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 public TrieNode[] children;
3 public boolean isEndOfWord;
4
5 public TrieNode() {
6 children = new TrieNode[26];
7 isEndOfWord = false;
8 }
9}
10
11class Trie {
12 private TrieNode root;
13
14 public Trie() {
15 root = new TrieNode();
16 }
17
18 public void insert(String word) {
19 TrieNode node = root;
20 for (char ch : word.toCharArray()) {
21 int index = ch - 'a';
22 if (node.children[index] == null) {
23 node.children[index] = new TrieNode();
24 }
25 node = node.children[index];
26 }
27 node.isEndOfWord = true;
28 }
29
30 public boolean search(String word) {
31 TrieNode node = root;
32 for (char ch : word.toCharArray()) {
33 int index = ch - 'a';
34 if (node.children[index] == null) {
35 return false;
36 }
37 node = node.children[index];
38 }
39 return node.isEndOfWord;
40 }
41
42 public boolean startsWith(String prefix) {
43 TrieNode node = root;
44 for (char ch : prefix.toCharArray()) {
45 int index = ch - 'a';
46 if (node.children[index] == null) {
47 return false;
48 }
49 node = node.children[index];
50 }
51 return true;
52 }
53}In Java, we define a TrieNode class with an array to hold pointers to child nodes, along with a boolean flag to determine if the node marks the end of a word. The Trie class utilizes a root node which is initialized in the constructor. The insertion method iterates over characters converting them into indices. The search method confirms both the presence of each character in the string and that the node marking the final character signifies the end of a word. The startsWith method similarly checks for the presence of each character in the prefix, validating their sequence up until the prefix ends.
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.