Collecting Chocolates

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class Solution:
    def minCost(self, nums: List[int], x: int) -> int:
        n = len(nums)
        result = []
        for k in range(n):
            result.append(x * k)

        for i in range(n):
            current = nums[i]
            for k in range(n):
                current = min(current, nums[i - k])
                result[k] += current

        return min(result)

“Collecting Chocolates” is a dynamic programming problem. Here are ten problems to prepare for it:

  1. 70. Climbing Stairs: This problem introduces you to the basics of dynamic programming, where you build up to the answer step by step.

  2. 198. House Robber: This problem is another introduction to dynamic programming. It involves a similar concept of optimizing a sequence of choices.

  3. 303. Range Sum Query - Immutable: This problem can help you understand how to create and use a prefix sum array, a common technique in dynamic programming problems.

  4. 322. Coin Change: This problem introduces the concept of dynamic programming in a more challenging context. It involves making optimal decisions about how to reach a target sum using a set of options.

  5. 509. Fibonacci Number: This problem provides a simple case of dynamic programming, which can help you get comfortable with the concept.

  6. 746. Min Cost Climbing Stairs: This problem involves a sequence of choices where each choice depends on the previous ones, similar to “Collecting Chocolates”.

  7. 121. Best Time to Buy and Sell Stock: This is a classic dynamic programming problem that can help you understand how to track and update the optimal solution as you go.

  8. 53. Maximum Subarray: This problem introduces Kadane’s algorithm, a technique closely related to dynamic programming.

  9. 139. Word Break: This problem requires you to track whether each index can be reached, a key technique in many dynamic programming problems.

  10. 416. Partition Equal Subset Sum: This problem is a more complex dynamic programming problem, where you need to track a range of sums.

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    def minCost(self, A: List[int], x: int) -> int:
        n = len(A)
        res = [x * k for k in range(n)]
        for i in range(n):
            cur = A[i]
            for k in range(n):
                cur = min(cur, A[i - k])
                res[k] += cur
        return min(res)

Problem Classification

This problem is a combinatorial optimization problem. It falls under the category of operations research or decision sciences. This is because it involves finding the best, or optimal, solution from a number of potential solutions under given constraints.

‘What’ Components:

  • An array of integers nums of size n, which represents the cost of collecting different chocolates.
  • Each chocolate has a unique type, represented by its index in the array.
  • An operation can be performed that changes the type of each chocolate to the next type, cycling back to the first type from the last type, at a cost equal to the chocolate’s original type.
  • The goal is to collect chocolates of all types at the minimum possible cost.

The problem can be classified as an optimization problem. Specifically, it can be seen as a problem of cyclic shift with cost consideration. It is a challenging problem that requires a comprehensive understanding of both array manipulation and cost optimization strategies. It could potentially be solved using dynamic programming or other advanced techniques to find the minimum cost of achieving a certain condition (collecting all types of chocolates). The solution should take into account the cyclical nature of the operations and their associated costs.

Clarification Questions

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

Identifying Problem Isomorphism

“Collecting Chocolates” can be mapped to “Jump Game II”.

In both these problems, the task is to reach the end of an array from the beginning, under certain constraints. In “Collecting Chocolates”, the player can only move right and needs to collect all the chocolates in the minimum amount of time, where time is the index multiplied by the number of chocolates. In “Jump Game II”, the player starts at the first index and can jump up to the number at that current index, and the aim is to reach the end in the minimum number of jumps.

The isomorphism is in the approach to the solution. Both problems can be solved using a greedy algorithm where at each step, you make the decision that looks best at the moment to reach the end of the array with the minimum cost.

The mapping is approximate. In “Jump Game II”, the goal is to minimize the number of jumps, while in “Collecting Chocolates”, the goal is to minimize the total time to collect all chocolates. Therefore, the specific conditions and goals are not identical.

“Jump Game II” is simpler as it only involves minimizing jumps. “Collecting Chocolates”, is more complex as it involves both collecting all chocolates and minimizing time, and the time for each move varies depending on the index and the number of chocolates.

Language Agnostic Coding Drills

  1. Dissecting the Code into Distinct Concepts:

    a. Basic Python Constructs: This includes creating a list, using a for loop, accessing and updating list elements. This is a fundamental concept in Python and most other programming languages.

    b. List Comprehension: The code initializes the res list with multiples of x using list comprehension. This is a slightly advanced Python concept, allowing for more concise and readable code when initializing lists based on a formula or condition.

    c. Nested Loops: The code contains nested for loops to iterate over all possible pairs of chocolates. Understanding how to control the flow of execution within nested loops is a fundamental concept.

    d. Arithmetic Operations: The code performs basic arithmetic operations, like multiplication and addition.

    e. Min Function: The code uses the min function to find the minimum cost among the current and previous costs.

  2. Coding Concepts or Drills in Order of Increasing Difficulty:

    a. Basic Python Constructs: This is a fundamental coding concept and is thus the easiest.

    b. Arithmetic Operations: Although this is also a fundamental concept, it requires a basic understanding of mathematics, making it slightly more difficult than basic Python constructs.

    c. List Comprehension: This concept requires a deeper understanding of Python syntax, as it is a more advanced way of initializing lists.

    d. Min Function: The min function is a built-in Python function. Understanding when and how to use it effectively can be a bit challenging.

    e. Nested Loops: Properly managing nested loops can be quite complex. It is important to keep track of the loop variables and understand how they interact.

  3. Problem-solving Approach:

The approach is to create a cost array with each index i representing the cost of performing i operations. Then, for each possible starting point in the array, find the minimum cost if starting at that point and perform operations from that starting point. The cost for each starting point is calculated by iterating over all the elements in the array (which is done in a circular manner) and for each iteration, updating the cost with the minimum value seen so far.

The Python drills corresponding to these concepts help in understanding and implementing the problem-solving approach. The ability to use basic Python constructs, arithmetic operations, and list comprehension is essential for setting up the problem and creating the initial cost array. Understanding the min function is crucial for comparing and choosing the minimum cost. Finally, the concept of nested loops is essential for iterating over all the pairs of chocolates and calculating the costs.

Targeted Drills in Python

  1. Python Drills for Each Concept:

    a. Basic Python Constructs: Write a Python script that creates a list, updates its elements, and uses a for loop to iterate over its elements. Here, you could create a list of integers and use a for loop to square each element in the list.

    b. Arithmetic Operations: Write a Python script that takes two numbers as input and performs basic arithmetic operations like addition, subtraction, multiplication, and division on them.

    c. List Comprehension: Write a Python script that uses list comprehension to generate a list of squares of all numbers from 0 to 9.

    d. Min Function: Write a Python script that takes a list of integers as input and uses the min function to find the smallest integer.

    e. Nested Loops: Write a Python script that uses nested for loops to generate all pairs of elements from a list.

  2. Drills for Specific Needs of the Problem:

    a. Understanding Modulo Arithmetic: Write a Python script that performs modulo operation on a set of numbers and understands its properties.

    b. Understanding Circular Iteration: Write a Python script that iterates over a list in a circular manner, i.e., after the last element, it goes back to the first element.

    c. Minimum Operation: Write a Python script that compares current element with the minimum element seen so far and updates the minimum.

  3. Merging These Drills for the Final Solution:

    a. Start with creating an array res using list comprehension and arithmetic operations. This array will represent the cost of performing ‘i’ operations where ‘i’ is the index.

    b. Next, use nested for loops to iterate over the array nums. The outer loop will iterate over all possible starting points, while the inner loop will iterate in a circular manner over the array from the starting point.

    c. Inside the inner loop, update the current minimum cost using the min function and arithmetic operations.

    d. Then, update the corresponding cost in the res array using arithmetic operations.

    e. After the nested loops, find the minimum cost from the res array using the min function and return it as the final solution.