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This is the most common implementation: - graph[i] contains a list of all vertices adjacent to vertex i - Works for both directed and undirected graphs """ def __init__(self, num_vertices: int): self.V = num_vertices # Create a list of empty lists, one per vertex self.adj = [[] for _ in range(num_vertices)] def add_edge(self, u: int, v: int, directed: bool = False): """Add edge from u to v (and v to u if undirected).""" self.adj[u].append(v) if not directed: self.adj[v].append(u) def neighbors(self, v: int) -> list[int]: """Return all neighbors of vertex v.""" return self.adj[v] def has_edge(self, u: int, v: int) -> bool: """Check if edge (u, v) exists.""" return v in self.adj[u] # O(degree) time! def degree(self, v: int) -> int: """Return the degree of vertex v.""" return len(self.adj[v]) # Example usage g = GraphArrayBased(5) g.add_edge(0, 1) g.add_edge(0, 4) g.add_edge(1, 2) g.add_edge(1, 3) g.add_edge(1, 4) g.add_edge(2, 3) g.add_edge(3, 4) print(f"Neighbors of vertex 1: {g.neighbors(1)}") # [0, 2, 3, 4] print(f"Degree of vertex 1: {g.degree(1)}") # 473:T4b6,class ListNode: """A node in the linked list of neighbors.""" def __init__(self, vertex: int): self.vertex = vertex self.next = None class GraphLinkedList: """ Adjacency list using linked lists. Classic textbook implementation. Pros: Efficient insertion at head Cons: Poor cache locality, more memory overhead """ def __init__(self, num_vertices: int): self.V = num_vertices # Array of linked list heads, one per vertex self.adj = [None] * num_vertices def add_edge(self, u: int, v: int, directed: bool = False): """Add edge by prepending to the linked list (O(1) insertion).""" # Add v to u's list new_node = ListNode(v) new_node.next = self.adj[u] self.adj[u] = new_node if not directed: # Add u to v's list new_node = ListNode(u) new_node.next = self.adj[v] self.adj[v] = new_node def neighbors(self, v: int): """Generator that yields all neighbors of vertex v.""" current = self.adj[v] while current: yield current.vertex current = current.next74:T41b,class GraphHashSet: """ Adjacency list using hash sets. Best for algorithms that frequently check edge existence. Trades memory for O(1) edge lookup. """ def __init__(self, num_vertices: int): self.V = num_vertices # Array of sets, one per vertex self.adj = [set() for _ in range(num_vertices)] def add_edge(self, u: int, v: int, directed: bool = False): """Add edge to sets (O(1) average time).""" self.adj[u].add(v) if not directed: self.adj[v].add(u) def remove_edge(self, u: int, v: int, directed: bool = False): """Remove edge (O(1) average time).""" self.adj[u].discard(v) if not directed: self.adj[v].discard(u) def has_edge(self, u: int, v: int) -> bool: """Check edge existence in O(1) average time!""" return v in self.adj[u] def neighbors(self, v: int) -> set[int]: """Return set of neighbors (iteration order may vary).""" return self.adj[v]75:T609,class DirectedGraphWithReverse: """ Maintains both forward (outgoing) and reverse (incoming) edges. Useful for algorithms that traverse edges in both directions. """ def __init__(self, num_vertices: int): self.V = num_vertices self.out_adj = [[] for _ in range(num_vertices)] # v → successors self.in_adj = [[] for _ in range(num_vertices)] # v → predecessors def add_edge(self, u: int, v: int): """Add directed edge u → v.""" self.out_adj[u].append(v) # u's outgoing neighbor self.in_adj[v].append(u) # v's incoming neighbor def successors(self, v: int) -> list[int]: """Return vertices that v points to.""" return self.out_adj[v] def predecessors(self, v: int) -> list[int]: """Return vertices that point to v.""" return self.in_adj[v] def out_degree(self, v: int) -> int: """Number of outgoing edges from v.""" return len(self.out_adj[v]) def in_degree(self, v: int) -> int: """Number of incoming edges to v.""" return len(self.in_adj[v]) # Example: Building dependency graph # Task 0 must complete before tasks 1 and 2 g = DirectedGraphWithReverse(4) g.add_edge(0, 1) # 0 → 1 g.add_edge(0, 2) # 0 → 2 g.add_edge(1, 3) # 1 → 3 g.add_edge(2, 3) # 2 → 3 # What depends on task 0? print(f"Tasks depending on 0: {g.successors(0)}") # [1, 2] # What must complete before task 3? print(f"Prerequisites of 3: {g.predecessors(3)}") # [1, 2]76:T653,# Approach 1: List of (neighbor, weight) tuples class WeightedGraphTuples: def __init__(self, num_vertices: int): self.V = num_vertices self.adj = [[] for _ in range(num_vertices)] def add_edge(self, u: int, v: int, weight: float, directed: bool = False): self.adj[u].append((v, weight)) if not directed: self.adj[v].append((u, weight)) def neighbors_with_weights(self, v: int): """Returns list of (neighbor, weight) tuples.""" return self.adj[v] # Approach 2: Dictionary of dictionaries (more Pythonic) class WeightedGraphDict: def __init__(self): self.adj = {} # {vertex: {neighbor: weight, ...}, ...} def add_vertex(self, v): if v not in self.adj: self.adj[v] = {} def add_edge(self, u, v, weight, directed=False): self.add_vertex(u) self.add_vertex(v) self.adj[u][v] = weight if not directed: self.adj[v][u] = weight def get_weight(self, u, v) -> float | None: """O(1) weight lookup for edge (u, v).""" return self.adj.get(u, {}).get(v) def neighbors(self, v): """Returns dict: {neighbor: weight, ...}""" return self.adj.get(v, {}) # Example: Road network roads = WeightedGraphDict() roads.add_edge("NYC", "Boston", 215) # 215 miles roads.add_edge("NYC", "Philadelphia", 95) roads.add_edge("Boston", "Portland", 110) print(f"NYC neighbors: {roads.neighbors('NYC')}") # {'Boston': 215, 'Philadelphia': 95} print(f"NYC to Boston distance: {roads.get_weight('NYC', 'Boston')}") # 21577:T4a0,import java.util.*; // Edge class for weighted graphs class Edge { int destination; double weight; Edge(int destination, double weight) { this.destination = destination; this.weight = weight; } } class WeightedGraph { private int V; private List> adj; WeightedGraph(int vertices) { V = vertices; adj = new ArrayList<>(V); for (int i = 0; i < V; i++) { adj.add(new ArrayList<>()); } } void addEdge(int u, int v, double weight, boolean directed) { adj.get(u).add(new Edge(v, weight)); if (!directed) { adj.get(v).add(new Edge(u, weight)); } } List getNeighbors(int v) { return adj.get(v); } // For Dijkstra's algorithm, we often need this interface: void relaxNeighbors(int u, double[] dist, PriorityQueue pq) { for (Edge e : adj.get(u)) { if (dist[u] + e.weight < dist[e.destination]) { dist[e.destination] = dist[u] + e.weight; pq.offer(new int[]{e.destination, (int)dist[e.destination]}); } } } }78:T654,class LabeledGraph: """ Graph supporting arbitrary vertex labels (strings, etc.) Internally maps labels to integer indices for efficiency. """ def __init__(self): self.label_to_index = {} # label → integer index self.index_to_label = [] # index → label self.adj = [] # adjacency list (by index) self.V = 0 def add_vertex(self, label) -> int: """Add a vertex with the given label, return its index.""" if label not in self.label_to_index: index = self.V self.label_to_index[label] = index self.index_to_label.append(label) self.adj.append([]) self.V += 1 return index return self.label_to_index[label] def add_edge(self, label_u, label_v, directed=False): """Add edge between labeled vertices.""" u = self.add_vertex(label_u) v = self.add_vertex(label_v) self.adj[u].append(v) if not directed: self.adj[v].append(u) def neighbors(self, label): """Return neighbor labels for the given vertex.""" idx = self.label_to_index.get(label) if idx is None: return [] return [self.index_to_label[i] for i in self.adj[idx]] # Example: Social network with usernames social = LabeledGraph() social.add_edge("alice", "bob") social.add_edge("alice", "carol") social.add_edge("bob", "dave") print(f"Alice's friends: {social.neighbors('alice')}") # ['bob', 'carol'] print(f"Bob's friends: {social.neighbors('bob')}") # ['alice', 'dave']79:T660,from collections import defaultdict class DynamicGraph: """ Fully dynamic graph using dictionaries. No fixed vertex count, arbitrary vertex labels. Tradeoff: Slightly more memory for dict overhead, but maximum flexibility for dynamic graphs. """ def __init__(self, directed=False): self.adj = defaultdict(set) # auto-creates empty sets self.directed = directed def add_edge(self, u, v): """Add edge (creates vertices implicitly).""" self.adj[u].add(v) if not self.directed: self.adj[v].add(u) def remove_edge(self, u, v): """Remove edge if it exists.""" self.adj[u].discard(v) if not self.directed: self.adj[v].discard(u) def remove_vertex(self, v): """Remove vertex and all its edges.""" # Remove v from all neighbor lists for neighbor in list(self.adj[v]): self.adj[neighbor].discard(v) # Remove v's entry del self.adj[v] def has_edge(self, u, v) -> bool: return v in self.adj[u] def has_vertex(self, v) -> bool: return v in self.adj def vertices(self): return self.adj.keys() def neighbors(self, v): return self.adj[v] # Example: Building graph incrementally g = DynamicGraph() g.add_edge("node_a", "node_b") g.add_edge("node_a", "node_c") g.add_edge("node_b", "node_c") print(f"All vertices: {list(g.vertices())}") # ['node_a', 'node_b', 'node_c'] g.remove_vertex("node_b") print(f"After removing node_b: {list(g.vertices())}") # ['node_a', 'node_c']7a:Te1c,from collections import defaultdict # ================================================== # Pattern 1: Edge List Input # Common in competitive programming # ================================================== def build_from_edge_list(n: int, edges: list[tuple[int, int]]) -> list[list[int]]: """ Input: number of vertices and list of (u, v) pairs Example input: n = 4 edges = [(0, 1), (0, 2), (1, 2), (2, 3)] """ adj = [[] for _ in range(n)] for u, v in edges: adj[u].append(v) adj[v].append(u) # remove for directed graph return adj # ================================================== # Pattern 2: Edge List with Number of Edges First # Very common format: first line has V E, then E edges # ================================================== def build_from_competitive_input(): """ Input format: 4 5 <- 4 vertices, 5 edges 0 1 <- edge 1 0 2 <- edge 2 1 2 <- edge 3 1 3 <- edge 4 2 3 <- edge 5 """ line = input().split() V, E = int(line[0]), int(line[1]) adj = [[] for _ in range(V)] for _ in range(E): edge = input().split() u, v = int(edge[0]), int(edge[1]) adj[u].append(v) adj[v].append(u) return adj # ================================================== # Pattern 3: Adjacency List Input (Direct Neighbors) # Each line lists neighbors of vertex i # ================================================== def build_from_neighbor_lists(neighbor_lists: list[list[int]]) -> list[list[int]]: """ Input: directly given neighbors for each vertex Example: neighbor_lists = [ [1, 2], # vertex 0 connects to 1, 2 [0, 2, 3], # vertex 1 connects to 0, 2, 3 [0, 1, 3], # vertex 2 connects to 0, 1, 3 [1, 2] # vertex 3 connects to 1, 2 ] """ # Input is already in adjacency list format! return [list(neighbors) for neighbors in neighbor_lists] # ================================================== # Pattern 4: Matrix to Adjacency List Conversion # Converting from matrix representation # ================================================== def matrix_to_adj_list(matrix: list[list[int]]) -> list[list[int]]: """ Convert adjacency matrix to adjacency list. Essential skill for choosing representation at runtime. """ n = len(matrix) adj = [[] for _ in range(n)] for i in range(n): for j in range(n): if matrix[i][j] != 0: # edge exists adj[i].append(j) return adj # ================================================== # Pattern 5: Grid to Graph Conversion # 2D grids are implicit graphs # ================================================== def grid_to_adj_list(grid: list[list[str]], blocked: str = '#') -> dict[tuple[int,int], list[tuple[int,int]]]: """ Convert a 2D grid to an adjacency list. Each cell (r, c) becomes a vertex. 4-directional neighbors become edges. Example grid: . . # . . . . . # . . . 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Learn how storing neighbors for each vertex provides space efficiency for sparse graphs, understand the time-space tradeoffs, and discover when adjacency lists are the optimal choice for graph-based problems.","difficulty":"INTERMEDIATE","estimatedMinutes":60,"pages":[{"id":"seed_d7ec08c729bd868c7ccaf8c31cbc4127","title":"List of Neighbors for Each Vertex","order":0,"content":[{"id":"opening-hook","title":"The Neighbor-Centric View of Graphs","content":[{"data":{"content":"In the previous module, we explored the **adjacency matrix**—a dense, two-dimensional representation that excels when graphs are nearly complete and when we need constant-time edge lookups. But real-world graphs tell a different story: they're typically **sparse**, with only a fraction of all possible edges actually present.\n\nConsider a social network with 1 billion users. Each user might have, on average, 500 friends. An adjacency matrix would require 10¹⁸ bits (over 100 petabytes) of storage, even though only about 250 billion actual connections exist. This astronomical waste demands a fundamentally different approach—one that stores only what exists, not the entire possibility space.\n\nWelcome to the **adjacency list representation**: the workhorse of graph algorithms in practice."},"type":"text"},{"data":{"icon":"Target","title":"What You Will Learn","content":"By the end of this page, you will understand how adjacency lists work at a fundamental level. You'll see how storing a list of neighbors for each vertex dramatically reduces space requirements for sparse graphs, explore multiple implementation strategies (arrays, linked lists, hash sets), and develop intuition for why this representation dominates real-world graph applications.","variant":"info"},"type":"callout"}]},{"id":"core-idea","title":"The Core Idea: Store Only What Exists","content":[{"data":{"content":"The adjacency list representation embodies a simple yet powerful principle: **for each vertex, maintain a collection of its neighbors**. Instead of reserving space for every possible edge (as the matrix does), we allocate space only for edges that actually exist in the graph.\n\n**The fundamental insight:**\n\nFor a graph G = (V, E) with |V| vertices and |E| edges, the total information we need to store is:\n- Which vertices exist (trivially, all of them: 0 through |V|-1)\n- For each vertex, which other vertices it connects to (its neighbors)\n\nThis neighbor-centric view naturally leads to a collection of lists—one list per vertex, containing that vertex's adjacent vertices."},"type":"text"},{"data":{"code":"Graph G = (V, E) where V = {0, 1, 2, 3, 4}\n\nEdges: (0,1), (0,4), (1,2), (1,3), (1,4), (2,3), (3,4)\n\nAdjacency List Representation:\n┌─────────┬─────────────────────────────────┐\n│ Vertex │ Neighbors (Adjacent Vertices) │\n├─────────┼─────────────────────────────────┤\n│ 0 │ → [1, 4] │\n│ 1 │ → [0, 2, 3, 4] │\n│ 2 │ → [1, 3] │\n│ 3 │ → [1, 2, 4] │\n│ 4 │ → [0, 1, 3] │\n└─────────┴─────────────────────────────────┘\n\nTotal storage: 14 entries (7 edges × 2 for undirected)\nMatrix would need: 25 cells (5 × 5)","title":"Conceptual Structure of an Adjacency List","language":"text"},"type":"code"},{"data":{"content":"**Contrast with the adjacency matrix:**\n\nIn a matrix, we ask: \"Is there an edge from vertex i to vertex j?\" and we store the answer for all possible (i, j) pairs.\n\nIn an adjacency list, we ask: \"Which vertices does vertex i connect to?\" and we store only those vertices that answer \"yes.\"\n\nThis shift in perspective—from \"all possible pairs\" to \"actual connections\"—is what enables adjacency lists to dramatically outperform matrices for sparse graphs."},"type":"text"},{"data":{"icon":"Info","title":"The Sparsity Advantage","content":"A graph is considered sparse when |E| is much smaller than |V|². Most real-world graphs fall into this category: road networks, social graphs, web links, dependency graphs, biological networks. For these, adjacency lists aren't just better—they're often the only feasible choice.","variant":"note"},"type":"callout"}],"titleType":"number","titleNumber":1},{"id":"implementation-strategies","title":"Implementation Strategies: Arrays, Lists, and Beyond","content":[{"data":{"content":"The \"list\" in adjacency list is somewhat misleading—the neighbor collection for each vertex can be implemented using various data structures, each with different tradeoffs. Let's examine the major options:\n\n**Option 1: Array of Dynamic Arrays (Most Common)**\n\nThis is the standard implementation in most programming languages. We maintain an array where index i contains a dynamic array (like Python's list, Java's ArrayList, or C++'s vector) of neighbors for vertex i."},"type":"text"},{"data":{"code":"$72","title":"Python Implementation: List of Lists","language":"python"},"type":"code"},{"data":{"content":"**Option 2: Array of Linked Lists (Classic Approach)**\n\nThe traditional textbook approach uses actual linked lists for each vertex's neighbors. While pedagogically important, this is rarely used in practice due to poor cache performance."},"type":"text"},{"data":{"code":"$73","title":"Linked List Implementation (Illustrative)","language":"python"},"type":"code"},{"data":{"content":"**Option 3: Array of Hash Sets (When Edge Lookup Matters)**\n\nIf we frequently need to check whether a specific edge exists (an operation that's O(degree) with arrays), we can use hash sets for each vertex's neighbors. This trades some memory for O(1) average-case edge lookup."},"type":"text"},{"data":{"code":"$74","title":"Hash Set Implementation for Fast Edge Lookup","language":"python"},"type":"code"},{"data":{"rows":[["Array of Arrays","O(degree)","O(1) amortized","Cache-friendly","Low","General purpose, traversals"],["Array of Linked Lists","O(degree)","O(1)","Cache-unfriendly","Higher (pointers)","Theoretical, historical"],["Array of Hash Sets","O(1) average","O(1) average","Unordered","Higher (hash table)","Frequent edge queries"]],"caption":"Comparison of Adjacency List Implementations","headers":["Implementation","Edge Lookup","Edge Insert","Iteration","Memory Overhead","Best For"]},"type":"table"}],"titleType":"number","titleNumber":2},{"id":"directed-vs-undirected","title":"Directed vs Undirected: How Storage Differs","content":[{"data":{"content":"The distinction between directed and undirected graphs has important implications for adjacency list storage:\n\n**Undirected Graphs: Symmetric Storage**\n\nIn an undirected graph, an edge (u, v) means both \"u is connected to v\" AND \"v is connected to u.\" We must store this relationship in both directions:"},"type":"text"},{"data":{"code":"Undirected graph with edges: {A-B, B-C, A-C}\n\nAdjacency List:\nA → [B, C] // A connects to B and C\nB → [A, C] // B connects to A and C \nC → [A, B] // C connects to A and B\n\nNotice: Edge A-B appears in A's list (as B) AND in B's list (as A)\nTotal entries: 6 (3 edges × 2)","title":"Undirected Graph: Each Edge Appears Twice","language":"text"},"type":"code"},{"data":{"content":"**Directed Graphs: Asymmetric Storage**\n\nIn a directed graph, an edge (u → v) means only \"u connects to v\"—there's no automatic reverse relationship. Each edge is stored exactly once:"},"type":"text"},{"data":{"code":"Directed graph with edges: A→B, B→C, C→A\n\nAdjacency List (Outgoing Edges):\nA → [B] // A has an outgoing edge to B\nB → [C] // B has an outgoing edge to C\nC → [A] // C has an outgoing edge to A\n\nTotal entries: 3 (exactly |E| entries for directed graphs)","title":"Directed Graph: Each Edge Appears Once","language":"text"},"type":"code"},{"data":{"content":"**Reverse Adjacency Lists for Directed Graphs**\n\nSometimes with directed graphs, we need to efficiently answer: \"Which vertices have edges pointing TO this vertex?\" (i.e., find predecessors, not successors). The standard adjacency list answers the question of outgoing edges efficiently, but incoming edges require a linear scan through all lists.\n\nFor algorithms that need both directions (like computing strongly connected components), we can maintain a **reverse adjacency list** alongside the regular one:"},"type":"text"},{"data":{"code":"$75","title":"Directed Graph with Reverse Adjacency List","language":"python"},"type":"code"},{"data":{"icon":"Lightbulb","title":"Space Tradeoff","content":"Maintaining both forward and reverse adjacency lists doubles the space requirement from O(V + E) to O(V + 2E), but enables O(1) access to both predecessors and successors. This is often worthwhile for algorithms like Kosaraju's SCC algorithm or topological sorting with special requirements.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":3},{"id":"weighted-graphs","title":"Representing Weighted Graphs with Adjacency Lists","content":[{"data":{"content":"Many graph algorithms require **edge weights**—numerical values associated with edges that might represent distances, costs, capacities, or probabilities. Adjacency lists accommodate weights naturally by storing (neighbor, weight) pairs instead of just neighbor vertices.\n\n**Common Approaches:**"},"type":"text"},{"data":{"code":"$76","title":"Weighted Graph: Multiple Implementation Styles","language":"python"},"type":"code"},{"data":{"code":"$77","title":"Java Implementation with Edge Objects","language":"java"},"type":"code"},{"data":{"icon":"Info","title":"Multiple Edge Attributes","content":"Real-world edges often have multiple attributes: distance, time, cost, reliability, capacity. The adjacency list naturally extends to store any edge metadata by using objects, dictionaries, or named tuples instead of simple (neighbor, weight) pairs.","variant":"note"},"type":"callout"}],"titleType":"number","titleNumber":4},{"id":"vertex-index-assumption","title":"The Vertex Index Assumption and Alternatives","content":[{"data":{"content":"The standard adjacency list assumes vertices are numbered 0 through V-1, allowing direct array indexing. But what if vertices have other identifiers—strings, UUIDs, or arbitrary objects?\n\n**Solution: Vertex Mapping**\n\nWe create a mapping between vertex identifiers and integer indices:"},"type":"text"},{"data":{"code":"$78","title":"Graph with String Vertex Labels","language":"python"},"type":"code"},{"data":{"content":"**Alternative: Pure Dictionary-Based Graphs**\n\nFor maximum flexibility—especially when vertices are added and removed dynamically—a pure dictionary approach avoids the index-mapping complexity entirely:"},"type":"text"},{"data":{"code":"$79","title":"Fully Dynamic Dictionary Graph","language":"python"},"type":"code"}],"titleType":"number","titleNumber":5},{"id":"building-from-input","title":"Building Graphs from Common Input Formats","content":[{"data":{"content":"In practice, graphs come from various sources: files, databases, APIs, or as part of coding problems. Mastering the skill of quickly constructing an adjacency list from raw input is essential. Here are the most common patterns:"},"type":"text"},{"data":{"code":"$7a","title":"Building Graphs from Different Input Formats","language":"python"},"type":"code"},{"data":{"icon":"Lightbulb","title":"Pro Tip: Recognize the Pattern Early","content":"Before writing any graph algorithm, always clarify the input format. Many bugs come from misinterpreting whether vertices are 0-indexed or 1-indexed, whether edges are directed or undirected, or whether the input is an edge list vs. adjacency list. Read the problem statement carefully!","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":6},{"id":"summary","title":"Summary: The Adjacency List Foundation","content":[{"data":{"content":"We've established the foundation of adjacency list representation—the primary graph storage method used in practice. Let's consolidate the key insights:"},"type":"text"},{"data":{"type":"ordered","items":["**Store only what exists**: Adjacency lists allocate space for actual edges, not all possible edges. This is fundamentally more efficient for sparse graphs.","**Implementation flexibility**: The 'list' can be dynamic arrays, linked lists, hash sets, or any collection—choose based on your algorithm's needs.","**Undirected edges appear twice**: In undirected graphs, each edge creates two entries (one in each endpoint's list).","**Directed graphs can use reverse lists**: When you need both predecessor and successor queries, maintain both forward and reverse adjacency lists.","**Weights integrate naturally**: Store (neighbor, weight) pairs or use dictionaries/objects to attach arbitrary edge metadata.","**Vertex mapping enables labels**: Map string/object labels to integer indices when needed, or use pure dictionary representations for maximum flexibility."],"title":{"icon":"CheckCircle","text":"Key Takeaways","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"content":"**What's Next:**\n\nWith the structural understanding in place, the next page analyzes the space complexity of adjacency lists—understanding precisely why O(V + E) space makes them the choice for sparse graphs, and exploring the constants hidden in that notation."},"type":"text"},{"data":{"icon":"CheckCircle","title":"Page Complete","content":"You now understand how adjacency lists store graphs by maintaining a collection of neighbors for each vertex. This neighbor-centric view forms the basis for nearly all practical graph implementations and algorithms.","variant":"success"},"type":"callout"}],"titleType":"number","titleNumber":7}],"isLocked":false},{"id":"seed_8b140fba90acda011dd350b10cf360cb","title":"Space Complexity — O(V + E)","order":1,"content":null,"isLocked":true},{"id":"seed_1c717310064d21d97f6c63eeb741e8fc","title":"Edge Lookup — O(degree) Time Complexity","order":2,"content":null,"isLocked":true},{"id":"seed_b4134cfb115c1ac627a76154349c7f00","title":"When Lists Are Appropriate","order":3,"content":null,"isLocked":true}]}},"dataUpdateCount":1,"dataUpdatedAt":1792287779218,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getModule"],{"input":{"slug":"graph-representation-adjacency-list"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getModule\"],{\"input\":{\"slug\":\"graph-representation-adjacency-list\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287779278,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792287779219,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getProgress"],{"input":{"moduleId":"seed_64ce626484afc3b731c22b34053bc338"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getProgress\"],{\"input\":{\"moduleId\":\"seed_64ce626484afc3b731c22b34053bc338\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287779278,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792287779219,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["feedback","get"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_1c717310064d21d97f6c63eeb741e8fc"},"type":"query"}],"queryHash":"[[\"feedback\",\"get\"],{\"input\":{\"contentId\":\"seed_1c717310064d21d97f6c63eeb741e8fc\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287779278,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792287779219,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["feedback","getUserLike"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_1c717310064d21d97f6c63eeb741e8fc"},"type":"query"}],"queryHash":"[[\"feedback\",\"getUserLike\"],{\"input\":{\"contentId\":\"seed_1c717310064d21d97f6c63eeb741e8fc\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287779278,"state":{"data":{"json":[]},"dataUpdateCount":1,"dataUpdatedAt":1792287779219,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["highlights","getHighlights"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_1c717310064d21d97f6c63eeb741e8fc"},"type":"query"}],"queryHash":"[[\"highlights\",\"getHighlights\"],{\"input\":{\"contentId\":\"seed_1c717310064d21d97f6c63eeb741e8fc\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287779278,"state":{"data":"$undefined","dataUpdateCount":0,"dataUpdatedAt":0,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"pending","fetchStatus":"fetching"},"queryKey":[["highlights","getPopularHighlights"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_1c717310064d21d97f6c63eeb741e8fc","minCount":2},"type":"query"}],"queryHash":"[[\"highlights\",\"getPopularHighlights\"],{\"input\":{\"contentId\":\"seed_1c717310064d21d97f6c63eeb741e8fc\",\"contentType\":\"LEARNING_PAGE\",\"minCount\":2},\"type\":\"query\"}]","promise":"$@7b"}]},"children":"$L7c"}]