feat(algorithms, trie, index pairs): index pairs of strings

Author: BrianLusina

algorithms/trie/index_pairs_of_a_string/README.md

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+# Index Pairs of a String
+
+Given a string text and an array of strings words, return a list of all index pairs [i, j] such that the substring 
+`text[i...j]` is present in words.
+
+Return the pairs [i, j] in a sorted order, first by the value of i, and if two pairs have the same i, by the value of j.
+
+## Constraints
+
+- 1 ≤ text.length ≤ 100
+- 1 ≤ words.length ≤ 20
+- 1 ≤ words[i].length ≤ 50
+- text and words[i] consist of lowercase English letters.
+- All the strings of words are unique.
+
+## Examples
+
+![Example 1](./images/examples/index_pairs_of_a_string_example_1.png)
+![Example 2](./images/examples/index_pairs_of_a_string_example_2.png)
+![Example 3](./images/examples/index_pairs_of_a_string_example_3.png)
+
+## Solution
+
+The algorithm uses a trie to find all pairs of start and end indexes for substrings in a given text that match words
+from a list. First, it builds the trie by inserting each word from the list. Then, it iterates over each starting index
+in the text to match substrings using the trie. For each character sequence, it checks if the current character exists
+in the trie and traverses accordingly. If a word’s end is found (marked by a flag), the start and end indexes of the
+matched substring are recorded in the result. This method optimizes substring searching using the trie structure to
+avoid redundant checks and efficiently match multiple words in the text.
+
+The algorithm to solve this problem is as follows:
+
+1. Insert each word from the list into the trie. Each character is added as a node, and isEndOfWord is set to mark the
+   end of a word.
+2. Loop through each character in text (starting at index i). For each starting index, try to find substrings that match
+   words in the trie by traversing them.
+3. For each character at position i, the algorithm begins traversing the trie from the root node. It then checks whether
+   each subsequent character (from index i to j) is a child node in the trie. If the character is found, the traversal
+   continues to the next character. A valid match has been found if the current node in the Trie marks the end of a word
+   (i.e., isEndOfWord is True). In that case, the index pair [i, j] is recorded, where i is the start index, and j is the
+   end index of the matched word.
+4. After checking all starting indexes, return the list of index pairs representing matched words’ start and end positions.
+
+### Time Complexity
+
+Inserting n words of average length m into the trie takes O(n∗m). For each index i in text, we perform a search that
+takes linear time in the length of the substring. This gives an overall time complexity of O(l∗k), where l is the length
+of the text, and k is the average length of a word.
+
+### Space Complexity
+
+The space the trie uses depends on the number of characters in the list words. If there are n words with an average length
+of m, the trie can take up to `O(n∗m)` space, assuming no overlapping prefixes among the words. In the worst case, if all
+words are unique and have no shared prefixes, each character is stored separately.
+
+The result list stores the index pairs [i, j]. In the worst case, every possible substring of text could be a word in
+the list words. This would result in `O(l^2)` pairs, where l is the length of the string text. So, the space complexity
+for the result list is `O(l^2)` in the worst case.
+
+Thus, the overall space complexity is `O(n∗m+l^2)`.

algorithms/trie/index_pairs_of_a_string/__init__.py

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+from typing import List
+from datastructures.trees.trie import Trie
+
+def index_pairs(text: str, words: List[str]) -> List[List[int]]:
+    trie = Trie()
+
+    for word in words:
+        trie.insert(word)
+
+    results = []
+
+    # loop through each character in the text
+    for idx, char in enumerate(text):
+        # start from the root of the Trie for each character in the text
+        node = trie.root
+
+        # Check each possible substring starting from index idx
+        for j in range(idx, len(text)):
+            ch = text[j]
+            # If the character is not in the current Trie Node's children, stop searching
+            if ch not in node.children:
+                break
+
+            # Move to the next node in the Trie
+            node = node.children[ch]
+
+            # If we reach the end of a word, record the indices
+            if node.is_end:
+                results.append([idx, j])
+
+    return results

algorithms/trie/index_pairs_of_a_string/images/examples/index_pairs_of_a_string_example_1.png

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+import unittest
+from typing import List
+from parameterized import parameterized
+from algorithms.trie.index_pairs_of_a_string import index_pairs
+
+INDEX_PAIRS_OF_A_STRING = [
+    (
+        "thestoryofeducativeandme",
+        ["story", "feduc", "educative"],
+        [[3, 7], [9, 13], [10, 18]],
+    ),
+    ("xyxyx", ["xyx", "xy"], [[0, 1], [0, 2], [2, 3], [2, 4]]),
+    ("howareyou", ["how", "are", "you"], [[0, 2], [3, 5], [6, 8]]),
+    ("weather", ["weather"], [[0, 6]]),
+    (
+        "aquickbrownfoxjumpsoverthelazydog",
+        ["quick", "fox", "dog"],
+        [[1, 5], [11, 13], [30, 32]],
+    ),
+]
+
+
+class IndexPairsOfAStringTestCase(unittest.TestCase):
+    @parameterized.expand(INDEX_PAIRS_OF_A_STRING)
+    def test_index_pairs_of_a_string(
+        self, text: str, words: List[str], expected: List[List[int]]
+    ):
+        actual = index_pairs(text, words)
+        self.assertEqual(expected, actual)
+
+
+if __name__ == "__main__":
+    unittest.main()