1813. Sentence Similarity III - Detailed Explanation

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Problem Statement

Description:
You are given two sentences represented as strings. A sentence is a list of words separated by a single space with no leading or trailing spaces. One sentence can be considered similar to the other if you can insert an arbitrary sentence (possibly empty) in the middle of one of them so that the two sentences become exactly equal. In other words, the words of one sentence should appear as a prefix and/or a suffix in the other sentence (in the same order).

Task:
Determine if the two sentences are similar according to the above definition.

Example Inputs & Outputs:

Example 1:
- Input:
  - sentence1 = "My name is Haley"
  - sentence2 = "My Haley"
- Process:
  - The first word "My" and the last word "Haley" appear in the same positions in both sentences.
  - The extra words "name is" in sentence1 can be seen as an inserted sentence.
- Output: True
- Explanation: The shorter sentence can be obtained by removing the middle words of the longer sentence.
Example 2:
- Input:
  - sentence1 = "of"
  - sentence2 = "A lot of words"
- Process:
  - There is no common prefix since "of" does not match "A".
  - Also, the suffixes do not align (e.g. "of" vs. "words").
- Output: False
- Explanation: No valid way exists to insert or remove words to make the sentences identical.
Example 3:
- Input:
  - sentence1 = "Eating right now"
  - sentence2 = "Eating"
- Process:
  - The word "Eating" at the beginning matches, and the extra words "right now" can be viewed as an inserted sentence in the longer one.
- Output: True
- Explanation: Removing the extra words from sentence1 gives sentence2.

Constraints:

The sentences contain only lowercase and uppercase English letters and spaces.
Words in a sentence are separated by a single space.
There are no leading or trailing spaces.
The number of words in each sentence is at least 1.

Hints

Hint 1:
Think about how you would compare two sentences word by word. Check for common words starting from the beginning (prefix) and then from the end (suffix).
Hint 2:
If you can match a prefix and a suffix in both sentences such that the total number of matched words is at least as many as the number of words in the shorter sentence, then the sentences are similar.

Approaches

Approach 1: Brute Force (Conceptual Discussion)

Idea:
Try removing different possible contiguous segments from the longer sentence and see if it matches the shorter sentence exactly.
Downside:
This approach would require checking many possibilities (potentially (O(n^2)) combinations), making it too slow for larger inputs.

Approach 2: Two-Pointer Matching (Optimal)

Idea:
Use two pointers to match words from the start (prefix) and from the end (suffix) of both sentences.
Steps:
1. Split each sentence into an array of words.
2. Initialize two pointers at the beginning of both arrays and count how many words match consecutively (prefix match).
3. Initialize two pointers at the end of both arrays and count how many words match consecutively (suffix match).
4. Check Condition:
  If the sum of the matched prefix and suffix words is at least equal to the length of the shorter sentence, then the sentences are similar; otherwise, they are not.
Why It’s Optimal:
This approach requires only two passes over the word arrays (one from the beginning and one from the end), making the solution linear with respect to the number of words.

Python Code

Python3

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Java Code

Java

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Complexity Analysis

Time Complexity:
- Splitting each sentence into words takes (O(n)), where (n) is the number of characters.
- The two-pointer matching (prefix and suffix) runs in (O(m)) where (m) is the number of words in the sentence.
- Overall: (O(n)) or (O(m)) depending on the representation (words versus characters), which is efficient.
Space Complexity:
- The extra space is used to store the arrays of words, which is (O(m)).
- No additional significant space is used.

Step-by-Step Walkthrough and Visual Example

Example Walkthrough for:
sentence1 = "My name is Haley"
sentence2 = "My Haley"

Splitting into Words:
- words1 = ["My", "name", "is", "Haley"]
- words2 = ["My", "Haley"]
Prefix Matching:
- Compare index 0: "My" vs. "My" → match.
  - i becomes 1.
- Compare index 1: "name" vs. "Haley" → no match.
  - Stop prefix matching.
Suffix Matching:
- Compare last element: "Haley" (from words1) vs. "Haley" (from words2) → match.
  - j becomes 1.
- Now, check if more suffix words can be matched.
  - For words2, there is no word before index 0 when (j = 1) (since length of words2 is 2, and (2 - i = 1)).
  - Stop suffix matching.
Validation:
- Total matched words = prefix match (i = 1) + suffix match (j = 1) = 2.
- The number of words in the shorter sentence = 2.
- Since (1 + 1 \geq 2), the sentences are similar.

Visual Representation:

Sentence1: My   name   is   Haley
             |             |
Sentence2: My             Haley

Prefix matched: "My"
Suffix matched: "Haley"
Total matched = 2, which equals the number of words in sentence2 → Similar.

Common Mistakes

Not Splitting Correctly:
Ensure that you split the sentence by a single space and handle cases with only one word properly.
Improper Suffix Matching:
When matching from the end, be cautious with the indices. The loop must ensure that you do not over-count (i.e., not matching overlapping words with the prefix).
Edge Conditions:
Not accounting for sentences that are exactly equal or when one sentence is a complete prefix or suffix of the other.

Edge Cases

Identical Sentences:
If both sentences are exactly the same, every word will match, and the function should return True.
One-Word Sentences:
When both sentences contain only one word, the function should compare the single words directly.
No Common Words:
If there is no match at the beginning or end, the function should quickly return False.

Sentence Similarity I & II:
Variations where the similarity might be defined by a set of allowed word pairs or where only adjacent words can be swapped.
Longest Common Prefix/Suffix Problems:
Problems that involve finding common prefixes or suffixes in arrays or strings.
Subsequence Matching:
Problems that require checking if one sequence is a subsequence of another.

Longest Common Subsequence:
Find the longest sequence that appears in both sequences.
String Matching Algorithms:
Problems that involve pattern matching (e.g., Knuth-Morris-Pratt).
Word Break Problems:
Determining if a string can be segmented into a space-separated sequence of dictionary words.