Prefix Search Using Trie in DSA Python - Time & Space Complexity

Time Complexity: Prefix Search Using Trie

O(m)

Understanding Time Complexity

We want to understand how fast we can find words that start with a given prefix using a Trie.

How does the search time change as the prefix or stored words grow?

Scenario Under Consideration

Analyze the time complexity of the following prefix search code in a Trie.

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = False

class Trie:
    def __init__(self):
        self.root = TrieNode()

    def starts_with(self, prefix: str) -> bool:
        node = self.root
        for char in prefix:
            if char not in node.children:
                return False
            node = node.children[char]
        return True
    

This code checks if any word in the Trie starts with the given prefix.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Looping through each character of the prefix.
• How many times: Once for each character in the prefix (length m).

How Execution Grows With Input

The search time grows with the length of the prefix, not the number of words stored.

Input Size (prefix length m)	Approx. Operations
10	About 10 character checks
100	About 100 character checks
1000	About 1000 character checks

Pattern observation: The operations increase linearly as the prefix length increases.

Final Time Complexity

Time Complexity: O(m)

This means the search time depends only on the prefix length, not on how many words are in the Trie.

Common Mistake

[X] Wrong: "The search time depends on the total number of words stored in the Trie."

[OK] Correct: The search only follows the prefix characters down the Trie nodes, so it depends on prefix length, not total words.

Interview Connect

Knowing how Trie prefix search scales helps you explain efficient word lookup in real apps like autocomplete or spell checkers.

Self-Check

"What if we changed the prefix search to return all words starting with the prefix? How would the time complexity change?"

Practice

1. What is the main purpose of using a Trie data structure in prefix search?

easy

A. To perform arithmetic operations quickly

B. To sort words alphabetically

C. To store numbers in sorted order

D. To efficiently find all words starting with a given prefix

Solution

1. Step 1: Understand Trie usage

   A Trie stores characters of words in a tree structure, allowing quick prefix lookups.

2. Step 2: Identify prefix search goal

   Prefix search means finding all words that start with a given prefix efficiently.

3. Final Answer:

   To efficiently find all words starting with a given prefix -> Option D

4. Quick Check:

   Trie = prefix search efficiency [OK]

Hint: Trie nodes represent prefixes for fast search [OK]

Common Mistakes:

• Confusing Trie with sorting data structure
• Thinking Trie is for arithmetic
• Assuming Trie stores whole words only

2. Which of the following is the correct way to insert a word into a Trie node in Python?

easy

A. node.children.add(char)

B. node.children[char] = TrieNode() if char not in node.children else node.children[char]

C. node.children.append(char)

D. node.children[char] = char

Solution

1. Step 1: Understand Trie children structure

   Trie children are usually stored in a dictionary mapping characters to TrieNode objects.

2. Step 2: Correct insertion syntax

   We check if character exists; if not, create new TrieNode and assign it to children dictionary.

3. Final Answer:

   node.children[char] = TrieNode() if char not in node.children else node.children[char] -> Option B

4. Quick Check:

   Insert word by mapping char to TrieNode [OK]

Hint: Use dictionary keys for children, not list methods [OK]

Common Mistakes:

• Using list methods like append or add on dictionary
• Assigning character instead of TrieNode
• Not checking if child node exists before insertion

3. Given the following code snippet, what will be the output of trie.starts_with('app')?

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = False

class Trie:
    def __init__(self):
        self.root = TrieNode()

    def insert(self, word):
        node = self.root
        for char in word:
            if char not in node.children:
                node.children[char] = TrieNode()
            node = node.children[char]
        node.is_end = True

    def starts_with(self, prefix):
        node = self.root
        for char in prefix:
            if char not in node.children:
                return False
            node = node.children[char]
        return True

trie = Trie()
trie.insert('apple')
trie.insert('app')
print(trie.starts_with('app'))

medium

A. Error

B. False

C. True

D. None

Solution

1. Step 1: Insert words 'apple' and 'app'

   Both words share prefix 'app', so nodes for 'a', 'p', 'p' exist in Trie.

2. Step 2: Check starts_with('app')

   Function checks each character in 'app' exists in children; all found, so returns True.

3. Final Answer:

   True -> Option C

4. Quick Check:

   Prefix 'app' exists in Trie = True [OK]

Hint: If prefix nodes exist, starts_with returns True [OK]

Common Mistakes:

• Confusing starts_with with full word search
• Expecting False because 'app' is prefix, not full word
• Ignoring that 'app' was inserted

4. Identify the bug in the following Trie starts_with method and select the correct fix:

def starts_with(self, prefix):
    node = self.root
    for char in prefix:
        if char in node.children:
            node = node.children[char]
        else:
            return True
    return False

medium

A. Return False inside else, return True after loop

B. Return True inside else, return True after loop

C. Return False inside else, return False after loop

D. Return True inside else, return False after loop

Solution

1. Step 1: Analyze current else return

   Currently, if char not found, method returns True incorrectly; should return False.

2. Step 2: Correct return values

   Return False inside else (prefix not found), and True after loop (prefix fully matched).

3. Final Answer:

   Return False inside else, return True after loop -> Option A

4. Quick Check:

   Prefix missing = False, prefix found = True [OK]

Hint: Return False if char missing, True if all found [OK]

Common Mistakes:

• Returning True when prefix char missing
• Returning False after loop instead of True
• Confusing else and after loop returns

5. You want to find all words in a Trie that start with the prefix "car". Which approach correctly extends the prefix search to list all such words?

hard

A. Traverse to node of 'car' prefix, then recursively collect all words from that node

B. Search each word in Trie and check if it starts with 'car' using string slicing

C. Insert 'car' as a word and then search for it

D. Use a hash map to store all words and filter by prefix

Solution

1. Step 1: Locate prefix node in Trie

   Traverse Trie nodes for 'c', 'a', 'r' to reach prefix node.

2. Step 2: Recursively collect words from prefix node

   From prefix node, explore all child nodes to build and collect complete words starting with 'car'.

3. Final Answer:

   Traverse to node of 'car' prefix, then recursively collect all words from that node -> Option A

4. Quick Check:

   Prefix node + recursion = all words with prefix [OK]

Hint: Find prefix node, then DFS to collect words [OK]

Common Mistakes:

• Checking all words instead of using Trie structure
• Inserting prefix as word incorrectly
• Using unrelated data structures like hash map