This approach involves creating a basic Trie data structure using a tree of nodes. Each node represents a character in a word. Starting from a root node, each character of the word is inserted sequentially, with branches representing the progression to subsequent characters. This forms a chain of nodes (linked in a tree-like manner) from the root to the nodes representing complete words. For searching, we traverse these nodes based on the characters of the input word or prefix to check for existence.
Time Complexity: Insert, search, and startsWith operations are O(m), where m is the length of the word/prefix.
Space Complexity: O(26 * n * m) in the worst case, where n is the number of inserted words and m is the average length of the words. Each insertion can potentially add m nodes with 26 possible children each.
1class TrieNode:
2 def __init__(self):
3 self.children = {}
4 self.is_end_of_word = False
5
6class Trie:
7 def __init__(self):
8 self.root = TrieNode()
9
10 def insert(self, word: str) -> None:
11 node = self.root
12 for char in word:
13 if char not in node.children:
14 node.children[char] = TrieNode()
15 node = node.children[char]
16 node.is_end_of_word = True
17
18 def search(self, word: str) -> bool:
19 node = self.root
20 for char in word:
21 if char not in node.children:
22 return False
23 node = node.children[char]
24 return node.is_end_of_word
25
26 def startsWith(self, prefix: str) -> bool:
27 node = self.root
28 for char in prefix:
29 if char not in node.children:
30 return False
31 node = node.children[char]
32 return TrueIn the Python implementation, we make use of a TrieNode class with a dictionary to store pointers to child nodes and a boolean attribute to denote if the node is the end of a word. The Trie class creates the root node as part of the constructor. The insert method processes the word character by character, adding nodes for characters that are missing in the current sequence. The search and startsWith methods traverse down the trie nodes based on the characters of word or prefix, verifying each node's presence accordingly and checking the is_end_of_word flag when necessary for search.
This approach relies on maps (or hashmaps) for storing children nodes dynamically. The advantage of using maps over arrays in this context arises from reduced space consumption when handling sparse trees since only existing characters are stored. Nodes manage their children using hash references, leading to more flexible branching.
Time Complexity: O(m) per operation where m is the word/prefix length.
Space Complexity: O(n * m), where n estimates the word count and m accounts for varying lengths since nodes only maintain necessary mappings.
1class TrieNode:
2 def __init__(self):
3 self.children = {}
4 self.is_end_of_word = False
5
6class Trie:
7 def __init__(self):
8 self.root = TrieNode()
9
10 def insert(self, word: str) -> None:
11 node = self.root
12 for char in word:
13 node = node.children.setdefault(char, TrieNode())
14 node.is_end_of_word = True
15
16 def search(self, word: str) -> bool:
17 node = self.root
18 for char in word:
19 if char not in node.children:
20 return False
21 node = node.children[char]
22 return node.is_end_of_word
23
24 def startsWith(self, prefix: str) -> bool:
25 node = self.root
26 for char in prefix:
27 if char not in node.children:
28 return False
29 node = node.children[char]
30 return TrueIn this Python solution, each TrieNode contains a dictionary (children) providing direct access to only required child characters. This approach optimizes storage, especially reducing overhead when branching is sparse. The root provides the base node in the Trie. The insert method applies setdefault for dictionary operation - simplifying conditional insertion logic. Both search and startsWith check the presence of each character in sequence traversing the root to target node, utilizing dictionary fast access and probing methods.