Prefix Search Using Trie in DSA C - 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.

The question is: how does the search time grow as the prefix length or number of words changes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Search for prefix in Trie
bool startsWith(TrieNode* root, const char* prefix) {
    TrieNode* node = root;
    for (int i = 0; prefix[i] != '\0'; i++) {
        int index = prefix[i] - 'a';
        if (!node->children[index]) {
            return false;
        }
        node = node->children[index];
    }
    return true;
}
    

This code checks if any word in the Trie starts with the given prefix by following nodes for each prefix character.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Loop over each character of the prefix.
• How many times: Exactly once per prefix character, so length of prefix (let's call it m).

How Execution Grows With Input

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

Input Size (prefix length m)	Approx. Operations
10	10 steps
100	100 steps
1000	1000 steps

Pattern observation: The search time grows linearly with the prefix length, independent of total words stored.

Final Time Complexity

Time Complexity: O(m)

This means the search time depends only on how long the prefix is, 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 nodes for the prefix characters, so it depends on prefix length, not total words.

Interview Connect

Knowing how Trie prefix search scales helps you explain efficient word lookups clearly and confidently in interviews.

Self-Check

"What if we changed the Trie to store uppercase and lowercase letters? 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 that start with a given prefix

Solution

1. Step 1: Understand the role of Trie in prefix search

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

2. Step 2: Identify the main use case

   It helps find all words starting with a prefix efficiently, unlike sorting or arithmetic.

3. Final Answer:

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

4. Quick Check:

   Trie = Prefix search efficiency [OK]

Hint: Trie is built for prefix-based word retrieval [OK]

Common Mistakes:

• Confusing Trie with sorting algorithms
• Thinking Trie is for arithmetic
• Assuming Trie stores numbers only

2. Which of the following is the correct way to declare a Trie node structure in C for prefix search?

easy

A. typedef struct TrieNode { int data; struct TrieNode* next; } TrieNode;

B. typedef struct TrieNode { char* word; struct TrieNode* child; } TrieNode;

C. typedef struct TrieNode { struct TrieNode* children[26]; int isEndOfWord; } TrieNode;

D. typedef struct TrieNode { int value; struct TrieNode* left; struct TrieNode* right; } TrieNode;

Solution

1. Step 1: Identify Trie node components

   A Trie node has an array of pointers for each alphabet letter and a flag for word end.

2. Step 2: Match with given options

   typedef struct TrieNode { struct TrieNode* children[26]; int isEndOfWord; } TrieNode; matches this structure with 26 children pointers and an end-of-word flag.

3. Final Answer:

   typedef struct TrieNode { struct TrieNode* children[26]; int isEndOfWord; } TrieNode; -> Option C

4. Quick Check:

   Trie node = children array + end flag [OK]

Hint: Trie nodes have 26 children pointers for letters a-z [OK]

Common Mistakes:

• Using linked list pointers instead of array for children
• Confusing Trie node with linked list or binary tree node
• Missing end-of-word flag

3. Given the following code snippet inserting words "cat", "car", and "dog" into a Trie, what will be the output of searching prefix "ca"?

insert("cat");
insert("car");
insert("dog");
searchPrefix("ca");

medium

A. ["cat", "car"]

B. ["dog"]

C. ["cat"]

D. []

Solution

1. Step 1: Insert words into Trie

   Words "cat", "car", and "dog" are inserted, building branches for 'c' and 'd'.

2. Step 2: Search prefix "ca"

   Both "cat" and "car" start with "ca", so they are returned by searchPrefix.

3. Final Answer:

   ["cat", "car"] -> Option A

4. Quick Check:

   Prefix "ca" matches cat and car [OK]

Hint: Words starting with prefix are returned [OK]

Common Mistakes:

• Returning only one matching word
• Returning words not matching prefix
• Empty result for existing prefix

4. Consider this buggy C code snippet for inserting a word into a Trie. What is the main error?

void insert(TrieNode* root, char* word) {
  TrieNode* node = root;
  for (int i = 0; word[i] != '\0'; i++) {
    int index = word[i] - 'a';
    if (node->children[index] == NULL) {
      node->children[index] = malloc(sizeof(TrieNode));
    }
    node = node->children[index];
  }
  node->isEndOfWord = 1;
}

medium

A. New nodes are not initialized after malloc

B. Index calculation is incorrect

C. Loop condition should be i < strlen(word)

D. isEndOfWord flag is set too early

Solution

1. Step 1: Analyze node creation

   malloc allocates memory but does not initialize children pointers or isEndOfWord flag.

2. Step 2: Identify impact

   Uninitialized children pointers can cause undefined behavior during traversal or insertion.

3. Final Answer:

   New nodes are not initialized after malloc -> Option A

4. Quick Check:

   malloc needs initialization [OK]

Hint: Always initialize new Trie nodes after malloc [OK]

Common Mistakes:

• Assuming malloc zeroes memory
• Ignoring initialization of isEndOfWord
• Misunderstanding loop condition

5. You want to find all words in a Trie that start with prefix "app" and have length exactly 5. Which approach correctly combines prefix search and length filtering?

hard

A. Collect all words in Trie, then filter those starting with "app" and length 5

B. Traverse Trie to node for "app", then collect words of length 5 from there

C. Search for "app" as a full word, then check length 5

D. Use binary search on sorted word list for prefix "app" and length 5

Solution

1. Step 1: Use Trie prefix traversal

   Traverse the Trie to the node representing the prefix "app" to limit search space.

2. Step 2: Collect words of length 5 from that node

   From this node, recursively collect words that complete to length 5.

3. Final Answer:

   Traverse Trie to node for "app", then collect words of length 5 from there -> Option B

4. Quick Check:

   Prefix traversal + length filter = Traverse Trie to node for "app", then collect words of length 5 from there [OK]

Hint: Start at prefix node, then filter by length [OK]

Common Mistakes:

• Filtering all words instead of prefix subtree
• Searching prefix as full word
• Using binary search on Trie data