Problem Description

A sentence is defined as a list of words that are separated by a single space, with no leading or trailing spaces. Words are made up of uppercase and lowercase English letters, and it's important to note that uppercase and lowercase letters are distinct from one another.

A sentence is considered "circular" if it meets two conditions:

1. The last character of a word is the same as the first character of the following word.
2. The last character of the last word is the same as the first character of the first word.

The task is to determine if a given sentence meets these criteria for being a circular sentence. In other words, we need to check if the sentence forms a continuous loop where the end of one word connects to the beginning of the next, and finally, the sentence loops from the end back to the beginning.

Intuition

To determine if a sentence is circular, a straightforward approach is adopted:

1. First, check if the first character of the sentence is the same as the last character. This fulfills the second condition for the sentence to be circular, forming the loop from end to beginning.

2. Then, inspect each space in the sentence to ensure that it is the separator between two words where the last character of the word before the space matches the first character of the word after the space. This is the first condition needed for a circular sentence.

3. Loop through each character in the sentence, and at every space character, compare the characters immediately before and after the space. If they are the same, we proceed; if not, the sentence cannot be circular.

4. If all spaces in the sentence have matching characters before and after, and the first and last characters of the sentence are the same, the sentence is circular. Otherwise, it is not.

The Python solution provided uses the all() function in combination with a list comprehension to efficiently check each space in the sentence for the required condition and to compare the first and last characters of the sentence. This performs the necessary checks in one pass, making it a simple and effective solution.

Solution Approach

The solution provided uses a simple yet effective approach to determine whether the given sentence is circular. The key aspects of the implementation are:

1. Checking the first and last characters: The condition s[0] == s[-1] checks whether the first character of the sentence (s[0]) is the same as the last character (s[-1]). This is essential for the sentence to loop back to the beginning after the last word.

2. Iterating through the sentence: We then utilize the enumerate() function, which allows us to loop through each character in the sentence while having access to both the character itself (c) and its index (i).

3. List comprehension with all(): A list comprehension is used in conjunction with the all() function. all() will return True only if all elements in the iterable are True. This is perfect for checking if every space in the sentence is followed by a word that starts with the same character that ends the previous word.

4. Condition within list comprehension: The condition c != " " or s[i - 1] == s[i + 1] within the list comprehension serves a dual purpose. It checks that:

   • For elements that are not spaces (where c != " "), we do not need to perform any action, so these cases should always return True by default to pass the all() condition.
   • For spaces (which means c is " "), it checks if the character before the space (s[i - 1]) is the same as the character after the space (s[i + 1]), thereby adhering to the rule that the last character of one word must match the first character of the next word.

5. Combining the conditions: The above two aspects are combined using an and operator to ensure that the sentence is circular only if both conditions are satisfied.

This approach is efficient because it only involves a single traversal of the sentence and does not require any extra data structures, thus keeping the space complexity to O(1). The code simplicity and the absence of nested loops also ensure that the time complexity is kept to O(n), where n is the length of the input sentence.

Example Walkthrough

Let's go through an example to illustrate the solution approach.

Consider the sentence: "cat toy yak kite energy"

To determine if this sentence is circular according to the description, we need to apply the provided solution.

1. First, we check if the sentence starts and ends with the same character, which are "c" and "y" in this case. Since they are not the same, we could conclude the sentence is not circular. However, for the sake of demonstration, we'll continue with the next steps as if they were the same to illustrate the full process.

2. Now we'll iterate through the sentence using enumerate() to examine each character and its position.

3. Our list comprehension will iterate over each character c along with its index i.

4. For each character that is not a space, we consider it valid as it doesn't contribute to the condition for circularity.

5. For each space character, we check the condition (s[i - 1] == s[i + 1]).

• After "cat", the space is followed by "toy". The last character of "cat" is 't' and the first character of "toy" is 't', so this condition is met.
• Following "toy", the space is followed by "yak". The last character of "toy" is 'y' and the first character of "yak" is 'y', so this condition is met as well.
• After "yak", the space is followed by "kite". The last character of "yak" is 'k' and the first character of "kite" is 'k', meeting the condition once again.
• Finally, after "kite", the last character 'e' should match the first character of "energy" 'e', which it does.

6. If the first and last characters matched and all the conditions for spaces were true as per the steps above, we could say the sentence is circular. The code would look like this:

sentence = "cat toy yak kite energy"
is_circular = sentence[0] == sentence[-1] and all(c != " " or sentence[i - 1] == sentence[i + 1] for i, c in enumerate(sentence))

Since the first character 'c' doesn't match the last character 'y', sentence[0] == sentence[-1] evaluates to False and therefore is_circular would be False even though every other condition is met. As such, the sentence "cat toy yak kite energy" is not circular.

This demonstrates how the provided solution approach can be applied to any sentence to efficiently determine its circularity.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(n) where n is the length of the string s. This is because the code iterates through each character in the string exactly once (via the enumerate function in the all()), and each operation inside the loop is O(1), including checking equality of characters.

The space complexity of the code is O(1). No additional space that scales with the size of the input string is required, as the evaluation is done in-place and the all() function does not create a new list or use any additional space that would depend on the length of the input string.

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