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Time: O(k × m × n × 8^L) where L = max word length Space: O(L) for recursion stack """ m, n = len(board), len(board[0]) directions = [(-1,-1), (-1,0), (-1,1), (0,-1), (0,1), (1,-1), (1,0), (1,1)] found = [] def dfs(row: int, col: int, word_index: int, word: str, visited: set) -> bool: # Base case: word completely matched if word_index == len(word): return True # Bounds check and character match if (row < 0 or row >= m or col < 0 or col >= n or (row, col) in visited or board[row][col] != word[word_index]): return False # Mark as visited visited.add((row, col)) # Try all 8 directions for dr, dc in directions: if dfs(row + dr, col + dc, word_index + 1, word, visited): visited.remove((row, col)) # Backtrack return True # Backtrack visited.remove((row, col)) return False # Check each word individually - THIS IS THE PROBLEM for word in words: word_found = False for i in range(m): for j in range(n): if board[i][j] == word[0]: # Start if first char matches if dfs(i, j, 0, word, set()): word_found = True break if word_found: break if word_found: found.append(word) return found73:T7d7,class TrieNode: """ A trie node optimized for word search problems. Key features: - 'children' dict for O(1) child lookup - 'word' stores the complete word at end nodes (for easy result collection) - 'count' tracks number of words passing through (for optimization) """ __slots__ = ['children', 'word', 'count'] # Memory optimization def __init__(self): self.children: dict[str, 'TrieNode'] = {} self.word: str | None = None # Non-None if this node marks end of a word self.count: int = 0 # Number of words passing through this node class Trie: """ Trie data structure for word search optimization. Supports: - insert(word): Add a word to the trie - Word storage at leaf nodes for result collection - Count tracking for optimization during search """ def __init__(self): self.root = TrieNode() def insert(self, word: str) -> None: """ Insert a word into the trie. Time: O(L) where L = len(word) Space: O(L) in worst case (no prefix sharing) """ node = self.root for char in word: if char not in node.children: node.children[char] = TrieNode() node = node.children[char] node.count += 1 # Track words passing through node.word = word # Store complete word at end node def build_from_words(self, words: list[str]) -> None: """ Build trie from a list of words. Time: O(sum of all word lengths) """ for word in words: self.insert(word) # Example usage: words = ["oath", "pea", "eat", "rain", "oat", "eta"] trie = Trie() trie.build_from_words(words) # The trie structure now looks like: # # (root) # / | \ # o p e... # | | # a e # | | # t* a* # / # h* # # Where * indicates node.word is set to the complete word74:T10a8,class TrieNode: __slots__ = ['children', 'word', 'count'] def __init__(self): self.children: dict[str, 'TrieNode'] = {} self.word: str | None = None self.count: int = 0 class Solution: """ Word Search II - Trie-Optimized Solution Time Complexity: O(m × n × 4^L) where L = max word length But with aggressive pruning, much faster in practice Space Complexity: O(k × L) for trie + O(L) for recursion where k = number of words """ def find_words(self, board: list[list[str]], words: list[str]) -> list[str]: if not board or not board[0] or not words: return [] # Step 1: Build trie from all dictionary words root = TrieNode() for word in words: node = root for char in word: if char not in node.children: node.children[char] = TrieNode() node = node.children[char] node.count += 1 node.word = word # Step 2: Set up grid traversal m, n = len(board), len(board[0]) # 4-directional movement (can be 8 for diagonal) directions = [(-1, 0), (1, 0), (0, -1), (0, 1)] result = [] def dfs(row: int, col: int, node: TrieNode) -> None: """ DFS traversal of grid, synchronized with trie navigation. Args: row, col: Current position in grid node: Current position in trie (represents prefix matched so far) """ char = board[row][col] # Pruning: If current char not in trie children, no word continues if char not in node.children: return # Move to child node in trie child_node = node.children[char] # Check if we've found a word if child_node.word is not None: result.append(child_node.word) # OPTIMIZATION: Remove word to prevent duplicates child_node.word = None # Decrement counts to enable pruning self._decrement_counts(root, child_node.word if child_node.word else result[-1]) # OPTIMIZATION: If no words remain through this path, prune if child_node.count == 0: # Remove this child from parent to speed up future searches del node.children[char] return # Mark cell as visited (in-place modification) original_char = board[row][col] board[row][col] = '#' # Use special marker # Explore all 4 directions for dr, dc in directions: new_row, new_col = row + dr, col + dc if 0 <= new_row < m and 0 <= new_col < n: if board[new_row][new_col] != '#': # Not visited dfs(new_row, new_col, child_node) # Backtrack: Restore cell board[row][col] = original_char # Step 3: Start DFS from every cell for i in range(m): for j in range(n): if board[i][j] in root.children: dfs(i, j, root) return result def _decrement_counts(self, root: TrieNode, word: str) -> None: """ Decrement counts along the path for a found word. This enables pruning of exhausted branches. """ node = root for char in word: if char in node.children: node.children[char].count -= 1 # If count reaches 0, we could delete the child if node.children[char].count == 0: del node.children[char] return node = node.children[char] # Example usage solution = Solution() board = [ ['o', 'a', 'a', 'n'], ['e', 't', 'a', 'e'], ['i', 'h', 'k', 'r'] ] words = ["oath", "pea", "eat", "rain", "oat", "eta"] result = solution.find_words(board, words) print(result) # ['oath', 'oat', 'eat', 'eta'] (order may vary)75:T81f,def find_words_optimized(board: list[list[str]], words: list[str]) -> list[str]: """ Fully optimized word search with all production-level enhancements. """ if not board or not board[0] or not words: return [] m, n = len(board), len(board[0]) # OPTIMIZATION 1: Pre-compute grid character frequency grid_chars = set() for row in board: grid_chars.update(row) # OPTIMIZATION 2: Filter words with impossible characters valid_words = [] for word in words: if all(c in grid_chars for c in word) and len(word) <= m * n: valid_words.append(word) if not valid_words: return [] # Build trie only with valid words root = build_trie(valid_words) # OPTIMIZATION 3: Pre-compute which cells can start a word starting_cells = [] for i in range(m): for j in range(n): if board[i][j] in root.children: starting_cells.append((i, j)) result = [] directions = [(-1, 0), (1, 0), (0, -1), (0, 1)] def dfs(row: int, col: int, node: TrieNode) -> None: char = board[row][col] if char not in node.children: return child = node.children[char] if child.word: result.append(child.word) child.word = None # Prevent duplicates # OPTIMIZATION 4: In-place marking with original char 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Your task is to find all words from the dictionary that can be formed by starting at any cell and traversing adjacent cells (horizontally, vertically, or diagonally) without revisiting cells.\n\nThis problem appears everywhere: word games, text mining, pattern matching, and is a **favorite in technical interviews** because it tests multiple skills simultaneously—graph traversal, backtracking, data structure selection, and algorithmic optimization.\n\nThe naive approach—check each word individually—quickly becomes catastrophically slow. But with tries, we transform this from an intractable problem into an elegant, efficient solution. This is one of the most powerful and instructive applications of the trie data structure."},"type":"text"},{"data":{"icon":"Target","title":"What You Will Learn","content":"By the end of this page, you will understand: (1) Why naive word search fails at scale, (2) How tries enable searching for multiple words simultaneously, (3) The complete implementation of trie-optimized grid search, (4) Time and space complexity analysis, and (5) Optimization techniques used in production systems.","variant":"info"},"type":"callout"}]},{"id":"understanding-the-problem","title":"Understanding the Problem: Word Search II","content":[{"data":{"content":"Let's formalize the **Word Search in Grid** problem, often called **Word Search II** in competitive programming contexts.\n\n**Problem Statement:**\n\nGiven:\n- An `m × n` grid `board` of characters\n- A list of `k` words from a dictionary\n\nFind all words that can be constructed by sequentially connecting adjacent cells. Adjacent cells are those horizontally, vertically, or diagonally neighboring (8 directions, or 4 directions in simpler variants). Each cell may only be used **once per word**.\n\n**Example:**"},"type":"text"},{"data":{"code":"Grid (3×4):\n┌───┬───┬───┬───┐\n│ o │ a │ a │ n │\n├───┼───┼───┼───┤\n│ e │ t │ a │ e │\n├───┼───┼───┼───┤\n│ i │ h │ k │ r │\n└───┴───┴───┴───┘\n\nDictionary: [\"oath\", \"pea\", \"eat\", \"rain\", \"oat\", \"eta\"]\n\nFound words: [\"oath\", \"eat\", \"oat\", \"eta\"]\n\nPaths:\n• \"oath\": (0,0)→(1,0)→(1,1)→(2,1) [o→a→t→h but wait, that's wrong]\n Actually: (0,0)→(0,1)→(1,1)→(2,1) [o→a→t→h] ✓\n• \"eat\": (1,0)→(0,1)→(1,1) [e→a→t] ✓\n• \"oat\": (0,0)→(0,1)→(1,1) [o→a→t] ✓\n• \"eta\": (1,0)→(1,1)→(0,1) [e→t→a] ✓","fileName":"Problem Example","showLineNumbers":false},"type":"code"},{"data":{"content":"**Why This Problem Is Challenging:**\n\nThe challenge lies in the **combinatorial explosion** of possible paths. From each cell, you can move to up to 8 neighbors. For a path of length `L`, the number of possible paths from a single starting cell is roughly `8^L`. Multiply this across all `m × n` starting cells and all `k` words in the dictionary.\n\nWith a 10×10 grid and 1000 words averaging 5 characters each, the naive approach performs approximately:\n\n`100 (cells) × 1000 (words) × 8^5 (paths) ≈ 3.3 billion operations`\n\nThis is clearly unacceptable. We need a smarter approach."},"type":"text"}],"titleType":"number","titleNumber":1},{"id":"naive-approach","title":"The Naive Approach and Its Limitations","content":[{"data":{"content":"Before appreciating the trie optimization, let's understand exactly why the straightforward approach fails.\n\n**The Naive Strategy:**\n\nFor each word in the dictionary:\n1. For each cell in the grid:\n - If the cell matches the word's first character, start DFS\n - Try to match the remaining characters by exploring adjacent cells\n - Backtrack when stuck or when the word is found"},"type":"text"},{"data":{"code":"$72","fileName":"naive_approach.py","languages":[{"label":"Python","value":"python"},{"label":"TypeScript","value":"typescript"}],"showLineNumbers":true},"type":"code"},{"data":{"icon":"AlertCircle","title":"Critical Inefficiency","content":"The naive approach performs a **complete DFS for every single word**. If 100 words share the prefix 'pre', we perform 100 separate, redundant searches to find 'pre' in the grid. This duplication of effort is catastrophic at scale.","variant":"warning"},"type":"callout"},{"data":{"content":"**Complexity Analysis of Naive Approach:**"},"type":"text"},{"data":{"rows":[["Words to check","O(k)"],["Starting positions","O(m × n)"],["Paths per position","O(8^L) where L = word length"],["**Total Time**","**O(k × m × n × 8^L)**"],["Space (recursion)","O(L)"]],"headers":["Component","Complexity"]},"type":"table"},{"data":{"content":"**The Fundamental Problem:**\n\nWe're treating each word as an isolated search. But words share structure—they share prefixes! The words 'cat', 'car', 'card', 'care' all start with 'ca'. When we find 'ca' in the grid, we should be able to check **all four words simultaneously**, not search four separate times."},"type":"text"}],"titleType":"number","titleNumber":2},{"id":"trie-insight","title":"The Trie Insight: Searching Multiple Words Simultaneously","content":[{"data":{"content":"The key insight that transforms this problem is recognizing that **tries naturally represent shared prefixes as shared paths**.\n\nInstead of searching for words one at a time, we:\n\n1. **Build a trie** from all dictionary words\n2. **Search the grid once**, carrying a trie node as we traverse\n3. **Simultaneously check** if the current path matches any word prefix\n4. **Prune immediately** when the current path isn't a prefix of any word\n\nThis is a paradigm shift: we go from **k separate word searches** to **one grid traversal with k-word awareness**."},"type":"text"},{"data":{"left":[{"data":{"items":["Search for 'apple': Full DFS","Search for 'apply': Full DFS (repeating 'appl')","Search for 'application': Full DFS (repeating 'appli')","3 complete searches","Massive redundancy"],"title":{"icon":"XCircle","text":"Naive: Per-Word Search","type":"icon","color":"destructive","variant":"outline-circle"},"variant":"accent"},"type":"list"}],"right":[{"data":{"items":["Build trie with all 3 words","One DFS traversal","When at 'appl', know all 3 words are possible","Find 'e' → found 'apple'","Find 'y' → found 'apply'","No redundant prefix exploration"],"title":{"icon":"CheckCircle","text":"Trie: Unified Search","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"}]},"type":"two-column"},{"data":{"content":"**Visual: Trie Enables Prefix Sharing**\n\nConsider the dictionary: `['cat', 'car', 'card', 'care', 'cards']`"},"type":"text"},{"data":{"code":" (root)\n │\n c\n │\n a ─────────┐\n │ │\n t* r*\n │\n ┌───┴───┐\n d* e*\n │\n s*\n\nNodes marked with * indicate end-of-word.\n\nWhen exploring the grid:\n- Find 'c' → Enter trie at 'c'\n- Find 'a' → Move to 'a' node (both 'cat' and 'car*' branches exist)\n- Find 't' → Move to 't*' → Found \"cat\"! Continue searching...\n- Find 'r' → Move to 'r*' → Found \"car\"! Continue for 'd' and 'e'...\n\nONE traversal checks ALL words simultaneously!","fileName":"Trie Structure","showLineNumbers":false},"type":"code"},{"data":{"icon":"Lightbulb","title":"The Power of Early Termination","content":"When the trie node has no child for the current character, we **immediately prune** that search path. If no dictionary word starts with 'cx', we never explore past 'cx' in the grid—regardless of how many cells we'd otherwise visit. This pruning is the source of the dramatic speedup.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":3},{"id":"building-trie","title":"Building the Trie for Word Search","content":[{"data":{"content":"The first step is constructing a trie from the dictionary. For word search problems, we need a trie that:\n\n1. **Stores the complete word** at end-of-word nodes (to collect results easily)\n2. **Supports efficient child lookup** (hash map for varied alphabets)\n3. **Allows node removal** for an important optimization (discussed later)\n\nLet's build this specialized trie:"},"type":"text"},{"data":{"code":"$73","fileName":"trie_for_word_search.py","languages":[{"label":"Python","value":"python"},{"label":"TypeScript","value":"typescript"}],"showLineNumbers":true},"type":"code"},{"data":{"icon":"Info","title":"Why Store the Complete Word?","content":"By storing the complete word at end-of-word nodes (`node.word`), we avoid reconstructing the word during DFS. When we reach an end-of-word node during grid traversal, we simply add `node.word` to our results. This is cleaner and more efficient than tracking the path.","variant":"note"},"type":"callout"},{"data":{"content":"**The `count` Field Explained:**\n\nThe `count` field tracks how many dictionary words \"pass through\" each node. This enables a crucial optimization:\n\n- When we find a word, we can **decrement counts** along its path\n- When a node's count reaches 0, **no remaining words use that prefix**\n- We can then **prune future searches** through that node\n\nThis prevents finding the same word multiple times and accelerates the search as words are discovered."},"type":"text"}],"titleType":"number","titleNumber":4},{"id":"trie-optimized-algorithm","title":"The Complete Trie-Optimized Algorithm","content":[{"data":{"content":"Now we combine everything into the complete, optimized word search algorithm. The key insight is that we **traverse the grid while simultaneously navigating the trie**. The trie node acts as our \"state\" of which words are still possible."},"type":"text"},{"data":{"code":"$74","fileName":"word_search_optimized.py","languages":[{"label":"Python","value":"python"},{"label":"TypeScript","value":"typescript"}],"showLineNumbers":true},"type":"code"},{"data":{"content":"**Algorithm Walkthrough:**\n\n1. **Build the Trie** (Lines 23-31): Insert all dictionary words. Each end-of-word node stores the complete word.\n\n2. **Start DFS from Each Cell** (Lines 79-82): For each cell whose character exists as a child of root, begin DFS.\n\n3. **Synchronized Traversal** (Lines 39-75): As we move through the grid, we simultaneously move through the trie. The trie node tells us which words are still possible.\n\n4. **Word Discovery** (Lines 53-59): When we reach a node with `word` set, we've found a dictionary word. Add it to results.\n\n5. **Backtracking** (Lines 69-74): Restore the cell and return, allowing other paths to use this cell.\n\n6. **Pruning** (Lines 61-65): When a branch is exhausted (count = 0), remove it from the trie to prevent future wasteful exploration."},"type":"text"}],"titleType":"number","titleNumber":5},{"id":"critical-optimizations","title":"Critical Optimizations for Production","content":[{"data":{"content":"The basic trie-optimized algorithm is already dramatically faster than the naive approach. However, several additional optimizations can make it production-ready:"},"type":"text"},{"data":{"items":["**1. In-Place Visited Marking**: Instead of a `visited` set (O(1) but high overhead), modify the board directly by replacing characters with a marker like '#'. Restore during backtracking. This avoids set operations entirely.","**2. Dynamic Trie Pruning**: When a word is found, remove it from the trie (set `word = None`) and decrement counts. When counts hit zero, delete the branch. This shrinks the trie as we find words.","**3. Early Termination on Root Children**: Before starting DFS from a cell, check if that character exists in `root.children`. Skip cells that can't start any word.","**4. Character Frequency Pre-check**: Before starting, verify that the grid contains all characters needed for the dictionary words. Skip words whose characters don't exist in the grid.","**5. Length-Based Pruning**: Skip words longer than `m × n` (impossible to form). Track max word length to limit DFS depth."],"title":{"icon":"Zap","text":"Key Optimizations","type":"icon","color":"primary","variant":"outline-circle"},"variant":"glass"},"type":"list"},{"data":{"code":"$75","fileName":"optimizations.py","showLineNumbers":true},"type":"code"},{"data":{"icon":"Lightbulb","title":"Why These Optimizations Matter","content":"In worst-case scenarios, these optimizations can provide 10-100x speedup. The character frequency pre-check alone can eliminate 50%+ of words in many real-world dictionaries. Dynamic pruning ensures the algorithm gets faster as it progresses—the trie shrinks with each found word.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":6},{"id":"complexity-analysis","title":"Time and Space Complexity Analysis","content":[{"data":{"content":"Understanding the complexity of the trie-optimized solution requires careful analysis of both the trie construction and the grid search phases."},"type":"text"},{"data":{"rows":[["Time: Build Phase","N/A","O(W × L) where W = total words, L = avg length"],["Time: Search Phase","O(k × m × n × 4^L)","O(m × n × 4^L)"],["Time: Overall","O(k × m × n × 4^L)","O(W × L + m × n × 4^L)"],["Space: Data Structure","O(1)","O(W × L) for trie"],["Space: Recursion","O(L)","O(L)"],["Pruning Benefit","None","Massive (early termination)"]],"caption":"Complexity Comparison: Naive vs Trie-Optimized","headers":["Aspect","Naive Approach","Trie-Optimized"]},"type":"table"},{"data":{"content":"**Detailed Analysis:**\n\n**Time Complexity:**\n\n1. **Trie Construction**: O(W × L) where W is the total number of characters across all words\n - Each character is processed exactly once\n\n2. **Grid Search**: O(m × n × 4^L) in the worst case\n - We start from each of m × n cells\n - Each path can extend up to 4^L ways (4 directions, L = max word length)\n\n3. **Key Improvement**: The factor of **k (number of words)** is eliminated!\n - Naive: O(**k** × m × n × 4^L) — scales with dictionary size\n - Optimized: O(m × n × 4^L) — independent of dictionary size!\n\n**In Practice:**\n\nThe 4^L factor is a **loose upper bound**. In practice:\n- Trie pruning eliminates most paths early\n- Visited cell marking prevents revisits\n- Dynamic branch removal accelerates as words are found\n- Average case is often O(m × n × L) for reasonable inputs"},"type":"text"},{"data":{"icon":"CheckCircle","title":"The Key Speedup","content":"Eliminating the k factor is the crucial win. With 10,000 dictionary words, the naive approach is 10,000× slower. The trie-optimized approach handles 10 words or 10 million words with the same grid traversal—only the trie build time increases linearly.","variant":"success"},"type":"callout"},{"data":{"content":"**Space Complexity:**\n\n1. **Trie Storage**: O(W × L) in the worst case\n - If all words are completely distinct, each character needs a node\n - In practice, prefix sharing reduces this significantly\n\n2. **Recursion Stack**: O(L) where L is the maximum word length\n - DFS depth is bounded by the longest possible word\n\n3. **Result Storage**: O(k) for storing found words\n\n**Space Optimization Note:**\n\nFor memory-constrained environments, consider processing words in batches—build smaller tries, search, then rebuild. This trades time for space."},"type":"text"}],"titleType":"number","titleNumber":7},{"id":"real-world-applications","title":"Real-World Applications","content":[{"data":{"content":"The trie-optimized word search pattern extends far beyond puzzle games. Understanding this pattern unlocks solutions to many practical problems:"},"type":"text"},{"data":{"items":["**Word Puzzle Games (Boggle, Scrabble)**: Find all valid words formable on a game board. Commercial games use this exact algorithm.","**Text Mining and OCR Validation**: When OCR produces uncertain character grids, search for valid dictionary words to improve accuracy.","**DNA Sequence Analysis**: Search for known gene patterns in nucleotide grids. The 4-character alphabet (A, C, G, T) is perfect for tries.","**Crossword Puzzle Validation**: Verify that all horizontal and vertical sequences form valid words.","**Name Entity Recognition**: Search document layouts for known entity names, handling 2D spatial relationships.","**Security Pattern Matching**: Detect known malicious patterns in 2D representations of binary data."],"title":{"icon":"Target","text":"Industry Applications","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"icon":"Info","title":"Pattern Recognition","content":"Whenever you face a problem involving: (1) a 2D grid of characters, (2) a dictionary of patterns to find, and (3) path-based matching, immediately consider the trie-optimized approach. It's the canonical solution for this problem class.","variant":"note"},"type":"callout"}],"titleType":"number","titleNumber":8},{"id":"summary","title":"Summary: Word Search with Tries","content":[{"data":{"content":"We've covered one of the most powerful applications of the trie data structure—optimizing word search in grids. Let's consolidate the key insights:"},"type":"text"},{"data":{"type":"ordered","items":["**The Problem**: Finding dictionary words in a character grid has exponential path complexity. Naive per-word search is O(k × m × n × 4^L).","**The Insight**: Tries represent shared prefixes as shared paths. This enables searching for all k words simultaneously.","**The Algorithm**: Build a trie from the dictionary. Traverse the grid once, carrying a trie node. When the trie has no child for a character, prune immediately.","**The Optimization**: Eliminate the k factor by searching for all words in one traversal—O(m × n × 4^L) regardless of dictionary size.","**Production Enhancements**: In-place marking, dynamic pruning, character pre-filtering, and length-based cutoffs provide 10-100× additional speedup.","**The Pattern**: This trie + DFS combination applies to any multi-pattern search in structured data."],"title":{"icon":"CheckCircle","text":"Key Takeaways","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"icon":"CheckCircle","title":"Page Complete","content":"You now understand how to use tries to optimize word search in grids—a classic and frequently-tested pattern. 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