Degree of an Array

Identifying Problem Isomorphism

“Degree of an Array” belongs to a category of problems that involve understanding the properties of an array and applying that understanding to find a solution.

A simpler problem in this category is “Majority Element”. In this problem, you have to find the element that appears more than n/2 times where n is the size of the array. This problem is simpler because it requires counting the frequency of elements in an array which is a more straightforward task.

A more complex problem in this category is “Subarray Sum Equals K”. This problem asks you to find the total number of continuous subarrays whose sum equals to a given integer. This is a more complex problem as it requires not just understanding the properties of an array but also the understanding of subarray sums.

The reasoning behind these choices:

  • “Majority Element” is simpler than “Degree of an Array” because it involves a straightforward counting of elements in an array.

  • “Subarray Sum Equals K” is more complex than “Degree of an Array” because it requires understanding the concept of subarray sums, and it also involves dealing with continuous subarrays, which adds a layer of complexity.

Therefore, the order of problems from simpler to more complex, based on understanding of array properties and their analysis:

  1. “Majority Element”
  2. “Degree of an Array”
  3. “Subarray Sum Equals K”

This mapping is an approximation. While these problems all deal with understanding and analyzing array properties, the exact problem constraints and details can significantly influence the complexity and solution approach.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12

class Solution:
    def findShortestSubArray(self, nums: List[int]) -> int:
        first, counter, res, degree = {}, {}, 0, 0
        for i, num in enumerate(nums):
            first.setdefault(num, i)  # set the first index if not yet set
            counter[num] = counter.get(num, 0) + 1  # count the frequency
            if counter[num] > degree:  # we have found a new maximum degree
                degree = counter[num]
                res = i - first[num] + 1
            elif counter[num] == degree:  # if degree is the same, minimize the length
                res = min(res, i - first[num] + 1)
        return res

Here are the building blocks of this solution:

  1. Python Dictionaries: Dictionaries are used to store the first occurrence index and the frequency count of each element. The setdefault method is used to set the first occurrence index if it’s not already set, and the get method is used to get the frequency count of an element, defaulting to 0 if the element is not yet in the dictionary.

  2. Iteration: A for loop is used to iterate over the array of numbers. The enumerate function is used to get both the index (i) and the value (num) in the list.

  3. Conditional Logic: If-else statements are used to check if the current element’s frequency count (counter[num]) is higher than the current degree. If it is, the degree and result are updated. If it’s equal to the degree, the result is updated only if the current subarray is smaller.

  4. Calculation of Subarray Length: The length of the current subarray is calculated as i - first[num] + 1, where i is the current index and first[num] is the first occurrence index of the current number.

  5. Minimum Function: The min function is used to update the result (res) to be the length of the smallest subarray found so far that meets the degree criteria.

  6. Degree Calculation: Degree is calculated as the maximum frequency of any element in the array. It is updated whenever a higher frequency is found.

At a higher level of abstraction, these building blocks collectively perform the following tasks:

  1. Tracking Element Information: The code tracks essential information about each element, such as the frequency of occurrence and the index of the first occurrence. This information is fundamental to the overall purpose of the problem, which involves identifying the degree of the array and subarrays with the same degree.

  2. Identifying the Degree of the Array: The code identifies the degree of the array, which is defined as the maximum frequency of any of its elements. The frequency of each element is tracked as the array is traversed.

  3. Finding Candidate Subarrays: The code identifies all candidate subarrays that have the same degree as the entire array. This is done by continually calculating the length of the subarray from the first occurrence of each element to the current index whenever the degree matches.

  4. Determining the Shortest Subarray: Among all candidate subarrays, the code identifies the shortest one. This is done by always maintaining the length of the smallest subarray that has been encountered so far.

By accomplishing these tasks, the solution successfully finds the smallest possible length of a contiguous subarray of nums that has the same degree as nums.

I see Frequency Counter, Running Maximum Degree. Can you see others?

In addition to “Frequency Counter” and “Running Maximum Degree”, here are some additional concepts that this problem touches upon:

  1. First Occurrence Index Tracking: We keep track of the first occurrence index of each element in the nums array. This is crucial in calculating the length of the subarray for each element when the degree matches.

  2. Minimum Subarray Length: We constantly maintain and update the smallest subarray length that has the same degree as the nums array. This allows us to identify the shortest possible subarray meeting the degree requirement.

  3. Array Traversal: This concept is used to walk through each element in the array and perform operations like frequency counting, tracking the first occurrence, and updating the degree.

  4. Element to Position Mapping: The usage of a dictionary to map each element to its first occurrence index is an application of this concept. The first occurrence index serves as a starting point in determining the length of subarrays.

  5. Degree Equality Check: This involves checking whether the frequency of the current number equals the maximum degree noted so far. If true, it indicates a potential subarray that meets the requirement.

Problem Classification

The problem falls into arrays and frequency counting, which are common themes in such problems.

Components:

  1. Non-empty array of non-negative integers (nums)
  2. Degree of an array (the maximum frequency of any one of its elements)
  3. Smallest possible length of a contiguous subarray of nums with the same degree as nums

This can be classified as an “Array Processing” problem with a focus on “Frequency Counting”. It involves finding a subarray with specific characteristics (matching the overall degree of the original array) which adds a dimension of “Sliding Window” or “Subarray” problem to it. The objective of finding the smallest such subarray introduces an “Optimization” aspect.

  1. Array Processing: The problem involves direct manipulation and processing of an array. The elements of the array are analyzed to find the frequency of each element and the degree of the array.
  2. Frequency Counting: The degree of an array is defined in terms of frequency count of the elements in the array.
  3. Sliding Window / Subarray: The problem asks for a contiguous subarray of the given array. This is a common subproblem in array processing tasks.
  4. Optimization: The task is to find not just any subarray with the same degree as the original array but the smallest such subarray, which introduces an optimization aspect to the problem.

Identifying Problem Isomorphism

“Degree of an Array” (LeetCode 697) is isomorphic to “Longest Substring with At Most Two Distinct Characters” (LeetCode 159) and “Fruit Into Baskets” (LeetCode 904).

In both these isomorphic problems, the task involves identifying a subarray under certain conditions, which aligns well with the “Degree of an Array” problem’s goal of finding a subarray of minimum length that has the same degree as the original array.

The difference is in the conditions to be satisfied. For instance, in “Longest Substring with At Most Two Distinct Characters”, the subarray must contain at most two distinct characters. In “Fruit Into Baskets”, the subarray represents baskets that can contain only two types of fruits.

The strategies to solve these problems bear similarities, as they all involve tracking frequencies and identifying suitable subarrays, thereby making them isomorphic to the “Degree of an Array” problem.

Language Agnostic Coding Drills

  1. Concepts Identified: a. Dictionary usage: This code involves the use of a dictionary to store the indices of each element as they appear in the array. b. Looping and condition checking: The code involves a loop through the array elements, with a condition check to update the dictionary. c. Enumerate function: This function is used to get both index and value from the list. d. List Comprehension: The code uses list comprehension to create lists of lengths of values and the differences between the first and last indices of values in the dictionary. e. Finding maximum and minimum: The built-in max and min functions are used to get the maximum frequency and minimum length of the subarray.

  2. Coding Concepts Ordered by Difficulty: a. Dictionary usage (Easy): Understanding and using dictionary data structure is one of the basic concepts in programming. b. Looping and condition checking (Easy): Iterating through a list and checking conditions are fundamental concepts in most programming languages. c. Enumerate function (Medium): The enumerate function, while common in Python, might not be intuitive for beginners. It allows iteration through elements along with their indices. d. List Comprehension (Medium): Although Python makes it relatively easy to use, understanding and using list comprehensions effectively can be considered intermediate level as it involves thinking in a more condensed and mathematical way. e. Finding maximum and minimum (Medium): Although max and min are built-in functions, using them with complex data structures or applying conditions (as in this code) can require a better understanding.

  3. Problem-solving approach: a. Create a dictionary where each key-value pair represents an element and its indices in the list respectively. b. Calculate the degree of the array by finding the maximum length of any value (list of indices) in the dictionary. c. Calculate the lengths of subarrays which have the same degree by finding the difference between the last and first index of values in the dictionary that have a length equal to the degree. d. Find the minimum length from the calculated lengths. This represents the smallest subarray with the same degree as the original array.

Targeted Drills in Python

  1. Python-based coding drills:

    a. Dictionary usage: We create a dictionary and add, update, and retrieve items from it. For a beginner, a simple exercise would be to create a dictionary that maps numbers to their squares.

    1
2
3
4
5

    nums = [1, 2, 3, 4, 5]
square_dict = {}
for num in nums:
    square_dict[num] = num ** 2
print(square_dict)

    b. Looping and condition checking: In this drill, we iterate over a list of numbers, and we check if the number is even or odd. If it’s even, we multiply it by 2, if it’s odd we raise it to the power of 2.

    1
2
3
4
5
6
7

    nums = [1, 2, 3, 4, 5]
for i in range(len(nums)):
    if nums[i] % 2 == 0:
        nums[i] *= 2
    else:
        nums[i] **= 2
print(nums)

    c. Enumerate function: In this drill, we use the enumerate function to get both the index and the value from the list. Then we print each number along with its position in the list.

    1
2
3

    nums = [1, 2, 3, 4, 5]
for i, num in enumerate(nums):
    print(f"Number {num} is at position {i}")

    d. List Comprehension: Here, we create a new list which contains the squares of all the numbers from the original list.

    1
2
3

    nums = [1, 2, 3, 4, 5]
squares = [num ** 2 for num in nums]
print(squares)

    e. Finding maximum and minimum: In this drill, we find the maximum and minimum numbers in a list.

    1
2
3
4

    nums = [1, 2, 3, 4, 5]
max_num = max(nums)
min_num = min(nums)
print(f"Max: {max_num}, Min: {min_num}")

  2. Problem-specific concepts:

    One of the problem-specific concepts in our task is maintaining the indices of each number in the list. We use a dictionary to do this. This concept is critical because we need to find the smallest subarray that has the same degree as the whole list, and for that, we need to know the first and last occurrence of the numbers.

    Here’s a simple drill to understand this concept: Given a list of numbers, create a dictionary that maps each number to a list of its indices.

    1
2
3
4
5
6
7
8

    nums = [1, 2, 2, 3, 1]
index_dict = {}
for i, num in enumerate(nums):
    if num in index_dict:
        index_dict[num].append(i)
    else:
        index_dict[num] = [i]
print(index_dict)

  3. Integration of drills:

    After understanding all the concepts, we can integrate them in order to solve our problem. Here’s the step-by-step integration:

    a. Loop over the list of numbers using the enumerate function to get both the index and the value. b. Use the concept of dictionary to keep track of indices of each number. If the number is already in the dictionary, append the index to its value list. If not, add a new entry in the dictionary with the number as the key and a new list containing the index as the value. c. Use

list comprehension and the max function to find the degree of the list, which is the maximum length of the value lists in the dictionary. d. Use list comprehension to create a list of lengths of the subarrays that have the same degree. This is done by finding the difference between the last and the first index in the value lists in the dictionary that have a length equal to the degree. e. Use the min function to find the minimum length from the lengths calculated in the previous step. This is the length of the smallest subarray that has the same degree as the list.