Problem Description

The problem provides us with the task of designing a data structure that must be able to compute the shortest distance between any two distinct strings within an array of strings. This implies that we first need to initialize the data structure with an array of strings, which we'll refer to as wordsDict. Once initialized, we should be able to query this data structure with two different strings, and it should return the smallest possible index difference between these two strings within the wordsDict array.

This could be visualized as an array where we want to find the minimum distance between two elements, where the elements are the positions of the words we're interested in. For example, if wordsDict is ["practice", "makes", "perfect", "coding", "makes"], and we query for the distance between "coding" and "practice", the returned value should be 3, as the closest "practice" to "coding" is three indices away.

Intuition

To efficiently find the shortest distance between two words in the dictionary, a preprocessing step is required during initialization. We traverse the wordsDict array once and create a hash map where keys are the distinct words from the array, and values are lists of indices where each key word is located in the original array. This preprocessing step allows for a quick lookup of the positions of any word, facilitating the computation of the distances between any two words.

Once the positions are mapped, to find the shortest distance between word1 and word2, we get their list of indices from our hash map. We need to find the minimum difference between any two indices from these lists. The lists are already sorted because the indices were appended in the order they were encountered during initialization.

A two-pointer approach efficiently solves the problem of finding the minimum difference. Start with the first index of each list, and at each step, move the pointer that points to the smaller index to the next index in its list. The intuition behind this is that we try to close the gap between word1 and word2 by moving forward in the list where the current index is smaller. This approach will traverse the two lists simultaneously and will always give the smallest possible difference in indices (thus the shortest distance) between word1 and word2. The process repeats until we have fully traversed one of the lists, ensuring that no potential shorter distance is missed.

The overall time complexity for the shortest operation, after the initial preprocessing, is O(N + M), where N and M are the number of occurrences of word1 and word2 in the dictionary, respectively. The preprocessing itself is O(K), where K is the total number of words in the dictionary, with the space complexity also being O(K) for storing the hash map.

Learn more about Two Pointers patterns.

Solution Approach

In the reference solution, the WordDistance class is defined, which implements the required functionality. The class uses a hash map (dictionary in Python) to store the indices of each word as they appear in wordsDict. Python's defaultdict from the collections module is used to handle the automatic creation of entries for new words, with each entry being a list that gets appended with the current index (i) every time the word (w) is encountered in the array.

Here is a breakdown of the implementation:

1. Initialization (__init__): This method takes a list of strings wordsDict as input. It iterates over wordsDict, enumerating each word with its index. For every word-index pair, it appends the index to the list that corresponds to the word in the hash map self.d. By the end of this process, we have a dictionary where each word is associated with a list of indices indicating its positions in the wordsDict array.

   self.d = defaultdict(list)
   for i, w in enumerate(wordsDict):
       self.d[w].append(i)

2. Finding Shortest Distance (shortest): This method calculates the shortest distance between two words word1 and word2. By accessing self.d[word1] and self.d[word2], it retrieves the two lists of indices for the given words. The algorithm then initializes two pointers, i and j, starting at 0. These pointers will traverse the index lists.

   The variable ans is initialized to infinity (inf). It will be updated with the minimum difference between indices found so far as we iterate through the lists of indices.

   Following a two-pointer approach, the algorithm compares the indices at the current pointers and updates ans with the smaller of the current difference and the previously stored ans. Based on which index is smaller, the corresponding pointer (i or j) is incremented to move to the next index in the respective list.

   The loop continues until one list is fully traversed, ensuring that the minimum distance has been found:

   a, b = self.d[word1], self.d[word2]
   ans = inf
   i = j = 0
   while i < len(a) and j < len(b):
       ans = min(ans, abs(a[i] - b[j]))
       if a[i] <= b[j]:
           i += 1
       else:
           j += 1
   return ans

In the above loop, at each step, because the two lists are sorted by the nature of their construction, we can guarantee that moving the pointer past the smaller index will give us a pair of indices that is potentially closer than the previous one. This algorithm effectively finds the minimum distance between the two words, which is the absolute difference between their closest indices.

The WordDistance class can then be instantiated with a string array wordsDict, and the shortest method can be called to find the shortest distance between any two words.

Example Walkthrough

Let's demonstrate how the WordDistance class and its methods work using a simple example.

Suppose our wordsDict is ["the", "quick", "brown", "fox", "quick"]. We initialize the WordDistance class with this list of strings. During the initialization process, the class builds the hash map self.d which will contain:

• "the" -> [0]
• "quick" -> [1, 4]
• "brown" -> [2]
• "fox" -> [3]

Now, let's say we want to find the shortest distance between the words "brown" and "quick". We call the shortest method with these two words. Here's what happens inside this method:

1. We access the two lists of indices for the words "brown" and "quick" from our hash map, which gives us a = [2] and b = [1, 4].

2. We initialize two pointers i and j and a variable ans which will keep track of the minimum distance. Initially, i = 0, j = 0, ans = inf.

3. We enter a while-loop, which will continue until we have fully traversed one of the lists.

4. Inside the loop, we compare a[i] with b[j]. At the start, a[i] = 2 and b[j] = 1. The difference |2 - 1| = 1 is less than ans, so we update ans = 1.

5. Since the current value at a[i] is greater than b[j], we increment j to move to the next index in list b.

6. Now, i = 0 and j = 1, which means a[i] = 2 and b[j] = 4. The difference |2 - 4| = 2 is not less than the current ans, so ans remains 1.

7. We've reached the end of list a, so the loop terminates.

Since we have examined all possible pairs of indices for "brown" and "quick", we have found the shortest distance to be 1, which is the minimum index difference between these words in the wordsDict.

Finally, we return ans which is 1, concluding that the shortest distance between "brown" and "quick" in the wordsDict array is 1.

Solution Implementation

Time and Space Complexity

Initialization (__init__)

• Time Complexity: O(N), where N is the length of wordsDict. This is because we're iterating through each word in the list and inserting the index into the dictionary.
• Space Complexity: O(N), as in the worst-case scenario, we're storing the index of each word within the dictionary, which would require space proportional to the number of words.

Shortest Method (shortest)

• Time Complexity: O(L + M), where L is the length of the list associated with word1 and M is the length of the list associated with word2. In the worst case, we traverse each list once in a single run of the while loop.
• Space Complexity: O(1), because we're only using a few variables (i, j, ans, a, b) for calculations and not utilizing any additional space that is dependent on the size of the input.

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