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Check root self-reference for (let i = 0; i < this.n; i++) { const root = this.findWithoutCompression(i); if (this.parent[root] !== root) { console.error(`Invariant violated: root ${root} not self-referential`); return false; } } // 2. Check reachability (no cycles other than root self-loops) for (let i = 0; i < this.n; i++) { const visited = new Set(); let current = i; while (!visited.has(current)) { visited.add(current); if (this.parent[current] === current) break; // Reached root current = this.parent[current]; } // Should have reached a root, not formed a cycle if (this.parent[current] !== current) { console.error(`Invariant violated: cycle detected from ${i}`); return false; } } // 3. Check unique root per set (implicitly guaranteed by structure) // Every element reaches exactly one root by following parents console.log("All invariants satisfied!"); return true; } private findWithoutCompression(x: number): number { while (this.parent[x] !== x) { x = this.parent[x]; } return x; } find(x: number): number { return this.findWithoutCompression(x); } union(x: number, y: number): void { const rootX = this.find(x); const rootY = this.find(y); if (rootX !== rootY) { this.parent[rootY] = rootX; } } }74:T687,interface IUnionFind { // Core operations /** * Find the representative (root) of the set containing x. * Elements in the same set have the same representative. */ find(x: number): number; /** * Merge the sets containing x and y. * Returns true if a merge occurred, false if x and y were already connected. */ union(x: number, y: number): boolean; /** * Check if x and y are in the same set. * Equivalent to: find(x) === find(y) */ connected(x: number, y: number): boolean; // Extended operations (optional but useful) /** * Return the size of the set containing x. * Requires tracking sizes during unions. */ getSize(x: number): number; /** * Return the total number of disjoint sets. * Decreases with each successful union. */ getCount(): number; /** * Get all elements in the same set as x. * Note: This is O(n) as we must scan all elements. */ getSetMembers(x: number): number[]; } // Example usage function demonstrateUnionFind() { const uf = new UnionFind(10); // 10 elements: 0-9 console.log(uf.getCount()); // 10 (each element is its own set) uf.union(0, 1); uf.union(2, 3); uf.union(4, 5); console.log(uf.getCount()); // 7 (three pairs, four singles) console.log(uf.connected(0, 1)); // true console.log(uf.connected(0, 2)); // false uf.union(1, 3); // Merges {0,1} with {2,3} console.log(uf.connected(0, 2)); // true (now in same set) console.log(uf.getSize(0)); // 4 (set is {0,1,2,3}) }75:T6bc,// When elements aren't contiguous integers from 0 to n-1 // Use a 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Structures & Algorithms","topic":"Union-Find (Disjoint Set Union)","title":"Union-Find (Disjoint Set Union) — The Connectivity Powerhouse","description":"Master the Union-Find data structure, a remarkably elegant solution for managing dynamic connectivity. Learn how this structure enables near-constant-time operations for grouping elements, detecting cycles, and solving connectivity problems across graphs, networks, and beyond.","difficulty":"INTERMEDIATE","estimatedMinutes":60,"pages":[{"id":"seed_61b3663e581c003d5af86e5e30a5a779","title":"The Union-Find Data Structure","order":0,"content":[{"id":"opening-hook","title":"When Connectivity Becomes the Core Problem","content":[{"data":{"content":"Imagine you're building a social network. Users arrive continuously and form friendships. At any moment, someone might ask: *\"Are Alice and David in the same friend network?\"* Or consider a computer network where cables connect machines—you need to quickly answer: *\"Can server A communicate with server B?\"*\n\nThese questions share a common structure. They're not asking about shortest paths, distances, or optimal routes. They're asking something simpler yet profound: **Are these two elements connected?**\n\nThis is the **dynamic connectivity problem**—one of the most fundamental questions in computer science. And while it sounds simple, solving it efficiently as connections form and dissolve requires a data structure of remarkable elegance: the **Union-Find** data structure, also known as **Disjoint Set Union (DSU)**."},"type":"text"},{"data":{"icon":"Target","title":"What You Will Learn","content":"By the end of this page, you will understand what Union-Find is, why it exists, what problems it solves, and how it represents disjoint sets internally. You'll see why this data structure is considered one of the most beautiful inventions in computer science—combining simplicity with extraordinary efficiency.","variant":"info"},"type":"callout"}]},{"id":"dynamic-connectivity-problem","title":"The Dynamic Connectivity Problem","content":[{"data":{"content":"Before we dive into the solution, let's precisely define the problem that Union-Find so elegantly addresses.\n\n**The Dynamic Connectivity Problem:**\n\nYou have `n` elements (numbered 0 to n-1). Initially, each element is in its own isolated set. You must support two operations:\n\n1. **Union(a, b)**: Merge the sets containing elements `a` and `b` into a single set\n2. **Find(a)**: Determine which set element `a` belongs to (typically by returning a representative element of that set)\n\nWith these two operations, you can answer connectivity queries: *\"Are a and b in the same set?\"* by checking if `Find(a) == Find(b)`."},"type":"text"},{"data":{"icon":"Info","title":"Why 'Disjoint Sets'?","content":"The sets in Union-Find are **disjoint**—no element can belong to more than one set at any time. When we union two sets, we merge them completely. This property is crucial: at any moment, the collection of sets forms a **partition** of all elements, where every element belongs to exactly one set.","variant":"note"},"type":"callout"},{"data":{"content":"**Why is this problem interesting?**\n\nAt first glance, you might think: *\"Just use a graph and run BFS or DFS for each connectivity query.\"* But consider the scale:\n\n- A social network with **1 billion users** making **millions of friend connections daily**\n- Each connectivity query with BFS/DFS: **O(V + E)** time\n- With millions of queries per day: **Catastrophic performance**\n\nWe need something dramatically faster—operations that approach constant time even as the network grows to billions of elements. This is exactly what Union-Find delivers."},"type":"text"},{"data":{"rows":[["Store list of connections, run BFS","O(1)","O(V + E)","Impractical for frequent queries"],["Maintain full adjacency matrix","O(1)","O(V²) for transitive closure","Memory prohibitive"],["Rebuild connected components","O(V + E)","O(1) after rebuild","Impractical for frequent unions"],["Union-Find with optimizations","α(n) ≈ O(1)*","α(n) ≈ O(1)*","Excellent at any scale"]],"caption":"Naive Approaches vs. Union-Find","headers":["Approach","Union Time","Find/Query Time","Practicality at Scale"]},"type":"table"},{"data":{"icon":"Lightbulb","title":"The Magic of α(n)","content":"The α(n) in the table is the **inverse Ackermann function**—a function that grows so slowly that for all practical purposes (any conceivable number of elements in the universe), α(n) ≤ 4. We'll explore this remarkable efficiency in later pages. For now, understand that Union-Find achieves essentially constant time per operation.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":1},{"id":"what-is-union-find","title":"What is Union-Find?","content":[{"data":{"content":"**Union-Find** (also called **Disjoint Set Union** or **DSU**) is a data structure that maintains a collection of disjoint (non-overlapping) sets. It provides near-constant-time operations to:\n\n1. **Find** which set an element belongs to\n2. **Union** two sets into one\n\nThe brilliance of Union-Find lies in its representation. Rather than explicitly storing set membership lists, it represents each set as a **tree**. Elements are nodes, and each node points to its parent. The **root** of each tree serves as the **representative** (or **canonical element**) of the set.\n\n**Key insight:** To find which set an element belongs to, follow parent pointers until you reach the root. Two elements are in the same set if and only if they have the same root."},"type":"text"},{"data":{"code":"Initially, 5 elements (0-4), each in its own set:\n\n [0] [1] [2] [3] [4]\n ↑ ↑ ↑ ↑ ↑\n root root root root root\n\nAfter Union(0, 1) - element 1 now points to 0:\n\n [0] [2] [3] [4]\n ↑ ↑ ↑ ↑\n 1\n \nElement 0 is root (representative) of set {0, 1}\nElements 2, 3, 4 are each their own root\n\nAfter Union(2, 3) and Union(3, 4):\n\n [0] [2]\n ↑ ↑\n 1 3\n ↑\n 4\n\nNow we have two sets: {0, 1} and {2, 3, 4}\n\nAfter Union(0, 2) - now all elements in one set:\n\n [0]\n ↑\n / \\\n [1] [2]\n ↑\n 3\n ↑\n 4\n\nAll elements connected through root 0","title":"Conceptual Visualization: Union-Find Tree Structure","language":"text"},"type":"code"},{"data":{"content":"**The representation is minimal and elegant:**\n\nAll we need is a **parent array** where `parent[i]` stores the parent of element `i`. If an element is a root, it is its own parent: `parent[i] = i`.\n\nThis single array encodes the entire partition structure:\n- **Space complexity: O(n)** — just one integer per element\n- **Finding the set:** Follow parent pointers to the root\n- **Unioning sets:** Make one root point to the other"},"type":"text"},{"data":{"code":"$72","title":"Union-Find: Basic Representation","language":"typescript"},"type":"code"},{"data":{"icon":"AlertCircle","title":"This Basic Version Has a Problem","content":"The naive implementation shown above can degenerate into a linked list in the worst case, making Find operations O(n). Imagine unioning elements in sequence: Union(0,1), Union(1,2), Union(2,3), ... This creates a chain where finding element 0 requires traversing n-1 pointers. We'll fix this with optimizations in later pages.","variant":"warning"},"type":"callout"}],"titleType":"number","titleNumber":2},{"id":"disjoint-set-abstraction","title":"The Disjoint Set Abstraction","content":[{"data":{"content":"Let's formalize what Union-Find represents mathematically. This abstraction helps us reason about correctness and understand why certain operations make sense.\n\n**Set Partition:**\n\nA **partition** of a set S is a collection of non-empty, pairwise disjoint subsets of S whose union equals S. In mathematical notation:\n\n- If P = {S₁, S₂, ..., Sₖ} is a partition of S, then:\n - Each Sᵢ ≠ ∅ (non-empty)\n - Sᵢ ∩ Sⱼ = ∅ for i ≠ j (disjoint)\n - S₁ ∪ S₂ ∪ ... ∪ Sₖ = S (complete coverage)\n\nUnion-Find maintains exactly such a partition. Every element belongs to exactly one set, and the collection of all sets covers all elements."},"type":"text"},{"data":{"items":["**Partition Invariant**: At all times, the structure represents a valid partition of elements. No element is orphaned; no element belongs to multiple sets.","**Monotonic Merging**: Unions only merge sets—they never split them. Once two elements are in the same set, they remain connected forever. This is crucial for correctness.","**Representative Consistency**: Within a set, all elements have the same representative (root). The representative uniquely identifies the set.","**Transitivity**: If a and b are connected, and b and c are connected, then a and c are connected. Union-Find naturally maintains transitive closure.","**Idempotency**: Union(a, b) when a and b are already connected is a no-op. Find(a) always returns the same value until a union involving a's set occurs."],"title":{"icon":"Layers","text":"Key Properties of Union-Find","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"content":"**Why Trees?**\n\nYou might wonder why we use trees rather than other representations. Consider the alternatives:\n\n**Alternative 1: Explicit set membership lists**\n- Store `setId[i]` for each element\n- Find: O(1) — just return setId[i]\n- Union: O(n) — must update setId for all elements in one set\n\n**Alternative 2: Linked lists with head pointers**\n- Each set is a linked list, with a map from element to head\n- Find: O(1) if we maintain the map\n- Union: O(smaller set size) — need to traverse one list\n\n**The tree representation:**\n- No explicit set membership storage\n- Find: O(tree height)\n- Union: O(tree height) — just link two roots\n\nWhen we add path compression and union by rank (covered in later pages), tree heights become nearly constant, giving us the best of all worlds: near-O(1) for both operations with O(n) space."},"type":"text"},{"data":{"icon":"Sparkles","title":"The Elegance of Implicit Structure","content":"Union-Find is a masterclass in implicit data structure design. Instead of maintaining complex explicit relationships, we let the tree structure emerge naturally. The parent array is all we need—everything else (set membership, connectivity, partitioning) is derived from following pointers. This simplicity is what enables the remarkable optimizations.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":3},{"id":"core-invariants","title":"Core Invariants and Properties","content":[{"data":{"content":"Understanding Union-Find's invariants is essential for both implementing it correctly and reasoning about its behavior. These invariants hold after every operation completes."},"type":"text"},{"data":{"items":["**Root Self-Reference**: For every root r, parent[r] = r. This is how we identify roots and terminate the Find operation.","**Reachability to Root**: From any element x, following parent pointers eventually reaches a root. There are no cycles other than root self-loops.","**Unique Root per Set**: Each connected component (set) has exactly one root. This root serves as the canonical representative.","**Union Preserves Connectivity**: After Union(a, b), Find(a) = Find(b). Elements united stay united.","**Find Stability**: Between unions, Find(x) returns the same value. Find operations do not change logical set membership (though they may restructure the tree)."],"title":{"icon":"Shield","text":"Fundamental Invariants","type":"icon","color":"primary","variant":"outline-circle"},"variant":"glass"},"type":"list"},{"data":{"content":"**Derived Properties:**\n\nFrom these invariants, we can derive several useful properties:\n\n**Property 1: Connectivity is an equivalence relation**\n- Reflexive: Every element is connected to itself\n- Symmetric: If a is connected to b, then b is connected to a\n- Transitive: If a↔b and b↔c, then a↔c\n\n**Property 2: The partition only becomes coarser**\n- We start with n sets (finest partition)\n- Each union reduces the number of sets by at most 1\n- We end with at least 1 set (coarsest partition)\n\n**Property 3: The number of unions is bounded**\n- At most n-1 meaningful unions can occur (each reduces set count by 1)\n- Starting with n sets, n-1 unions leave exactly 1 set"},"type":"text"},{"data":{"code":"$73","title":"Verifying Invariants (for debugging)","language":"typescript"},"type":"code"}],"titleType":"number","titleNumber":4},{"id":"real-world-applications","title":"Real-World Applications","content":[{"data":{"content":"Union-Find isn't just a theoretical construct—it's a workhorse data structure that appears in countless real-world applications. Understanding these applications helps you recognize when Union-Find is the right tool for a problem.\n\n**Why Union-Find appears everywhere:**\n\nAny problem that involves grouping elements, detecting when elements become connected, or maintaining equivalence classes is a potential Union-Find application. The structure's efficiency makes it practical even for massive datasets."},"type":"text"},{"data":{"rows":[["Graph Algorithms","Kruskal's MST Algorithm","Detect if adding an edge would create a cycle"],["Graph Algorithms","Connected Components","Track and query component membership dynamically"],["Network Design","Network Connectivity","Determine if two nodes can communicate"],["Image Processing","Percolation Simulation","Model fluid flow through porous materials"],["Social Networks","Friend Circles","Find and count distinct friend networks"],["Compilers","Type Unification","Merge type variables during type inference"],["Gaming","Dynamic Terrain","Track connected regions as terrain changes"],["Clustering","Single-Linkage Clustering","Build hierarchical clusters bottom-up"]],"caption":"Union-Find Applications Across Domains","headers":["Domain","Application","How Union-Find Helps"]},"type":"table"},{"data":{"content":"**Deep Dive: Kruskal's Algorithm**\n\nPerhaps the most famous application of Union-Find is in **Kruskal's algorithm** for finding Minimum Spanning Trees. The algorithm:\n\n1. Sort all edges by weight\n2. For each edge (u, v) in sorted order:\n - If u and v are in different components: add edge to MST, union(u, v)\n - If u and v are in the same component: skip (would create cycle)\n\nUnion-Find answers the critical question *\"Are u and v already connected?\"* in near-constant time, making Kruskal's algorithm efficient even for dense graphs."},"type":"text"},{"data":{"code":"interface Edge {\n u: number;\n v: number;\n weight: number;\n}\n\nfunction kruskalMST(n: number, edges: Edge[]): Edge[] {\n // Sort edges by weight\n edges.sort((a, b) => a.weight - b.weight);\n \n const uf = new UnionFind(n); // Assume optimized Union-Find\n const mst: Edge[] = [];\n \n for (const edge of edges) {\n // Key question: Are u and v already connected?\n // Union-Find answers this in near O(1) time\n if (!uf.connected(edge.u, edge.v)) {\n uf.union(edge.u, edge.v);\n mst.push(edge);\n \n // MST complete when we have n-1 edges\n if (mst.length === n - 1) break;\n }\n }\n \n return mst;\n}\n\n// Without Union-Find, we'd need O(V + E) per connectivity check\n// With Union-Find, the entire algorithm runs in O(E log E) time\n// (dominated by sorting, not connectivity checks)","title":"Kruskal's Algorithm with Union-Find","language":"typescript"},"type":"code"},{"data":{"content":"**Deep Dive: Percolation**\n\nPercolation is a fascinating application in computational physics. Imagine a porous material represented as an n×n grid. Each cell can be open (permeable) or closed (blocked). We want to know: *Can fluid flow from the top to the bottom?*\n\nThis happens if and only if there's a path of open cells connecting any top-row cell to any bottom-row cell. As cells randomly open, we use Union-Find to efficiently track when percolation occurs.\n\n**The trick:** Create two virtual nodes—one connected to all top-row cells, one connected to all bottom-row cells. Percolation occurs when these virtual nodes become connected."},"type":"text"},{"data":{"icon":"Lightbulb","title":"The Virtual Node Technique","content":"Adding virtual nodes that connect to multiple real nodes is a powerful Union-Find pattern. Instead of checking connectivity between many pairs, you check connectivity between two virtual nodes. This reduces multiple queries to a single query—a technique applicable to many problems.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":5},{"id":"interface-api-design","title":"Interface and API Design","content":[{"data":{"content":"A well-designed Union-Find interface is clean, intuitive, and expressive. Let's examine the standard API and some useful extensions.\n\n**Core Operations:**"},"type":"text"},{"data":{"rows":[["`find(x)`","Find representative of x's set","`find(x: number): number`"],["`union(x, y)`","Merge sets containing x and y","`union(x: number, y: number): boolean`"],["`connected(x, y)`","Check if x and y are in same set","`connected(x: number, y: number): boolean`"]],"headers":["Operation","Purpose","Typical Signature"]},"type":"table"},{"data":{"code":"$74","title":"Complete Union-Find Interface (Preview of Full Implementation)","language":"typescript"},"type":"code"},{"data":{"content":"**Design Considerations:**\n\n**Return value of union():**\n\nSome implementations return `void`, others return `boolean` (whether a merge occurred). The boolean version is convenient—it tells you if this was a \"new\" connection or a redundant one.\n\n**Zero-indexed vs One-indexed:**\n\nMost implementations use 0-indexed elements (0 to n-1). Be consistent and match your problem's indexing.\n\n**Dynamic resizing:**\n\nThe basic Union-Find has a fixed size set at construction. For dynamic applications, you might need a map-based variant that can grow arbitrarily."},"type":"text"},{"data":{"code":"$75","title":"Map-Based Union-Find for Dynamic Elements","language":"typescript"},"type":"code"},{"data":{"icon":"Info","title":"Choosing the Right Variant","content":"Use array-based Union-Find when elements are integers in a known range—it's faster and uses less memory. Use Map-based Union-Find when elements are sparse, non-integer, or the range is unknown. The algorithmic complexity is the same, but array operations have lower constants.","variant":"note"},"type":"callout"}],"titleType":"number","titleNumber":6},{"id":"summary-looking-ahead","title":"Summary and Looking Ahead","content":[{"data":{"content":"We've established the foundation for understanding Union-Find. Let's consolidate what we've learned:"},"type":"text"},{"data":{"type":"ordered","items":["**Union-Find solves dynamic connectivity** — It efficiently tracks which elements are connected as connections form over time.","**Tree-based representation** — Each set is a tree; elements point to parents; roots identify sets. A single parent array is all we need.","**Core operations: Find and Union** — Find traverses to root; Union links two roots. Connected queries reduce to comparing roots.","**Maintains a set partition** — The collection of sets is always a valid partition: disjoint, complete, with every element in exactly one set.","**Widely applicable** — From Kruskal's MST to network connectivity to percolation, Union-Find appears across domains wherever connectivity matters.","**The naive version has issues** — Without optimizations, tree height can become O(n), making operations slow. This motivates the optimizations we'll study next."],"title":{"icon":"CheckCircle","text":"Key Takeaways","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"content":"**What's next:**\n\nWe've defined *what* Union-Find is. Now we need to understand *how* it works efficiently. In the next page, we'll dive deep into the **Union and Find operations**—examining exactly how they work, tracing through examples, and understanding why the naive approach can degrade to linear time per operation."},"type":"text"},{"data":{"icon":"CheckCircle","title":"Page Complete","content":"You now understand what Union-Find is, why it exists, and how it represents disjoint sets. You've seen the basic implementation and its applications. Next, we'll explore the Union and Find operations in detail, setting the stage for the powerful optimizations that make Union-Find truly remarkable.","variant":"success"},"type":"callout"}],"titleType":"number","titleNumber":7}],"isLocked":false},{"id":"seed_592bb72539fba96cecc8aa07f7fa55ad","title":"Union and Find Operations","order":1,"content":null,"isLocked":true},{"id":"seed_f01aa80b1a0942978762b0285a5592d0","title":"Path Compression Optimization","order":2,"content":null,"isLocked":true},{"id":"seed_0f7aea16aae77d4b20681ee0ae4919ef","title":"Union by Rank Optimization","order":3,"content":null,"isLocked":true},{"id":"seed_551957aeec1bf647ee5efbd2d71d4b5c","title":"Nearly O(1) Amortized Operations","order":4,"content":null,"isLocked":true}]}},"dataUpdateCount":1,"dataUpdatedAt":1792303953885,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getModule"],{"input":{"slug":"union-find"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getModule\"],{\"input\":{\"slug\":\"union-find\"},\"type\":\"query\"}]"},{"dehydratedAt":1792303953947,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792303953885,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getProgress"],{"input":{"moduleId":"seed_ea81e69a7b1c3faa8455bae76c553e87"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getProgress\"],{\"input\":{\"moduleId\":\"seed_ea81e69a7b1c3faa8455bae76c553e87\"},\"type\":\"query\"}]"},{"dehydratedAt":1792303953947,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792303953885,"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_f01aa80b1a0942978762b0285a5592d0"},"type":"query"}],"queryHash":"[[\"feedback\",\"get\"],{\"input\":{\"contentId\":\"seed_f01aa80b1a0942978762b0285a5592d0\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792303953947,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792303953885,"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_f01aa80b1a0942978762b0285a5592d0"},"type":"query"}],"queryHash":"[[\"feedback\",\"getUserLike\"],{\"input\":{\"contentId\":\"seed_f01aa80b1a0942978762b0285a5592d0\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792303953947,"state":{"data":{"json":[]},"dataUpdateCount":1,"dataUpdatedAt":1792303953885,"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_f01aa80b1a0942978762b0285a5592d0"},"type":"query"}],"queryHash":"[[\"highlights\",\"getHighlights\"],{\"input\":{\"contentId\":\"seed_f01aa80b1a0942978762b0285a5592d0\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792303953947,"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_f01aa80b1a0942978762b0285a5592d0","minCount":2},"type":"query"}],"queryHash":"[[\"highlights\",\"getPopularHighlights\"],{\"input\":{\"contentId\":\"seed_f01aa80b1a0942978762b0285a5592d0\",\"contentType\":\"LEARNING_PAGE\",\"minCount\":2},\"type\":\"query\"}]","promise":"$@76"}]},"children":"$L77"}]