Problem Description

The problem requires us to find the k most frequent strings in an array of strings words. This entails two primary tasks: calculating the frequency of each word and then finding the top k based on this frequency. The problem adds a layer of complexity by asking for these top k strings to be in a specific order. The words should first be sorted based on frequency, with the most frequent words appearing first. If there are multiple words with the same frequency, those words should be ordered lexicographically (like in a dictionary), which means alphabetically in ascending order.

The expected outcome is a list of strings representing the k most frequent words in the order described above.

Intuition

The intuition behind the solution is to employ data structures and sorting mechanisms that can efficiently handle the two-level sorting criteria described in the problem statement.

The solution uses the following steps to arrive at the desired result:

1. Count the occurrences of each word using a data structure that maps words to their frequencies. Python's Counter class from the collections module is well-suited for this and helps us quickly determine the frequency of each word in words.
2. Sort the words based on the two criteria using a custom sorting function:
   • First, by the frequency of each word in descending order so that the most frequent words come first.
   • Second, by the word itself in lexicographical order for words with the same frequency. The lexicographical order ensures that if two words have the same frequency, the word that comes first alphabetically will come first in the sorted list.
3. To sort by multiple criteria, we can use a tuple where the first element is the negative frequency (to sort in descending order) and the second element is the word itself (to sort lexicographically as a tiebreaker).
4. Finally, we slice the sorted list to only include the top k elements.

Using this approach, we can leverage the powerful built-in sorting available in Python, ensuring that both sorting criteria are met, and only k elements are returned as the final result.

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

Solution Approach

The implementation of the solution takes advantage of Python's libraries and language features to accomplish the task of finding the top k most frequent strings. Here's a step-by-step explanation of the methods used:

Step 1: Counting Frequencies

We first use the Counter class from Python's collections module, which is a specialized dictionary designed for counting hashable objects. It takes an iterable (in this case, the words list) and creates a Counter object that maps each unique word to its frequency count.

cnt = Counter(words)

Step 2: Custom Sorting

Once we have the frequency counts, the next step is to sort the items based on our two criteria. Python's sorted function is utilized here, which allows for custom sorting logic through its key parameter. The key parameter accepts a function that returns a tuple containing the sorting criteria for each element in the iterable.

sorted(cnt, key=lambda x: (-cnt[x], x))[:k]

In the lambda function:

• x is each word from the Counter keys.
• -cnt[x] is the frequency of the word, negated so that the list is sorted in descending order (since Python sorts in ascending order by default).
• x is the word itself, which ensures that words with identical frequencies are sorted lexicographically.

The tuple (-cnt[x], x) ensures that the sorting happens first by frequency and then, as a tiebreaker, by lexicographical order.

Step 3: Slicing

Lastly, after sorting, we slice the list to only select the first k elements, as we're only interested in the top k most frequent words.

[:k]

This represents the final step where the method only returns the first k sorted elements, effectively giving us the top k most frequent words in the required order.

Combining these steps, the solution applies well-known algorithms for counting (hash table via Counter) and sorting (with custom criteria via sorted) to efficiently resolve the problem at hand. The use of lambda expressions and list slicing demonstrates the expressive power of Python's syntax, allowing for concise and readable code.

Example Walkthrough

Let's consider the following small example array of strings words and a value for k to illustrate the solution approach:

words = ["apple", "banana", "cherry", "apple", "banana", "apple"] k = 2

We want to find the 2 most frequent strings in the words array.

Step 1: Counting Frequencies

Using Python's Counter from the collections module, calculate the frequency of each word.

from collections import Counter
cnt = Counter(words)
# cnt = {'apple': 3, 'banana': 2, 'cherry': 1}

The Counter object now holds the frequency of each word: "apple" appears 3 times, "banana" 2 times, and "cherry" 1 time.

Step 2: Custom Sorting

Now we need to sort the words based on two criteria: the frequency (higher to lower) and lexicographic order. We use a custom sorting function with Python's sorted:

sorted_words = sorted(cnt, key=lambda x: (-cnt[x], x))
# sorted_words = ['apple', 'banana', 'cherry']

Here, sorted_words is sorted by frequency in descending order. Since "apple" and "banana" have different frequencies, no need for lexicographical sorting between them. "Cherry" has a lower frequency so comes last.

Step 3: Slicing

Finally, we only want the top k elements, so we slice the sorted list:

result = sorted_words[:k]
# result = ['apple', 'banana']

The result list ['apple', 'banana'] represents the 2 most frequent words in the words array, adhering to the frequency then lexicographical ordering rules.

Combining Step 1 through Step 3, the complete code to accomplish finding the k most frequent strings would look like this:

from collections import Counter

def k_most_frequent(words, k):
    cnt = Counter(words)
    sorted_words = sorted(cnt, key=lambda x: (-cnt[x], x))
    return sorted_words[:k]

# Usage
words = ["apple", "banana", "cherry", "apple", "banana", "apple"]
k = 2
print(k_most_frequent(words, k))  # Output: ['apple', 'banana']

This example walkthrough demonstrates how the combined steps of counting, sorting, and slicing give us a clean solution using Python's powerful built-in features.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code can be analyzed in a few steps:

1. Counter(words) has a time complexity of O(n) where n is the number of words, as it requires one pass to count the frequency of each word.
2. sorted(cnt, key=lambda x: (-cnt[x], x)) has a time complexity of O(n log n) on average, as it sorts the list of words by their frequencies, and in case of ties, it sorts them lexicographically. Here, "n" refers to the unique words in the list. 3.[:k]slicing has a time complexity ofO(k)` because it needs to copy the first k elements from the sorted array.

Overall, the time complexity of the function is dominated by the sorting step, which, in the worst case, is O(n log n).

Space Complexity

For space complexity:

1. The Counter uses O(u) space where u is the number of unique words in the input list, to store the word counts.
2. The sorted list will also use O(u) space.
3. The slice of the top k elements will require O(k) space, but this does not add to the overall space complexity since k is no larger than u and so is subsumed by it.

Therefore, the overall space complexity of the function is O(u).

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