Maximum Average Subarray I

An invariant in computer science and mathematics refers to a property that remains unchanged while an algorithm is being executed.

In the context of this problem, where you’re finding a contiguous subarray whose length is equal to ( k ) that has the maximum average value, the invariant could be related to the process of calculating the averages for different contiguous subarrays of length ( k ).

Here’s how you might think of an invariant in this problem:

  1. Length of Subarray: The length of the subarray you’re considering must always be ( k ).
  2. Sliding Window Technique: You can use a sliding window of size ( k ) to calculate the sum of contiguous elements. The window’s size remains invariant, and you slide it through the array to find the subarray with the maximum average.

Here’s a breakdown of what those invariants mean in this context:

  • Length of Subarray: While you’re iterating through the array and calculating averages for different subarrays, the length of the subarrays you’re considering should always be ( k ). This is a property that does not change, regardless of the specific subarray you’re looking at.

  • Sliding Window Technique: The sliding window technique is a common approach for solving problems like this. You maintain a “window” of fixed size ( k ), which you slide through the array. The sum within this window is updated efficiently as you slide it through the array, by adding the new number that comes into the window and subtracting the number that leaves the window. The size of the window is the invariant here, as it stays the same throughout the algorithm.

By maintaining these invariants, you can create an efficient algorithm for solving the problem. The time complexity of the algorithm would be ( O(n) ), where ( n ) is the length of the input array.

Loop Invariant

In the context of the given problem, the loop invariant will be a condition that holds true before and after each iteration of the loop that you would use to find the contiguous subarray with the maximum average value.

Here’s what the loop invariant would be for this problem:

  1. Window Sum: If you are using a loop to iterate through the array with a sliding window of size ( k ), then the loop invariant would be that the sum inside the window accurately represents the sum of the ( k ) elements in the current window. This must be true at the start of each iteration, and it must still be true at the end of each iteration.

Here’s how you might describe this loop invariant in more detail:

  • Before Iteration: Before each iteration, the sum inside the window should be equal to the sum of the ( k ) elements that the window currently covers.
  • After Iteration: After each iteration, when you slide the window one position to the right, you must update the sum inside the window by adding the new element that comes into the window and subtracting the element that leaves the window. The sum inside the window should still accurately represent the sum of the ( k ) elements that the window now covers.

By maintaining this loop invariant, you can ensure that your algorithm is working correctly, as the sum inside the window will always accurately represent the sum of the ( k ) elements currently inside the window. This invariant is essential for the correct functioning of the sliding window technique in this problem.

The terms “invariant” and “loop invariant” are not the same, and they serve different purposes in the context of this problem.

  1. Invariant: An invariant is a broader concept and refers to a property or condition that remains unchanged throughout the execution of the entire algorithm. In this problem, the invariant might refer to the fact that the length of the subarray being considered must always be ( k ).

  2. Loop Invariant: A loop invariant is a specific type of invariant that relates to a particular loop within an algorithm. It is a property or condition that holds true before and after each iteration of that loop. In this problem, the loop invariant refers to the sum inside the sliding window, accurately representing the sum of the ( k ) elements currently inside the window.

While both concepts deal with properties that remain constant, the loop invariant is specific to a particular loop and must be true at each iteration of that loop. The invariant might be a more general property that applies throughout the entire algorithm.

So in summary, while both the invariant and the loop invariant are related to constant properties within the algorithm, they are not the same thing. The loop invariant is a more specific concept that applies to a particular loop within the algorithm.

To solve this problem, we will use the sliding window approach.

The idea is to calculate the sum of the first k elements in the array and then slide the window through the array, subtracting the first element of the previous window and adding the next element in the array. At each step, we keep track of the maximum sum. Finally, we return the maximum sum divided by k to get the maximum average.

Python code:

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class Solution:
    def findMaxAverage(self, nums: List[int], k: int) -> float:
        window_sum = sum(nums[:k])  # Calculate the sum of first 'k' elements
        max_sum = window_sum  # Initialize max_sum as window_sum

        for i in range(k, len(nums)):
            window_sum = window_sum - nums[i - k] + nums[i]  # Slide the window
            max_sum = max(max_sum, window_sum)  # Update max_sum if needed

        return max_sum / k  # Return the maximum average

In this code, window_sum - nums[i - k] + nums[i] slides the window by subtracting the first element of the previous window and adding the next element in the array. max_sum = max(max_sum, window_sum) updates max_sum if the sum of the current window is greater. At the end, max_sum / k returns the maximum average.

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# @param {Integer[]} nums
# @param {Integer} k
# @return {Float}
def find_max_average(nums, k)
   sum = 0

   for i in (0..k-1)
      sum += nums[i]
   end

   result = sum
   for i in (k..nums.size-1)
      sum = sum + nums[i] - nums[i - k]
      result = [sum, result].max
   end
   result.to_f/k
end

Mistake #1.

Using the k instead of k -1 for the first loop.

Mistake #2.

Integer division instead of using to_f to get floating point result for calculating average.

Mistake #3.

Using two pointers and trying to make it work in one loop.

Mistake #4.

Not knowing the trick of subtracting the element leaving the window and adding the new element entering the window.

10 Prerequisite LeetCode Problems

Here are ten problems to prepare you to tackle the “643. Maximum Average Subarray I” problem:

  1. “Maximum Subarray” (LeetCode Problem #53): This problem requires you to find a contiguous subarray that has the maximum sum. It helps with understanding contiguous array problems.

  2. “Moving Average from Data Stream” (LeetCode Problem #346): This problem provides practice for understanding and implementing the sliding window concept.

  3. “Range Sum Query - Immutable” (LeetCode Problem #303): This problem helps with understanding how to deal with subarray sums which is a sub-problem of the main problem.

  4. “Subarray Product Less Than K” (LeetCode Problem #713): This problem uses a sliding window to find a subarray product, similar conceptually to finding a subarray average.

  5. “Find All Anagrams in a String” (LeetCode Problem #438): This problem also involves sliding windows and can help you understand more advanced applications of the concept.

  6. “Subarray Sum Equals K” (LeetCode Problem #560): This problem can provide further practice on the sliding window concept, along with the idea of maintaining a running total.

  7. “Maximum Size Subarray Sum Equals k” (LeetCode Problem #325): This problem can help solidify your understanding of subarray sums and the running total concept.

  8. “Minimum Size Subarray Sum” (LeetCode Problem #209): This problem introduces the concept of a sliding window with a variable size, which is a useful technique for the main problem.

  9. “Sliding Window Median” (LeetCode Problem #480): Although it is more complex, it uses a similar sliding window approach and can be solved using a double-ended queue (deque).

  10. “Binary Subarrays With Sum” (LeetCode Problem #930): This problem is about finding the number of subarrays with a given sum, which is an important concept for problems dealing with subarrays.

These cover sliding window concept, running totals, and subarray sums. This should help you tackle the “Maximum Average Subarray I” problem.

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class Solution:
    def findMaxAverage(self, nums: List[int], k: int) -> float:
    	M = d = 0
    	for i in range(len(nums)-k):
    		d += nums[i+k] - nums[i]
    		if d > M: 
            M = d

    	return (sum(nums[:k])+M)/k

Problem Classification

The problem involves handling an array (a fundamental data structure) and finding an optimal subarray, which is an algorithmic problem.

‘What’ Components:

  1. Input: An integer array ’nums’ consisting of n elements, and an integer ‘k’.
  2. Output: A maximum average value of a contiguous subarray of length ‘k’.
  3. Constraints: The length of the subarray should be equal to ‘k’. The answer can have a calculation error of less than 10^-5.

This is an Optimization Problem because the goal is to find a maximum (or optimal) average value under certain constraints (subarray length must equal ‘k’). It involves exploring a solution space (all possible contiguous subarrays of length ‘k’) and finding an optimal solution within that space. The problem requires a good understanding of array manipulation and sliding window algorithms to solve efficiently.

This problem involves finding a solution that maximizes a certain metric, which is why it falls under the category of optimization problems. Additionally, the problem deals with an array, which is a fundamental data structure in computer science, and involves a search for the optimal subarray that satisfies the given conditions, hence it is classified under the sub-domain of Data Structures and Algorithms.

Language Agnostic Coding Drills

  1. Dissecting the code:

    a. Basic syntax and structure: Understanding the class structure, method definition, and basic Python syntax.

    b. List slicing and indexing: Here, the list ’nums’ is sliced and indexed to retrieve elements or subarrays. This concept is useful for manipulating arrays.

    c. Mathematical operations: The code involves basic mathematical operations like addition, subtraction, and division.

    d. Control flow - if condition: The if condition is used to check if a certain condition is met and to perform certain operations when it is.

    e. Looping through list: A for loop is used to iterate over the list of integers. Understanding how loops work is crucial for many programming tasks.

    f. Variables initialization and updating: The variables ‘M’ and ’d’ are initialized and updated throughout the loop. It requires an understanding of how variable assignment and updating work.

    g. Sliding window technique: This is a more complex concept that involves manipulating a ‘window’ of elements in an array to solve certain types of problems efficiently.

  2. Order of increasing difficulty:

    a. Basic syntax and structure: This is fundamental and often one of the first things learned in any programming language.

    b. Mathematical operations: These are common operations that are simple to understand.

    c. List slicing and indexing: This concept requires an understanding of how arrays work and how to access their elements.

    d. Variables initialization and updating: This involves understanding the process of assigning and manipulating values stored in variables.

    e. Control flow - if condition: This involves the ability to implement and understand conditional logic in code.

    f. Looping through list: Although a fundamental concept, it requires an understanding of how iteration works in Python.

    g. Sliding window technique: This is a more advanced concept that requires understanding how to manipulate a subset of an array to solve problems more efficiently.

  3. Problem-solving approach:

    The problem involves finding the maximum average of a subarray of length ‘k’ in the given array. This could be achieved by brute force by calculating the average for every possible subarray of length ‘k’, but this would be inefficient.

    The code uses a sliding window technique to solve the problem more efficiently. The initial average is calculated for the first ‘k’ elements, and this is stored. Then, for each subsequent element, we ‘slide’ the window one element to the right by subtracting the value of the element leaving the window and adding the value of the element entering the window.

    The maximum average is tracked throughout this process. By using the sliding window technique, the code avoids having to recalculate the average for overlapping subarrays, thus improving efficiency.

Targeted Drills in Python

  1. Python coding drills for each identified concept:

    a. Basic syntax and structure:

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    class MyClass:
    def myMethod(self):
    pass

    This represents a basic class definition and a method within the class.

    b. Mathematical operations:

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    a = 5
b = 3
sum_ab = a + b
diff_ab = a - b
div_ab = a / b

    This code demonstrates addition, subtraction, and division operations.

    c. List slicing and indexing:

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    my_list = [1, 2, 3, 4, 5]
first_element = my_list[0]
first_three_elements = my_list[:3]

    This code shows how to access an element of a list and how to slice a list.

    d. Variables initialization and updating:

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    var = 0
var = var + 5
var += 3

    This code demonstrates how to initialize a variable and then update its value.

    e. Control flow - if condition:

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    a = 5
b = 3
if a > b:
    print("a is greater than b")

    This code shows how to use a basic if condition.

    f. Looping through list:

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    my_list = [1, 2, 3, 4, 5]
for i in my_list:
    print(i)

    This code demonstrates how to iterate over each element in a list.

    g. Sliding window technique:

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    arr = [1, 2, 3, 4, 5, 6, 7]
k = 3
window_sum = sum(arr[:k])
for i in range(k, len(arr)):
    window_sum = window_sum - arr[i - k] + arr[i]

    This drill demonstrates the sliding window technique, by maintaining the sum of ‘k’ elements and sliding this window through the array.

  2. Problem-specific concept:

    a. Maintaining a running maximum: This is an important concept in this problem as we have to find the maximum average. For each new average, we compare it with the current maximum and update the maximum if the new average is greater.

    Python coding drill:

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    numbers = [1, 2, 3, 4, 5, 6, 7]
max_number = numbers[0]
for num in numbers:
    if num > max_number:
        max_number = num

    This code goes through a list of numbers and keeps track of the maximum number encountered so far.

  3. Assembly of the drills to solve the problem:

    The initial step is to understand the basic syntax and structure, mathematical operations, list slicing and indexing, and control flow. These are fundamental coding concepts that form the basis of the problem-solving approach.

    The sliding window technique is then applied to solve the problem more efficiently. The sum of the first ‘k’ elements is calculated and then, for each subsequent element, the window is slid one element to the right, updating the sum as per the sliding window technique.

    For each new sum (which when divided by ‘k’ gives the average), it’s checked if it’s greater than the current maximum average. If it is, the maximum average is updated. This uses the concept of maintaining a running maximum.

    Thus, starting from the basic coding concepts, and proceeding to the sliding window technique, and then the problem-specific concept of maintaining a running maximum, the drills can be integrated to form the final solution for the problem.