Problem Description

The problem is about reducing the size of an integer array by removing some subsets of integers to make sure at least half of the elements in the original array are removed. The task is to find the minimum number of distinct integers needed for these subsets such that once their occurrences are removed, at least half of the array elements are gone.

To clarify with an example, if we have an array [3,3,3,3,5,5,5,2,2,7], a possible solution would be choosing the subset {3,5} which contains 2 integers. Removing all the instances of 3 and 5 from the array would remove 3 + 3 = 6 elements from the 10-element array, thereby removing more than half of the array, which satisfies the condition.

Intuition

The intuition behind the solution approach involves the following steps:

1. Identify that we want to maximize the number of elements removed by picking the least number of distinct integers.
2. Understand that to remove the most numbers, we should start by removing the integers that appear most frequently.
3. Use a frequency counter to count the occurrences of each integer in the array.
4. Sort these integers by their frequencies in descending order, so we can remove the integers with higher frequencies first.
5. Start removing integers from the sorted list one by one, adding their frequencies to a cumulative sum (m in the code).
6. Each time an integer is added to the set (the subset we're removing), increase the count (ans in the code) that represents the size of our set.
7. Once we've removed enough numbers such that the cumulative sum is at least half of the original array's size, we've found our minimum set size and can stop.

The provided Python code uses a Counter from the collections module to tally the integer frequencies and most_common() method to sort them by frequency in descending order. It iterates through these frequencies while keeping a running total and a count until at least half the array size is reached.

Learn more about Greedy, Sorting and Heap (Priority Queue) patterns.

Solution Approach

The solution uses a hash map and sorting technique to solve the problem efficiently. Here's a breakdown of the implementation steps referring to the Python code provided:

1. Hash Map (Counter):

   • Use the Python Counter from the collections module to create a hash map that counts the occurrences of each integer in the input array arr. This operation has a complexity of O(n), where n is the size of the array.

   cnt = Counter(arr)

2. Sorting:

   • The most_common() function is called on the Counter object which under the hood performs a sort operation, arranging the counted integers in descending order based on their frequency. The complexity of this operation is O(u log u), where u is the number of unique elements in arr.

   for _, v in cnt.most_common():

3. Iterative Accumulation:

   • Initialize two variables, ans to count the number of elements in the required set and m to keep track of the cumulative frequency of elements removed from the array arr.

   ans = m = 0

   • Iteratively add the frequency v of the most common element to m. Increment ans by 1 to reflect that an element is added to the set.

   m += v
   ans += 1

   • Check if the cumulative frequency m is at least half the length of the original array. If so, break out of the loop since the requirement of removing at least half of the array elements is met.

   if m * 2 >= len(arr):
       break

4. Return the Result:

   • After the loop terminates, return the value of ans, which represents the minimum size of the set to remove at least half of the array elements.

   return ans

This approach is efficient as it targets the frequency of elements which is a key insight for achieving the task with a minimal set size. The heavy-lifting operations are counting and sorting which are both done optimally using Python's built-in data structures and algorithms.

Example Walkthrough

Let's use a small example to illustrate the solution approach. Suppose we have an array arr = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4].

1. Hash Map (Counter):

   • First, we'll count the occurrences of each integer:

   from collections import Counter
   cnt = Counter(arr)
   # cnt = Counter({4: 4, 3: 3, 2: 2, 1: 1})

   In cnt, the key is the integer and the value is its frequency in the array.

2. Sorting:

   • Next, we sort these by frequency using most_common():

   # This will sort the integers by frequency: [4, 3, 2, 1]
   sorted_integers_by_freq = [item for item, count in cnt.most_common()]

3. Iterative Accumulation:

   • Initialize the variables ans and m:

   ans = m = 0

   • Remove integers one by one, starting from the most frequent. In our list, 4 occurs the most, so we start with that:

   m += cnt[4]  # m = 4 (since 4 appears 4 times)
   ans += 1     # ans = 1 (we have added one integer to our set)

   • Now we move to the next most frequent integer, 3:

   m += cnt[3]  # m = 7 (4 from '4's and 3 from '3's)
   ans += 1     # ans = 2 (we have added two integers to our set)

4. Checking the Cumulative Frequency:

   • We need to check if we have removed at least half of the elements in the array. The original array has 10 elements, so we need to remove at least 5. With m = 7, we have surpassed this:

   if m * 2 >= len(arr):  # 7*2 >= 10 which is True
       # Break the loop as we have reached our requirement.

5. Return the Result:

   • We've found our answer, which is 2, because we were able to remove at least half of the elements by removing numbers 4 and 3.

   return ans
   # return 2

In this example, the subsets {4, 3} contain the minimum number of distinct integers (2 integers) that, once removed, reduce the original array size at least by half. Following this procedure guarantees that we achieve the goal with the smallest possible set, efficiently utilizing the insights about frequency of elements.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code can be broken down into the following operations:

1. Creating the Counter: The Counter(arr) operation has a time complexity of O(N) where N is the length of the array. It goes through each element exactly once.

2. Sorting: The most_common() method on the counter returns elements in decreasing frequency, which implies an internal sort. The complexity of this sorting is O(K log K) where K is the number of unique elements in the array.

3. Iterating through sorted frequencies: In the worst case, this can iterate up to K times. However, since it stops once the removed elements count is at least half of the array size, and considering the frequency of the elements, in practice, it might be less than K. Nevertheless, we consider the worst case, so this step has a complexity of O(K).

When combined, the time complexity is dominated by the time complexity of the sorting step, resulting in a total time complexity of O(N) + O(K log K) + O(K), which simplifies to O(N + K log K) since K ≤ N.

Space Complexity

The space complexity can be analyzed as follows:

1. Counter: Storing the frequency of each element in the array requires O(K) space, where K is the number of unique elements.

2. Most Common List: The list of tuples returned by most_common() also takes up O(K) space.

Combined, the overall space complexity of the algorithm is O(K), but since K can be at most N (when all elements are unique), the space complexity simplifies to O(N).

Learn more about how to find time and space complexity quickly using problem constraints.