Smallest Rotation with Highest Score

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class Solution:
    def bestRotation(self, nums: List[int]) -> int:
        n = len(nums)
        change = [1] * n

        for i in range(n):
            change[(i - nums[i] + 1) % n] -= 1

        for i in range(1, n):
            change[i] += change[i - 1]

        return change.index(max(change))

Problem Classification

This problem belongs to the domain of “Arrays and Rotation”. It includes the operations on an array like rotation and scoring points based on a specific condition.

The ‘What’ components include:

  1. An input array of non-negative integers ’nums’.
  2. The rotation of this array by a non-negative integer ‘k’.
  3. The calculation of points for each rotation based on a given condition: any entry that is less than or equal to its index earns one point.
  4. Returning the rotation index ‘k’ which yields the highest score, and if there are multiple such indices, returning the smallest index.

This problem can be further categorized as a “Search” problem since we are searching for a specific rotation that maximizes the score. It also contains elements of “Combinatorial Search” as we are examining all possible rotations (combinations).

It could also be viewed as a “Simulation” problem because we rotate the array and compute the scores following the given rules, which simulates a specific process.

To summarize, the problem involves array manipulations, search, simulation and possibly optimization (finding max score) tasks. It is a complex problem that combines these different concepts, requiring a good understanding of arrays and algorithmic thinking to solve.

Thought Process

The problem asks us to find a rotation index ‘k’ for a given array such that it yields the maximum score. For each rotation, elements less than or equal to their index will be considered for scoring.

Approach

A naive solution would be to compute the scores for all possible rotations and return the rotation index that gives the maximum score. However, this could be time-consuming for large arrays. Therefore, we aim to devise an efficient approach.

We observe that rotating the array essentially moves elements from the end of the array to the front, and each rotation increases the indices of the elements still at their original place.

We start by finding the initial score (before any rotations). Then, we go through the array from left to right, rotating it one step at a time. For each rotation, we update the score: We subtract 1 if the element leaving the window (nums[i]) is less than or equal to its old index (i), and we add 1 if the element entering the window (nums[i]) is less than or equal to its new index (i).

The problem is asking to find the rotation index k that provides the maximum score from an array, where each element that is less than or equal to its index after rotation scores a point. The input array will have at most 20000 elements, hinting that we should aim for a solution with linear time complexity, O(N).

The score changes in the following way when k increments:

  • Gain Point: For every rotation, the element at index 0 moves to index N-1, gaining one additional point. This is certain and doesn’t need explicit calculation.

  • Lose Point: An element A[i] loses a point when k = (i - A[i] + 1) % N, because this makes A[i]’s index greater than A[i]. So, we need to record these k values where we start to lose points.

  • Elements with value A[i] = 0 always gain points, since 0 is always less than or equal to the index.

The approach to solve the problem is as follows:

  1. Identify and record the k values where score decrement happens into a list named change.

  2. Use a simple for loop to calculate the score for every possible k value. Here, score[K] = score[K-1] + change[K]. The changes are accumulated to get the changed score for each k value relative to k=0.

  3. Find the index of the best score, which is the required rotation index k. If there are multiple indices with the same best score, return the smallest one.

Code

Here is the Python code based on the above approach:

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def bestRotation(nums):
    n = len(nums)
    change = [1] * n
    for i in range(n):
        change[(i - nums[i] + 1) % n] -= 1
    for i in range(1, n):
        change[i] += change[i - 1]
    return change.index(max(change))

The function bestRotation first initializes a list change to keep track of the score changes. Then, for each element in nums, it decreases the corresponding score change value where the element would cause the score to decrease if the array were rotated. Next, it calculates the prefix sum of change to get the actual score change for each rotation. Finally, it returns the index with the maximum score change.

This approach ensures that we only need to traverse the array twice, making it an O(n) solution. This problem is a great example of how understanding the nature of the problem and careful observations can lead to efficient solutions.

Language Agnostic Coding Drills

  1. Dissecting the Code

The code contains several distinct concepts:

  • List manipulation: This involves creating lists, manipulating them and understanding list properties.
  • Looping: The code involves iterating over elements in a list.
  • Modulo operation: It is used to make sure indices are within bounds.
  • Indexing: Retrieving and modifying values at certain positions in the list.
  • List methods: Using built-in list methods like index() and max() to find the maximum value and its index.
  1. Ordering Coding Concepts

a. List manipulation: This is a basic concept, the foundation for solving problems involving sequences or arrays.

b. Indexing: Slightly more advanced than basic list manipulation, as it requires understanding how indices work in a list.

c. Looping: This is a core concept in programming that allows you to perform an action for each item in a list.

d. Modulo operation: The modulo operation, though mathematically simple, may be somewhat confusing when used in combination with other operations like it is in this case.

e. List methods: Knowledge about built-in methods is not hard but requires familiarity with Python. Understanding how and when to use these methods can significantly simplify your code.

  1. Problem-Solving Approach

The given problem is about finding a rotation of a given list that maximizes a certain score. This score is defined as the number of elements which are smaller or equal to their index. The problem can be solved by tracking the changes in score for every possible rotation, and then finding the rotation that gives the maximum score.

The coding drills help in implementing this approach as follows:

  • List manipulation and indexing are used to initialize a change list and update its values based on the elements of nums.
  • Looping is used to iterate over the nums list and update the change list.
  • The modulo operation is used to wrap around the indices when updating the change list, simulating the rotation of the list.
  • Finally, the max() and index() methods are used to find the maximum score and the rotation that gives this score.

Using these drills, the problem can be solved in a step-by-step manner. The first step involves initializing the change list and updating its values based on nums. Then for each possible rotation, we calculate the change in score and add it to the current score. Finally, we find the rotation with the maximum score by finding the index of the maximum value in the change list.

Targeted Drills in Python

  1. Coding Drills for Identified Concepts

    a. List Manipulation

    Python provides a wide range of operations that can be performed on lists. Here, we demonstrate the creation of a list and the assignment of elements to it.

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    # Create a list
my_list = [1, 2, 3, 4, 5]
print(my_list)  # Output: [1, 2, 3, 4, 5]

    b. Indexing

    In Python, you can access any item by using its index. Indexing in Python starts from 0.

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    # Access the first element
print(my_list[0])  # Output: 1

    c. Looping

    Looping is used to iterate over a sequence. Here we are using a for loop to iterate over our list.

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    for i in my_list:
    print(i)

    d. Modulo Operation

    The modulo operator (%) is used to get the remainder of a division.

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    print(10 % 3)  # Output: 1

    e. List Methods

    Python provides many methods for list manipulation. Here we are demonstrating the use of index() and max() methods.

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    # index() method
print(my_list.index(3))  # Output: 2

# max() method
print(max(my_list))  # Output: 5

  2. Problem-Specific Concepts

    For this problem, we also need to understand how to rotate a list, which involves removing elements from the end and adding them to the front.

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    # Rotation of a list
k = 2
rotated_list = my_list[-k:] + my_list[:-k]
print(rotated_list)  # Output: [4, 5, 1, 2, 3]

    This is crucial for our problem as we are required to find the best rotation that can maximize our score.

  3. Assembling the Drills

    Once all drills have been understood and coded, these can be integrated to solve our initial problem.

    Start with list manipulation to create the change array. Use indexing and modulo operation while iterating over the nums list to update the change list. Finally, use list methods to find the maximum score and the rotation that gives this score.