Autocomplete System with Trie in DSA C# - Time & Space Complexity

Time Complexity: Autocomplete System with Trie

O(p + m)

Understanding Time Complexity

We want to understand how fast an autocomplete system using a Trie can find suggestions.

How does the time to find words grow as we add more words or type longer prefixes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public List<string> GetWordsWithPrefix(string prefix) {
    TrieNode node = root;
    foreach (char c in prefix) {
        if (!node.Children.ContainsKey(c)) return new List<string>();
        node = node.Children[c];
    }
    return CollectAllWords(node, prefix);
}

private List<string> CollectAllWords(TrieNode node, string prefix) {
    List<string> results = new List<string>();
    if (node.IsWord) results.Add(prefix);
    foreach (var child in node.Children) {
        results.AddRange(CollectAllWords(child.Value, prefix + child.Key));
    }
    return results;
}
    

This code finds all words in the Trie that start with a given prefix.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Traversing the prefix characters once, then recursively visiting all child nodes under that prefix.
• How many times: The prefix traversal runs once per character in the prefix (length p). The recursive calls visit all descendant nodes under the prefix node, depending on how many words share that prefix.

How Execution Grows With Input

The time to find the prefix node grows linearly with prefix length.

After that, the time depends on how many words share that prefix, because we collect all those words.

Input Size (n)	Approx. Operations
10 words, prefix length 3	3 steps to find prefix + visits to matching words (few)
100 words, prefix length 3	3 steps + more recursive visits if many words share prefix
1000 words, prefix length 3	3 steps + many recursive visits if prefix is common

Pattern observation: The prefix search is fast and depends only on prefix length, but collecting words depends on how many words share that prefix.

Final Time Complexity

Time Complexity: O(p + m)

This means the search takes time proportional to the prefix length plus the number of matching words found.

Common Mistake

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

[OK] Correct: Actually, the search depends on prefix length and how many words share that prefix, not all words in the Trie.

Interview Connect

Understanding this time complexity helps you explain how autocomplete systems stay fast even with many words, which is a common real-world problem.

Self-Check

"What if we limited the number of suggestions returned to a fixed number k? How would the time complexity change?"

Practice

1. What is the main advantage of using a Trie over a HashSet for implementing an autocomplete system?

easy

A. Trie uses less memory than HashSet for all cases

B. Trie supports prefix-based search efficiently, while HashSet requires checking all words

C. Trie is faster for exact word lookup than HashSet

D. Trie automatically sorts words alphabetically unlike HashSet

Solution

1. Step 1: Understand autocomplete requirement

   Autocomplete needs to find all words that start with a given prefix quickly.

2. Step 2: Compare Trie vs HashSet

   Trie stores characters level by level, so prefix traversal is O(prefix length). HashSet requires scanning all words to check which ones start with the prefix.

3. Final Answer:

   Trie supports prefix-based search efficiently, while HashSet requires checking all words -> Option B

4. Quick Check:

   Trie = prefix search in O(m), HashSet = O(n*m) for prefix [OK]

Hint: Trie shares common prefixes among words - natural fit for autocomplete [OK]

Common Mistakes:

• Assuming Trie always uses less memory - it can use more for sparse word sets
• Thinking exact lookup is why Trie beats HashSet - HashSet is actually faster for exact lookup
• Assuming Trie sorts automatically - it doesn't sort, traversal order depends on children insertion

2. Which C# code correctly inserts a word into a Trie node structure where each node has a Dictionary<char, TrieNode> Children and a bool IsEndOfWord?

easy

A. node.Children[c] = new TrieNode(); node = node.Children[c];

B. if (!node.Children.ContainsKey(c)) node.Children[c] = new TrieNode(); node = node.Children[c];

C. node.Children.Add(c); node = node.Children[c];

D. node.Children[c] = c; node = node.Children[c];

Solution

1. Step 1: Check if child exists

   Before creating a new node, check if the character key already exists in Children to avoid overwriting existing subtrees.

2. Step 2: Create only if missing

   Using ContainsKey check ensures we reuse existing nodes and only create new ones when the character path doesn't exist yet.

3. Final Answer:

   if (!node.Children.ContainsKey(c)) node.Children[c] = new TrieNode(); node = node.Children[c]; -> Option B

4. Quick Check:

   Always check ContainsKey before adding to avoid overwriting existing paths [OK]

Hint: Always guard with ContainsKey before inserting a new Trie node [OK]

Common Mistakes:

• Option A overwrites existing children, destroying previously inserted words
• Option C uses Add which throws if key already exists
• Option D stores char as value instead of TrieNode

3. What will the following C# Trie prefix search return when searching for prefix "ca" in a Trie containing ["cat", "car", "card", "dog"]?

List<string> results = new List<string>();
void DFS(TrieNode node, string current) {
    if (node.IsEndOfWord) results.Add(current);
    foreach (var kvp in node.Children)
        DFS(kvp.Value, current + kvp.Key);
}
TrieNode prefixNode = GetPrefixNode("ca");
if (prefixNode != null) DFS(prefixNode, "ca");

medium

A. ["cat", "car", "card", "dog"]

B. ["ca"]

C. ["cat", "car", "card"]

D. []

Solution

1. Step 1: Find prefix node

   GetPrefixNode("ca") traverses c→a and returns the node at 'a'. "dog" is unreachable from this node.

2. Step 2: DFS from prefix node

   DFS explores all children from 'a': t (cat), r (car), r→d (card). All three have IsEndOfWord=true.

3. Final Answer:

   ["cat", "car", "card"] -> Option C

4. Quick Check:

   Only words under prefix "ca" subtree are returned, "dog" excluded [OK]

Hint: DFS from prefix node returns only words in that subtree - "dog" is in a different branch [OK]

Common Mistakes:

• Including 'dog' - it's in a completely separate Trie branch
• Returning only 'ca' - 'ca' itself is not marked IsEndOfWord
• Returning empty - prefix 'ca' exists in the Trie so DFS finds words

4. Identify the bug in this C# autocomplete prefix search:

bool StartsWith(string prefix) {
    TrieNode node = root;
    foreach (char c in prefix) {
        if (!node.Children.ContainsKey(c))
            return true;
        node = node.Children[c];
    }
    return false;
}

medium

A. Should return false when character not found, and true at the end

B. Should use a List instead of Dictionary for Children

C. Should check IsEndOfWord instead of ContainsKey

D. Should start from the last character of the prefix

Solution

1. Step 1: Trace the logic

   When a character is not found in Children, it means the prefix doesn't exist - should return false, not true. And after traversing all prefix characters successfully, the prefix exists - should return true.

2. Step 2: Identify the swap

   The return values are swapped: 'return true' and 'return false' are in each other's place.

3. Final Answer:

   Should return false when character not found, and true at the end -> Option A

4. Quick Check:

   Missing prefix char = false, full prefix traversed = true [OK]

Hint: If you can't follow the path, prefix is absent = false; if path complete = true [OK]

Common Mistakes:

• Thinking Dictionary is wrong - Dictionary is correct for Children
• Checking IsEndOfWord - StartsWith only checks if prefix path exists, not word end
• Starting from last character - prefix traversal always goes left to right

5. In a Trie-based autocomplete system with 100,000 words averaging 8 characters each, what is the time complexity to retrieve all words matching a prefix of length p, where k is the number of matching words and m is the average word length?

hard

A. O(n) where n is total number of words

B. O(p + k * m) - traverse prefix then DFS over matching words

C. O(p * n) - check prefix against each word

D. O(1) - Trie provides constant time autocomplete

Solution

1. Step 1: Cost of reaching prefix node

   Traversing p characters to reach the prefix node costs O(p).

2. Step 2: Cost of DFS to collect results

   DFS visits all nodes under the prefix subtree. With k matching words of average length m, this costs O(k * m) in the worst case.

3. Final Answer:

   O(p + k * m) -> Option B

4. Quick Check:

   Prefix traversal O(p) + result collection O(k*m) = O(p + k*m) [OK]

Hint: Trie autocomplete = prefix path O(p) + collect results O(k*m), independent of total word count n [OK]

Common Mistakes:

• O(n) - Trie doesn't scan all words, only the prefix subtree
• O(1) - result collection depends on number and length of matching words
• O(p*n) - that's the HashSet approach, not Trie