Algorithm Practice: Analyze

1. Problem-solving strategies​

Prompt​

info

You only need to change the text inside Context

Context:

"""
Question Description:

There are several cards arranged in a row, and each card has an associated number of points. The points are given in the integer array cardPoints.
In one step, you can take one card from the beginning or from the end of the row. You have to take exactly k cards.
Your score is the sum of the points of the cards you have taken.
Given the integer array cardPoints and the integer k, return the maximum score you can obtain.

"""
Action:
"""
Help me use the folling problem-solving strategy, ensuring each strategy is under 100 words and contains no code:
    

  1. Suggest a brute force solution.
    2. 

  2. Provide three similar LeetCode questions.
    3. 

  3. Create three simpler versions of the question for example constraint Relaxation.
    4. 

  4. List some subproblems that seem already solved.
    5. 

  5. Break down the main problem into subproblems.
    6. 

  6. Use reverse thinking to propose a solution to a reverse version of the problem.
    7. 

"""

Response From AI Coach​

Brute Force Solution​
A brute force solution involves generating all possible combinations of taking ( k ) cards from the start and end of the array. For each combination, calculate the sum of points and keep track of the maximum score found. This approach has a time complexity of ( O(n^k) ), where ( n ) is the number of cards.

Similar LeetCode Questions​
    

  1. LeetCode 1006: Clumsy Factorial
    2. 

  2. LeetCode 1234: Replace the Substring for Balanced String
    3. 

  3. LeetCode 1423: Maximum Points You Can Obtain from Cards
    4. 

Simpler Versions​
    

  1. Take only one card (k=1).
    2. 

  2. All cards have the same point value.
    3. 

  3. You can take cards from only one end (either left or right).
    4. 

Subproblems Already Solved​
    

  1. Finding the maximum sum of a subarray.
    2. 

  2. Sliding window techniques for maximum sums.
    3. 

  3. Dynamic programming for optimal substructure problems.
    4. 

Breakdown of Main Problem into Subproblems​
    

  1. Calculate the maximum sum of points from the first ( k ) cards.
    2. 

  2. Calculate the maximum sum of points from the last ( k ) cards.
    3. 

  3. Combine results from both ends while ensuring exactly ( k ) cards are taken.
    4. 

Reverse Thinking Solution​
Consider the problem of minimizing the points from the remaining cards after taking ( k ). Calculate the sum of points from all cards and subtract the minimum sum of the ( n-k ) cards left. This approach efficiently finds the maximum score by focusing on minimizing what remains.

2. Ask for hint​

Sometimes, knowing the category of the problem may not be enough to solve it. When this happens, you can let the AI Coach use Socratic questioning to help you solve this problem.

Prompt​

Context:

"""
I don't know how to solve this problem.
"""
Requirements:
    

  • Do not give me the answer directly.
    • 

  • Use scocratic questioning.
    • 

Action:
"""
    

  • Ask me three question to help me sovle this problem
    • 

"""

Response from AI Coach​

Certainly! Let's use Socratic questioning to guide you towards solving the problem of finding two indices in an array whose values sum up to a given target. Here are three questions that can help you develop your understanding and approach:
    

  • Can you think of a way to track which numbers you have already seen as you go through the array, and how might this help you determine if the complement of a current number (i.e., target - current number) has been seen before? (This question aims to lead you toward considering the use of a data structure that allows for fast access and lookup, such as a hash table.)
    • 

  • If you were to sort the array first, how would you then use two pointers to find the two numbers that add up to the target? (This question encourages you to visualize the process and utility of sorting the array and then using two pointers to efficiently find the pair that meets the target sum.)
    • 

  • Considering the need for an efficient solution, what are the trade-offs between using a sorting method followed by the two-pointer technique versus using a hash table in terms of time and space complexity? (Here, you're prompted to analyze and compare the efficiency of different approaches, helping you to choose the most suitable method based on the problem's constraints and requirements.)
    • 

These questions are designed to help you think through the problem and consider various methods and their implications, enabling you to develop a deeper understanding and select the best approach for solving the problem.

3. Review your solution​

Try to write down a step-by-step solution based on your understanding (if you can't solve it, you can provide blank steps), and then let the AI Coach review your solution.

Prompt​

Context:
"""
Here is my solution:
    

  • Tags: Sorting, Two Pointers
    • 

  • Time Complexity: O(nlogn)
    • 

  • Space Complexity: O(1)
    • 

  • Steps:

      

    1. Sort the array from small to large.
      2. 

    2. Using two pointers, left pointers points to the first element and right pointer points to the last element.
      3. 

    3. If the sum of these elements equals to the target, return the index of these two pointers.
      4. 

    4. If the sum is larger than the target, decrease the right pointer by one.
      5. 

    5. If the sum is smaller than the target, increase the left pointer by one.
      6. 

    

    • 

"""
Requirements:
"""
    

  • Limit the number of steps within 5
    • 

  • Explain your solution with a testcase
    • 

  • Provide a better solution if you can, including tags, time complexity and space complexity of the solution.
"""
    • 

Action:
"""
    

  • Ignore the steps I've provided. Instead, through reading the problem, try to write down a step-by-step solution without code using the tag I provided.
    • 

  • Compare the differences between your step and mine. Point out my mistakes or where I'm missing.
    • 

"""

Response From AI Coach​

Your solution using the Sorting and Two Pointers approach is well-structured and follows a logical sequence to solve the problem. However, I'll provide a detailed explanation of a potentially more efficient approach using a Hash Table and compare it with your solution.
Alternative Solution Using Hash Table:
    

  • Tags: Hash Table
    • 

  • Time Complexity: O(n), as each element is processed exactly once.
    • 

  • Space Complexity: O(n), due to storing elements and their indices in the hash table.
    • 

  • Steps:

      

    1. Create an empty hash table to keep track of each number and its index as you iterate through the array.
      2. 

    2. Iterate over the array. For each element, calculate the complement (i.e., target - current element).
      3. 

    3. Check if the complement exists in the hash table. If it does, return the index of the current element and the index of the complement from the hash table as they sum up to the target.
      4. 

    4. If the complement is not found, store the current element and its index in the hash table.
      5. 

    5. Continue the process until a pair is found or the array is fully traversed.
      6. 

    

    • 

Test Case Example:
    

  • Input: array = [2, 7, 11, 15], target = 9
    • 

  • Expected Output: [0, 1] (since 2 + 7 = 9)
    • 

Using the Hash Table Approach:
    

  1. Hash Table is {}.
    2. 

  2. Element 2, Complement 7. Hash Table becomes {2: 0}.
    3. 

  3. Element 7, Complement 2. Found 2 in Hash Table. Return [0, 1].
    4. 

Comparison and Analysis, Your Solution (Sorting and Two Pointers):
    

  • Steps Correctness: You've outlined the steps correctly for the sorting and two-pointer technique. This method works well and is efficient for sorted arrays.
    • 

  • Efficiency: The time complexity of O(n log n) is primarily due to the sorting step. While efficient, it is not the fastest possible for this problem.
    • 

  • Space Complexity: Correctly noted as O(1) assuming an in-place sort.
    • 

Potential Issues:
    

  • Index Tracking: After sorting, the original indices of the elements are lost, which can be problematic if the original indices need to be returned. This requires additional steps to track indices, which were not mentioned in your steps.
    • 

Advantages of the Hash Table Method:
    

  • Faster Time Complexity: Avoids the sorting step and operates in linear time, making it more efficient for large inputs.
    • 

  • Simplicity and Direct Access: The hash table allows for immediate lookup of complement values, simplifying the logic and speeding up the solution process.
    • 

  • Handling of Indices: Directly maintains and utilizes the original indices without additional complexity.
    • 

In summary, while your method is valid and well-structured, the hash table approach offers a faster and potentially more straightforward solution, especially when considering the need to maintain original indices. This makes it particularly suitable for scenarios where time efficiency is critical.