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Structures & Algorithms","topic":"Fenwick Trees (Binary Indexed Trees)","title":"Fenwick Trees — Binary Indexed Trees","description":"Master Fenwick Trees (Binary Indexed Trees), an elegant and space-efficient data structure for cumulative frequency tables and prefix sum queries. Learn their construction, operations, time complexity, and practical applications compared to segment trees.","difficulty":"ADVANCED","estimatedMinutes":90,"pages":[{"id":"seed_6b105881bcc74d00410fe49ac5824828","title":"Simpler Alternative to Segment Trees","order":0,"content":[{"id":"opening-hook","title":"The Quest for Elegance in Range Queries","content":[{"data":{"content":"In the previous modules, you mastered **segment trees**—a powerful data structure capable of answering arbitrary range queries and supporting both point and range updates in O(log n) time. Segment trees are versatile, supporting operations like sum, minimum, maximum, GCD, and more.\n\nBut with great power comes great complexity. Segment trees require:\n- **4n space** in the worst case (or 2n with careful implementation)\n- Complex tree-building logic with careful index management\n- Intricate lazy propagation for range updates\n- Recursive or iterative implementations with multiple edge cases\n\nWhat if there existed a **simpler** data structure—one that sacrifices some generality for elegance, requiring less code, less memory, and simpler mental models while still achieving O(log n) operations?\n\n**Enter the Fenwick Tree**, also known as the **Binary Indexed Tree (BIT)**—an ingeniously simple data structure that solves a specific but extremely common class of range query problems."},"type":"text"},{"data":{"icon":"Target","title":"What You Will Learn","content":"By the end of this page, you will understand:\n• The historical origin and motivation behind Fenwick Trees\n• Why simpler sometimes means better in algorithm design\n• The exact problem class where BITs excel\n• A comprehensive comparison between Fenwick Trees and Segment Trees\n• When to choose each data structure in practice","variant":"info"},"type":"callout"}]},{"id":"historical-context","title":"Historical Context — Peter Fenwick's Insight","content":[{"data":{"content":"The Fenwick Tree was invented by **Peter M. Fenwick** in 1994, published in his seminal paper *\"A New Data Structure for Cumulative Frequency Tables\"* in the journal Software: Practice and Experience.\n\nFenwick's motivation was practical: he was working on **data compression algorithms** (specifically arithmetic coding) which required frequent updates to cumulative frequency tables. The existing solutions—either naive O(n) prefix sums or complex balanced trees—were either too slow or too complicated for his purposes.\n\nFenwick's brilliant insight was to exploit the **binary representation of array indices** to create a data structure that:\n1. Uses exactly **n elements of storage** (compared to 4n for segment trees)\n2. Supports **O(log n)** prefix sum queries\n3. Supports **O(log n)** point updates\n4. Has an **extraordinarily simple implementation** (under 20 lines of code)\n5. Has **excellent cache locality** due to sequential memory access patterns"},"type":"text"},{"data":{"icon":"Info","title":"Why \"Binary Indexed Tree\"?","content":"The name \"Binary Indexed Tree\" comes from the fact that the structure exploits the binary representation of indices. Each index i is responsible for a range of elements determined by the lowest set bit in i's binary representation. We'll explore this in detail in subsequent pages.","variant":"note"},"type":"callout"},{"data":{"content":"**The compression connection:**\n\nIn arithmetic coding (a lossless data compression technique), you need to maintain cumulative frequency counts. For each symbol in the input:\n1. **Query** the cumulative frequency up to that symbol (to determine its coding interval)\n2. **Update** that symbol's frequency (increment by 1 after processing)\n\nWith n possible symbols and m characters to process, a naive approach gives O(n × m) total time. Fenwick's structure brought this down to **O(m × log n)**—a dramatic improvement for large alphabets.\n\nThough arithmetic coding was Fenwick's original application, the structure's elegance has made it a favorite for competitive programmers and systems designers working with:\n- Cumulative frequency tables\n- Range sum queries with point updates\n- Inversion count problems\n- Order statistics\n- Computational geometry problems"},"type":"text"}],"titleType":"number","titleNumber":1},{"id":"problem-class","title":"The Problem Class — Where BITs Excel","content":[{"data":{"content":"Before diving into implementation details, let's precisely define the problem class where Fenwick Trees shine. This clarity will help you recognize when to reach for a BIT versus a segment tree or other data structure.\n\n**The Canonical BIT Problem:**\n\nGiven an array A of n elements, support the following operations:\n1. **Prefix Query**: Compute sum(A[0..i]) for any index i\n2. **Point Update**: Set A[i] = A[i] + delta for any index i and value delta\n\nBoth operations should complete in O(log n) time."},"type":"text"},{"data":{"icon":"AlertCircle","title":"Range Query Derivation","content":"Notice that BITs natively support PREFIX queries, not ARBITRARY range queries. To compute sum(A[l..r]), you compute prefix(r) - prefix(l-1). This works for SUM because addition has an inverse (subtraction). For MIN/MAX, this trick doesn't work—you cannot derive range MIN from two prefix MINs.","variant":"warning"},"type":"callout"},{"data":{"content":"**Why this problem is so common:**\n\nThe prefix sum + point update pattern appears everywhere in computer science:\n\n**1. Frequency Counting with Queries**\n- Count how many elements less than x have been seen so far\n- Dynamic order statistics (rank queries)\n\n**2. Range Sum with Updates**\n- Real-time analytics dashboards\n- Inventory tracking systems\n- Financial transaction logs\n\n**3. Inversion Counting**\n- Count pairs (i, j) where i < j but A[i] > A[j]\n- Measuring sortedness of an array\n\n**4. 2D Problems**\n- Counting points in a rectangle (with 2D BIT)\n- Image processing with dynamic updates\n\n**5. Computational Geometry**\n- Counting points below a certain y-coordinate\n- Sweep line algorithms"},"type":"text"},{"data":{"rows":[["\"Prefix sum\" or \"cumulative\" in problem","Sum of first k elements","BIT's native operation"],["\"Point update + range query\" with SUM","Update one cell, query range sum","Range = prefix(r) - prefix(l-1)"],["\"Count elements less than x\"","Number of smaller elements seen","Use value as index, query prefix"],["\"Inversion count\"","Count out-of-order pairs","Classic BIT application"],["\"Dynamic order statistics\"","k-th smallest element","Binary search + prefix queries"]],"caption":"Problem Pattern Recognition for BITs","headers":["Pattern Signal","Example","Why BIT Works"]},"type":"table"}],"titleType":"number","titleNumber":2},{"id":"segment-trees-comparison","title":"Segment Trees — The Heavy Artillery","content":[{"data":{"content":"To appreciate Fenwick Trees, we must first understand what segment trees offer—and at what cost.\n\n**Segment Tree Capabilities:**\n\n1. **Arbitrary Range Queries**: Compute f(A[l..r]) for any associative function f (sum, min, max, GCD, XOR, etc.)\n2. **Point Updates**: Modify a single element and propagate changes\n3. **Range Updates**: Modify entire ranges (with lazy propagation)\n4. **Non-Invertible Operations**: Works for min/max where you can't \"undo\"\n\n**Segment Tree Costs:**\n\n1. **Space**: 4n elements (or 2n with careful indexing)\n2. **Implementation Complexity**: ~50-100 lines for basic version, 150+ with lazy propagation\n3. **Mental Overhead**: Tree indexing, merge functions, push-down logic\n4. **Cache Performance**: Tree traversal has less predictable memory access patterns"},"type":"text"},{"data":{"code":"$72","title":"Segment Tree Implementation — The Complexity Cost","language":"typescript"},"type":"code"},{"data":{"content":"**The cognitive burden:**\n\nNotice the segment tree implementation requires:\n- Understanding of implicit tree representation\n- Careful management of three indices (node, start, end)\n- Correct base cases and range checks\n- Proper merge operation placement\n- Easy-to-introduce off-by-one errors\n\nFor competitive programming or quick implementations, this complexity can lead to bugs. For production systems, it means more code to maintain and more opportunities for subtle errors.\n\n**When is this complexity justified?**\n\nSegment trees are worth their complexity when you need:\n- Non-invertible operations (min, max)\n- Range updates (increment all elements in [l..r])\n- Complex merge operations (like finding the count of minimum)\n- Maximum flexibility in supported operations"},"type":"text"}],"titleType":"number","titleNumber":3},{"id":"fenwick-promise","title":"The Fenwick Tree Promise — Elegant Simplicity","content":[{"data":{"content":"Now let's see what Fenwick Trees offer—the entire data structure in remarkably few lines:\n\n**Fenwick Tree Capabilities:**\n\n1. **Prefix Queries**: Compute sum(A[0..i]) in O(log n)\n2. **Point Updates**: Add delta to A[i] in O(log n)\n3. **Range Queries**: Compute sum(A[l..r]) as prefix(r) - prefix(l-1)\n\n**Fenwick Tree Advantages:**\n\n1. **Space**: Exactly n elements\n2. **Implementation**: Under 20 lines of code\n3. **Mental Model**: Once understood, nearly impossible to implement incorrectly\n4. **Cache Performance**: Sequential memory access patterns\n5. **Constant Factor**: ~2-3x faster than segment trees in practice"},"type":"text"},{"data":{"code":"class FenwickTree {\n private tree: number[];\n private n: number;\n\n constructor(n: number) {\n this.n = n;\n this.tree = new Array(n + 1).fill(0); // 1-indexed for simpler bit manipulation\n }\n\n // Add delta to index i (1-indexed)\n update(i: number, delta: number): void {\n for (; i <= this.n; i += i & (-i)) {\n this.tree[i] += delta;\n }\n }\n\n // Get prefix sum from 1 to i\n query(i: number): number {\n let sum = 0;\n for (; i > 0; i -= i & (-i)) {\n sum += this.tree[i];\n }\n return sum;\n }\n\n // Range query [l, r] (1-indexed)\n rangeQuery(l: number, r: number): number {\n return this.query(r) - this.query(l - 1);\n }\n}\n\n// Usage — intuitive and simple\nconst bit = new FenwickTree(6);\nbit.update(1, 1); // A[1] += 1\nbit.update(2, 3); // A[2] += 3\nbit.update(3, 5); // A[3] += 5\nconsole.log(bit.query(3)); // Prefix sum 1..3 = 9\nconsole.log(bit.rangeQuery(2, 3)); // Range sum 2..3 = 8","title":"Fenwick Tree Implementation — Elegant Simplicity","language":"typescript"},"type":"code"},{"data":{"icon":"Lightbulb","title":"The Magic of i & (-i)","content":"The expression i & (-i) extracts the lowest set bit of i. This single operation is the heart of Fenwick Trees. For example:\n• 12 in binary: 1100, so 12 & (-12) = 0100 = 4\n• 6 in binary: 0110, so 6 & (-6) = 0010 = 2\nWe'll explore why this works in the next page.","variant":"tip"},"type":"callout"},{"data":{"content":"**Compare the implementations:**\n\nThe Fenwick Tree implementation is:\n- **Half the lines** of the segment tree\n- **No recursion** needed (simple loops)\n- **No complex index management** (just i and bit operations)\n- **No tree structure** to visualize (just an array)\n- **No build function** needed (updates construct the structure)\n\nThis simplicity isn't just aesthetic—it translates to:\n- Fewer bugs in implementation\n- Faster debugging when issues arise\n- Lower cognitive load when reading code\n- Easier optimization and modification"},"type":"text"}],"titleType":"number","titleNumber":4},{"id":"comparison-table","title":"Comprehensive Comparison — BIT vs Segment Tree","content":[{"data":{"content":"Let's systematically compare these two data structures across all relevant dimensions:"},"type":"text"},{"data":{"rows":[["Space Complexity","O(n) — exactly n elements","O(n) — typically 4n elements"],["Prefix Query Time","O(log n)","O(log n)"],["Range Query Time","O(log n) — via prefix difference","O(log n) — direct support"],["Point Update Time","O(log n)","O(log n)"],["Range Update Time","O(log n) — with two BITs trick","O(log n) — lazy propagation"],["Supported Operations","Only invertible (sum, XOR)","Any associative operation"],["Min/Max Queries","❌ Not supported","✓ Fully supported"],["Implementation Lines","~15-20 lines","~50-100+ lines"],["Bug Probability","Very low — simple logic","Moderate — complex indices"],["Constant Factor","~2-3x faster","Baseline"],["Cache Performance","Excellent — sequential access","Good — tree traversal"],["Learning Curve","Low — one key insight","Moderate — tree concepts"],["Memory Footprint","n × element_size","4n × element_size (typical)"],["2D Extension","Straightforward","Possible but complex"]],"caption":"Fenwick Tree vs Segment Tree — Complete Comparison","headers":["Aspect","Fenwick Tree (BIT)","Segment Tree"]},"type":"table"},{"data":{"icon":"Info","title":"The 4x Memory Difference","content":"Why 4n for segment trees? Segment trees are typically implemented as complete binary trees. For n leaves, you need roughly 2n nodes total. But to use simple 2i+1, 2i+2 indexing with a non-power-of-2 n, you often allocate 4n to be safe. BITs use exactly n elements regardless of n's value.","variant":"info"},"type":"callout"}],"titleType":"number","titleNumber":5},{"id":"decision-framework","title":"Decision Framework — When to Use What","content":[{"data":{"content":"With both tools in your arsenal, how do you choose? Here's a systematic decision framework:"},"type":"text"},{"data":{"items":["**Operation is invertible** — Sum, XOR, or any operation where you can compute f(l, r) from f(0, r) and f(0, l-1)","**Prefix queries dominate** — Your queries naturally fit the prefix sum pattern","**Memory is constrained** — You need minimal space overhead","**Implementation simplicity matters** — Competitive programming time limits, or production reliability","**Performance is critical** — The ~2-3x constant factor improvement matters","**2D extensions needed** — 2D BITs are simpler than 2D segment trees","**You need to explain the code** — BIT's simplicity aids code review and maintenance"],"title":{"icon":"CheckCircle","text":"Use Fenwick Tree (BIT) When:","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"items":["**Non-invertible operations** — Min, max, GCD queries require segment trees","**Complex merge operations** — Finding minimum AND its count, for example","**Lazy propagation needed** — Efficient range updates on arbitrary operations","**Arbitrary range queries** — When prefix subtraction trick doesn't apply","**Dynamic trees** — Need to merge or split trees (persistent segment trees)","**Query structure is complex** — Problems with multiple aggregate functions"],"title":{"icon":"XCircle","text":"Use Segment Tree When:","type":"icon","color":"destructive","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"code":"$73","title":"Decision Helper Function","language":"typescript"},"type":"code"}],"titleType":"number","titleNumber":6},{"id":"real-world-tradeoffs","title":"Real-World Trade-off Analysis","content":[{"data":{"content":"Let's examine concrete scenarios where the BIT vs Segment Tree decision matters:\n\n**Scenario 1: Real-Time Analytics Dashboard**\n\nYou're building a dashboard that shows:\n- Cumulative sales up to any date\n- Point updates when new sales are recorded\n- 10 million data points, 1000 queries/second\n\n**Recommendation: Fenwick Tree**\n- Prefix sum is the natural query pattern\n- 4x memory savings (40MB vs 160MB for the structure alone)\n- Better cache performance under high query load\n- Simple implementation reduces production bugs"},"type":"text"},{"data":{"content":"**Scenario 2: Stock Trading Minimum Price Query**\n\nYou need to track:\n- Minimum stock price in any date range\n- Price updates as trades occur\n- Must handle 50 million records\n\n**Recommendation: Segment Tree**\n- Minimum is not invertible — can't derive range min from prefix mins\n- No BIT-based solution exists for this problem\n- Segment tree's O(4n) = 200MB is acceptable for the use case"},"type":"text"},{"data":{"content":"**Scenario 3: Competitive Programming Contest**\n\nProblem requires:\n- Range sum queries\n- Point updates\n- 30-minute time limit to implement\n\n**Recommendation: Fenwick Tree**\n- Under 20 lines of code\n- Almost impossible to introduce subtle bugs\n- Faster to type and debug\n- Performance sufficient for any contest constraints"},"type":"text"},{"data":{"icon":"CheckCircle","title":"The Right Tool for the Job","content":"Neither data structure is universally superior. Fenwick Trees are a laser-focused tool for prefix-sum-like problems, while Segment Trees are a Swiss Army knife for range queries. Knowing both—and knowing when to use each—makes you a more effective engineer.","variant":"success"},"type":"callout"}],"titleType":"number","titleNumber":7},{"id":"simplicity-advantage","title":"The Simplicity Advantage in Production","content":[{"data":{"content":"Beyond algorithmic considerations, Fenwick Trees exemplify an important engineering principle: **simpler solutions are often better solutions**.\n\n**Why simplicity matters:**"},"type":"text"},{"data":{"items":["**Fewer lines = fewer bugs** — Every line of code is a potential failure point. BIT's 15 lines have far fewer places to hide bugs than segment tree's 100 lines.","**Easier code review** — Reviewers can understand the entire implementation at a glance, making thorough review practical.","**Faster onboarding** — New team members can understand BIT code quickly, reducing knowledge silos.","**Simpler debugging** — When issues arise, diagnosing a 15-line structure is dramatically easier than a 100-line one.","**Predictable performance** — Simple loop-based code has more predictable performance characteristics than recursive tree walking.","**Easier testing** — Fewer code paths mean exhaustive testing is actually achievable."],"title":{"icon":"Code","text":"Production Code Benefits","type":"icon","color":"primary","variant":"outline-circle"},"variant":"glass"},"type":"list"},{"data":{"content":"**The KISS Principle Applied:**\n\nKISS (Keep It Simple, Stupid) is a design principle stating that most systems work best when kept simple. Fenwick Trees embody this:\n\n> *If a simpler solution solves your problem, choose the simpler solution.*\n\nFor prefix sum queries with point updates—one of the most common range query patterns—Fenwick Trees are that simpler solution. Reaching for a segment tree when a BIT suffices is like using a crane to move a chair."},"type":"text"},{"data":{"icon":"Lightbulb","title":"Complexity Budget","content":"Every engineering project has a complexity budget. Saving complexity on well-understood components (like range queries) leaves more budget for novel, problem-specific code. Using a BIT when appropriate is good complexity stewardship.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":8},{"id":"summary","title":"Summary — The Elegance of Constraints","content":[{"data":{"content":"This page introduced Fenwick Trees as a simpler alternative to segment trees for a specific but extremely common problem class. Here are the key takeaways:"},"type":"text"},{"data":{"type":"ordered","items":["**Fenwick Trees (BITs)** were invented by Peter Fenwick in 1994 for cumulative frequency tables in data compression.","**BITs excel at prefix queries with point updates** — the most common range query pattern in practice.","**Trade-off**: BITs sacrifice generality (only invertible operations) for simplicity and efficiency.","**Space advantage**: BITs use exactly n elements vs segment trees' 4n typical allocation.","**Implementation advantage**: ~15 lines vs ~100 lines, dramatically reducing bug probability.","**Performance advantage**: ~2-3x faster due to better cache locality and simpler code paths.","**Decision framework**: Use BIT for sum/XOR with point updates; use segment tree for min/max/GCD or range updates."],"title":{"icon":"CheckCircle","text":"Key Takeaways","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"content":"**What's next:**\n\nNow that you understand *why* Fenwick Trees exist and *when* to use them, we'll explore *how* they work. The next page dives deep into the BIT structure and representation—revealing the ingenious insight that makes this data structure possible."},"type":"text"},{"data":{"icon":"CheckCircle","title":"Page Complete","content":"You now understand the motivation, historical context, and trade-offs of Fenwick Trees compared to segment trees. You can identify problems where BITs are the optimal choice and articulate why simpler solutions are often better. Next, we'll explore the elegant binary index structure that makes BITs work.","variant":"success"},"type":"callout"}],"titleType":"number","titleNumber":9}],"isLocked":false},{"id":"seed_2945561d4c1738733b2709b5058aa5f3","title":"BIT Structure and Representation","order":1,"content":null,"isLocked":true},{"id":"seed_4ebaa3f2c5d034350063a6480ba4fae8","title":"Prefix Sum Queries","order":2,"content":null,"isLocked":true},{"id":"seed_52d711a94610157988c5990b0e94ade5","title":"Point Updates","order":3,"content":null,"isLocked":true},{"id":"seed_be71da421af1bd99410b168cee265cf2","title":"Time and Space Efficiency","order":4,"content":null,"isLocked":true}]}},"dataUpdateCount":1,"dataUpdatedAt":1792313303127,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getModule"],{"input":{"slug":"fenwick-trees-binary-indexed-trees"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getModule\"],{\"input\":{\"slug\":\"fenwick-trees-binary-indexed-trees\"},\"type\":\"query\"}]"},{"dehydratedAt":1792313303348,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792313303318,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getProgress"],{"input":{"moduleId":"seed_7c5188e54c2d8465543e276b8b9ecf5f"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getProgress\"],{\"input\":{\"moduleId\":\"seed_7c5188e54c2d8465543e276b8b9ecf5f\"},\"type\":\"query\"}]"},{"dehydratedAt":1792313303348,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792313303318,"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_be71da421af1bd99410b168cee265cf2"},"type":"query"}],"queryHash":"[[\"feedback\",\"get\"],{\"input\":{\"contentId\":\"seed_be71da421af1bd99410b168cee265cf2\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792313303348,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792313303318,"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_be71da421af1bd99410b168cee265cf2"},"type":"query"}],"queryHash":"[[\"feedback\",\"getUserLike\"],{\"input\":{\"contentId\":\"seed_be71da421af1bd99410b168cee265cf2\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792313303348,"state":{"data":{"json":[]},"dataUpdateCount":1,"dataUpdatedAt":1792313303318,"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_be71da421af1bd99410b168cee265cf2"},"type":"query"}],"queryHash":"[[\"highlights\",\"getHighlights\"],{\"input\":{\"contentId\":\"seed_be71da421af1bd99410b168cee265cf2\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792313303348,"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_be71da421af1bd99410b168cee265cf2","minCount":2},"type":"query"}],"queryHash":"[[\"highlights\",\"getPopularHighlights\"],{\"input\":{\"contentId\":\"seed_be71da421af1bd99410b168cee265cf2\",\"contentType\":\"LEARNING_PAGE\",\"minCount\":2},\"type\":\"query\"}]","promise":"$@74"}]},"children":"$L75"}]