feat(patterns): pattern taxonomy + is_optimal

backend/data/patterns/backtracking.yaml

@@ -1,6 +1,8 @@
 name: Backtracking
 slug: backtracking
 difficulty_level: 3
+pattern_type: algorithm
+display_order: 10
 
 description: >
   Build solutions incrementally, exploring choices one at a time and abandoning

backend/data/patterns/bfs.yaml

@@ -1,6 +1,8 @@
 name: BFS (Breadth-First Search)
 slug: bfs
 difficulty_level: 3
+pattern_type: algorithm
+display_order: 6
 
 description: >
   Level-by-level traversal using a queue, exploring all neighbors at the current

backend/data/patterns/binary-search.yaml

@@ -1,6 +1,8 @@
 name: Binary Search
 slug: binary-search
 difficulty_level: 2
+pattern_type: algorithm
+display_order: 4
 
 description: >
   Efficiently search sorted data by repeatedly dividing the search space in half.
@@ -200,3 +202,172 @@ related_patterns:
   - two-pointers
 
 prerequisite_patterns: []
+
+visualization_examples:
+  - id: binary-search-find-target
+    title: Find Target in Sorted Array
+    input:
+      nums: [1, 3, 5, 7, 9, 11, 13]
+      target: 9
+    code: |
+      def binary_search(nums, target):
+          left, right = 0, len(nums) - 1
+
+          while left <= right:
+              mid = left + (right - left) // 2
+
+              if nums[mid] == target:
+                  return mid
+              elif nums[mid] < target:
+                  left = mid + 1
+              else:
+                  right = mid - 1
+
+          return -1
+    steps:
+      - id: step-1
+        description: "Initialize pointers: left=0, right=6. Search space is entire array."
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 1, state: default }
+              - { value: 3, state: default }
+              - { value: 5, state: default }
+              - { value: 7, state: default }
+              - { value: 9, state: default }
+              - { value: 11, state: default }
+              - { value: 13, state: default }
+            pointers:
+              left: 0
+              right: 6
+        variables:
+          target: 9
+          left: 0
+          right: 6
+        codeHighlight:
+          startLine: 2
+          endLine: 2
+
+      - id: step-2
+        description: "Calculate mid=3. nums[3]=7 < target=9, so search right half."
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 1, state: visited }
+              - { value: 3, state: visited }
+              - { value: 5, state: visited }
+              - { value: 7, state: comparing }
+              - { value: 9, state: default }
+              - { value: 11, state: default }
+              - { value: 13, state: default }
+            pointers:
+              left: 0
+              mid: 3
+              right: 6
+        variables:
+          target: 9
+          left: 0
+          right: 6
+          mid: 3
+        codeHighlight:
+          startLine: 9
+          endLine: 10
+
+      - id: step-3
+        description: "Update left=4. New search space is indices 4-6."
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 1, state: visited }
+              - { value: 3, state: visited }
+              - { value: 5, state: visited }
+              - { value: 7, state: visited }
+              - { value: 9, state: default }
+              - { value: 11, state: default }
+              - { value: 13, state: default }
+            pointers:
+              left: 4
+              right: 6
+        variables:
+          target: 9
+          left: 4
+          right: 6
+        codeHighlight:
+          startLine: 4
+          endLine: 4
+
+      - id: step-4
+        description: "Calculate mid=5. nums[5]=11 > target=9, so search left half."
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 1, state: visited }
+              - { value: 3, state: visited }
+              - { value: 5, state: visited }
+              - { value: 7, state: visited }
+              - { value: 9, state: default }
+              - { value: 11, state: comparing }
+              - { value: 13, state: visited }
+            pointers:
+              left: 4
+              mid: 5
+              right: 6
+        variables:
+          target: 9
+          left: 4
+          right: 6
+          mid: 5
+        codeHighlight:
+          startLine: 11
+          endLine: 12
+
+      - id: step-5
+        description: "Update right=4. Search space narrowed to index 4 only."
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 1, state: visited }
+              - { value: 3, state: visited }
+              - { value: 5, state: visited }
+              - { value: 7, state: visited }
+              - { value: 9, state: active }
+              - { value: 11, state: visited }
+              - { value: 13, state: visited }
+            pointers:
+              left: 4
+              right: 4
+        variables:
+          target: 9
+          left: 4
+          right: 4
+        codeHighlight:
+          startLine: 4
+          endLine: 4
+
+      - id: step-6
+        description: "Calculate mid=4. nums[4]=9 == target=9. Found at index 4!"
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 1, state: visited }
+              - { value: 3, state: visited }
+              - { value: 5, state: visited }
+              - { value: 7, state: visited }
+              - { value: 9, state: found }
+              - { value: 11, state: visited }
+              - { value: 13, state: visited }
+            pointers:
+              mid: 4
+        variables:
+          target: 9
+          mid: 4
+          result: 4
+        codeHighlight:
+          startLine: 7
+          endLine: 8

backend/data/patterns/bit-manipulation.yaml

@@ -1,6 +1,8 @@
 name: Bit Manipulation
 slug: bit-manipulation
 difficulty_level: 2
+pattern_type: technique
+display_order: 19
 
 description: >
   Techniques using binary operations (AND, OR, XOR, NOT, shifts) to solve

backend/data/patterns/counting-sort.yaml

@@ -0,0 +1,92 @@
+name: Counting / Bucket Sort
+slug: counting-sort
+difficulty_level: 2
+pattern_type: technique
+display_order: 20
+
+description: >
+  Exploit bounded value ranges to achieve linear time sorting or selection
+  by using values as array indices.
+
+when_to_use: |
+  - Finding top K elements when frequencies are bounded
+  - Sorting when values are in a known, limited range
+  - Problems involving frequency counting with bounded inputs
+  - Color sorting (Dutch National Flag)
+
+metaphor: |
+  Imagine sorting mail into numbered PO boxes. Instead of comparing letters
+  to each other, you simply look at the box number and drop it in. If you
+  have 100 boxes, sorting 1000 letters takes 1000 steps, not 1000 x log(1000).
+
+core_concept: |
+  When values are bounded within a known range [0, k], you can use the value
+  itself as an index into an array of "buckets." This converts comparison-based
+  O(n log n) sorting into O(n + k) counting operations.
+
+  The key insight: **bounded values = direct addressing is possible**.
+
+code_template: |
+  def bucket_sort_approach(nums: list[int], k: int) -> list[int]:
+      # Create buckets indexed by value/frequency
+      n = len(nums)
+      buckets = [[] for _ in range(n + 1)]  # n+1 for frequency 0 to n
+
+      # Count frequencies
+      count = {}
+      for num in nums:
+          count[num] = count.get(num, 0) + 1
+
+      # Place elements in frequency buckets
+      for num, freq in count.items():
+          buckets[freq].append(num)
+
+      # Collect from highest frequency
+      result = []
+      for i in range(n, 0, -1):
+          for num in buckets[i]:
+              result.append(num)
+              if len(result) == k:
+                  return result
+      return result
+
+recognition_signals:
+  - "top k frequent"
+  - "sort colors"
+  - "values in range [0, n]"
+  - "frequency bounded by array size"
+  - "O(n) time required"
+  - "counting occurrences"
+
+common_mistakes:
+  - title: Using Heap When Bucket Sort is Optimal
+    description: |
+      Heap gives O(n log k) but bucket sort gives O(n) when frequencies
+      are bounded. Always check if values/frequencies have a known upper bound.
+    fix: |
+      Ask: "What's the maximum possible value/frequency?" If bounded by n,
+      use bucket sort.
+
+  - title: Off-by-One in Bucket Array
+    description: |
+      Creating `n` buckets for frequencies 0 to n-1 misses frequency `n`
+      (when all elements are identical).
+    fix: |
+      Create `n + 1` buckets to handle frequencies from 0 to n inclusive.
+
+variations:
+  - name: Top K Frequent Elements
+    description: Use frequency as bucket index, collect from highest
+    example: "top-k-frequent-elements"
+  - name: Sort Colors (Dutch National Flag)
+    description: Three buckets for 0, 1, 2
+    example: "sort-colors"
+  - name: H-Index
+    description: Citation count buckets
+    example: "h-index"
+
+related_patterns:
+  - heap
+  - two-pointers
+
+prerequisite_patterns: []

backend/data/patterns/cyclic-sort.yaml

@@ -0,0 +1,96 @@
+name: Cyclic Sort
+slug: cyclic-sort
+difficulty_level: 2
+pattern_type: algorithm
+display_order: 21
+
+description: >
+  Place each number at its "correct" index when dealing with arrays containing
+  numbers in the range [1, n] or [0, n-1].
+
+when_to_use: |
+  - Array contains numbers from 1 to n (or 0 to n-1)
+  - Finding missing/duplicate numbers in such arrays
+  - Problems asking for O(1) space and O(n) time
+  - "First missing positive" type problems
+
+metaphor: |
+  Imagine a row of n chairs numbered 1 to n, and n people each holding a
+  ticket with their seat number. Instead of searching for each person's seat,
+  you ask everyone to swap positions until they're in their ticketed seat.
+  After at most n swaps, everyone is seated correctly.
+
+core_concept: |
+  For an array where values should map directly to indices (e.g., value 3
+  belongs at index 2 if 1-indexed), repeatedly swap each element to its
+  correct position until the array is "sorted."
+
+  Key insight: Each swap places at least one element correctly, so at most
+  n swaps are needed -> O(n) time with O(1) space.
+
+code_template: |
+  def cyclic_sort(nums: list[int]) -> list[int]:
+      """Sort array where values are in range [1, n]."""
+      i = 0
+      while i < len(nums):
+          # Correct index for value nums[i] (1-indexed value -> 0-indexed position)
+          correct_idx = nums[i] - 1
+
+          # If not in correct position and not a duplicate, swap
+          if nums[i] != nums[correct_idx]:
+              nums[i], nums[correct_idx] = nums[correct_idx], nums[i]
+          else:
+              i += 1
+      return nums
+
+  def find_missing(nums: list[int]) -> int:
+      """Find missing number after cyclic sort."""
+      cyclic_sort(nums)
+      for i, num in enumerate(nums):
+          if num != i + 1:
+              return i + 1
+      return len(nums) + 1
+
+recognition_signals:
+  - "array of length n with values 1 to n"
+  - "find the missing number"
+  - "find the duplicate"
+  - "first missing positive"
+  - "O(1) extra space"
+  - "in-place rearrangement"
+
+common_mistakes:
+  - title: Infinite Loop on Duplicates
+    description: |
+      Swapping endlessly when current value equals value at target index
+      (both are duplicates).
+    fix: |
+      Check `nums[i] != nums[correct_idx]` before swapping, not just
+      `nums[i] != correct_idx + 1`.
+
+  - title: Wrong Index Calculation
+    description: |
+      Confusing 0-indexed vs 1-indexed. If values are 1 to n, correct
+      index is `value - 1`. If values are 0 to n-1, correct index equals value.
+    fix: |
+      Clearly define the mapping: `correct_idx = nums[i] - 1` for 1-indexed values.
+
+variations:
+  - name: Find Missing Number
+    description: After cyclic sort, scan for index where value != index + 1
+    example: "missing-number"
+  - name: Find Duplicate
+    description: After cyclic sort, find where value != expected
+    example: "find-the-duplicate-number"
+  - name: First Missing Positive
+    description: Ignore negatives and values > n, then cyclic sort
+    example: "first-missing-positive"
+  - name: Find All Missing Numbers
+    description: Collect all indices where value != expected
+    example: "find-all-numbers-disappeared-in-an-array"
+
+related_patterns:
+  - counting-sort
+  - fast-slow-pointers
+
+prerequisite_patterns: []

backend/data/patterns/dfs.yaml

@@ -1,6 +1,8 @@
 name: DFS (Depth-First Search)
 slug: dfs
 difficulty_level: 3
+pattern_type: algorithm
+display_order: 7
 
 description: >
   Explore as deep as possible along each branch before backtracking, using

backend/data/patterns/divide-and-conquer.yaml

@@ -0,0 +1,97 @@
+name: Divide and Conquer
+slug: divide-and-conquer
+difficulty_level: 3
+pattern_type: algorithm
+display_order: 23
+
+description: >
+  Break a problem into smaller subproblems, solve them independently,
+  and combine results. Unlike DP, subproblems don't overlap.
+
+when_to_use: |
+  - Problems naturally split into independent halves
+  - Merge sort style combining
+  - Finding kth element efficiently
+  - Tree-structured recursion without memoization needs
+
+metaphor: |
+  Imagine sorting a deck of cards by splitting it in half, having two
+  friends each sort their half, then merging the sorted halves. Each
+  friend can recursively split their half too. The key: halves are
+  independent and combining sorted halves is easy.
+
+core_concept: |
+  Three steps:
+  1. **Divide:** Split problem into smaller subproblems
+  2. **Conquer:** Solve subproblems recursively (base case when trivial)
+  3. **Combine:** Merge subproblem solutions into final answer
+
+  Time complexity often follows: T(n) = aT(n/b) + O(n^c)
+  Solved by Master Theorem.
+
+code_template: |
+  def merge_sort(arr: list[int]) -> list[int]:
+      """Classic divide and conquer example."""
+      if len(arr) <= 1:
+          return arr
+
+      # Divide
+      mid = len(arr) // 2
+      left = merge_sort(arr[:mid])
+      right = merge_sort(arr[mid:])
+
+      # Combine (merge)
+      return merge(left, right)
+
+  def merge(left: list[int], right: list[int]) -> list[int]:
+      result = []
+      i = j = 0
+      while i < len(left) and j < len(right):
+          if left[i] <= right[j]:
+              result.append(left[i])
+              i += 1
+          else:
+              result.append(right[j])
+              j += 1
+      result.extend(left[i:])
+      result.extend(right[j:])
+      return result
+
+recognition_signals:
+  - "merge sort"
+  - "find kth largest/smallest"
+  - "count inversions"
+  - "closest pair of points"
+  - "problems on sorted arrays that can be split"
+
+common_mistakes:
+  - title: Confusing with Dynamic Programming
+    description: |
+      D&C subproblems are independent. DP subproblems overlap and need
+      memoization. Using D&C on overlapping subproblems causes exponential time.
+    fix: |
+      Check: Do subproblems share computation? If yes, use DP. If no, use D&C.
+
+  - title: Inefficient Combine Step
+    description: |
+      If combining takes O(n^2), overall might not be better than brute force.
+    fix: |
+      Design O(n) or O(n log n) combine step. Merge sort's merge is O(n).
+
+variations:
+  - name: Merge Sort
+    description: Sort by splitting, sorting halves, merging
+    example: "sort-an-array"
+  - name: Quick Select
+    description: Find kth element by partitioning
+    example: "kth-largest-element-in-an-array"
+  - name: Count Inversions
+    description: Count pairs where larger element precedes smaller
+    example: "count-of-smaller-numbers-after-self"
+
+related_patterns:
+  - binary-search
+  - dynamic-programming
+
+prerequisite_patterns:
+  - binary-search

backend/data/patterns/dynamic-programming.yaml

@@ -1,6 +1,8 @@
 name: Dynamic Programming
 slug: dynamic-programming
 difficulty_level: 4
+pattern_type: technique
+display_order: 11
 
 description: >
   Break problems into overlapping subproblems, storing results to avoid

backend/data/patterns/fast-slow-pointers.yaml

@@ -1,6 +1,8 @@
 name: Fast & Slow Pointers
 slug: fast-slow-pointers
 difficulty_level: 2
+pattern_type: algorithm
+display_order: 3
 
 description: >
   Use two pointers moving at different speeds to detect cycles, find midpoints,
@@ -296,3 +298,188 @@ related_patterns:
 
 prerequisite_patterns:
   - two-pointers
+
+visualization_examples:
+  - id: find-middle-of-list
+    title: Find Middle of Linked List
+    input:
+      list: [1, 2, 3, 4, 5]
+    code: |
+      def find_middle(head):
+          slow = fast = head
+
+          while fast and fast.next:
+              slow = slow.next
+              fast = fast.next.next
+
+          return slow
+    steps:
+      - id: step-1
+        description: "Initialize slow and fast pointers at head (node 1)."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: active }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+              - { value: 4, state: default }
+              - { value: 5, state: default }
+            pointers:
+              slow: 0
+              fast: 0
+        variables: {}
+        codeHighlight:
+          startLine: 2
+          endLine: 2
+
+      - id: step-2
+        description: "Move slow 1 step (to 2), fast 2 steps (to 3)."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited }
+              - { value: 2, state: active }
+              - { value: 3, state: active }
+              - { value: 4, state: default }
+              - { value: 5, state: default }
+            pointers:
+              slow: 1
+              fast: 2
+        variables: {}
+        codeHighlight:
+          startLine: 5
+          endLine: 6
+
+      - id: step-3
+        description: "Move slow 1 step (to 3), fast 2 steps (to 5)."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited }
+              - { value: 2, state: visited }
+              - { value: 3, state: active }
+              - { value: 4, state: visited }
+              - { value: 5, state: active }
+            pointers:
+              slow: 2
+              fast: 4
+        variables: {}
+        codeHighlight:
+          startLine: 5
+          endLine: 6
+
+      - id: step-4
+        description: "fast.next is null, loop ends. slow is at middle (node 3)."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited }
+              - { value: 2, state: visited }
+              - { value: 3, state: found }
+              - { value: 4, state: visited }
+              - { value: 5, state: visited }
+            pointers:
+              slow: 2
+        variables:
+          result: 3
+        codeHighlight:
+          startLine: 8
+          endLine: 8
+
+  - id: detect-cycle
+    title: Detect Cycle in Linked List
+    input:
+      list: [1, 2, 3, 4, 5]
+      cycle_at: 2
+    code: |
+      def has_cycle(head):
+          slow = fast = head
+
+          while fast and fast.next:
+              slow = slow.next
+              fast = fast.next.next
+
+              if slow == fast:
+                  return True
+
+          return False
+    steps:
+      - id: step-1
+        description: "List has a cycle: 5 points back to 3. Initialize pointers at head."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: active }
+              - { value: 2, state: default }
+              - { value: 3, state: default, annotations: ["cycle start"] }
+              - { value: 4, state: default }
+              - { value: 5, state: default, annotations: ["→3"] }
+            pointers:
+              slow: 0
+              fast: 0
+        variables: {}
+        codeHighlight:
+          startLine: 2
+          endLine: 2
+
+      - id: step-2
+        description: "slow moves to 2, fast moves to 3."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited }
+              - { value: 2, state: active }
+              - { value: 3, state: active, annotations: ["cycle start"] }
+              - { value: 4, state: default }
+              - { value: 5, state: default, annotations: ["→3"] }
+            pointers:
+              slow: 1
+              fast: 2
+        variables: {}
+        codeHighlight:
+          startLine: 5
+          endLine: 6
+
+      - id: step-3
+        description: "slow moves to 3, fast moves to 5."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited }
+              - { value: 2, state: visited }
+              - { value: 3, state: active, annotations: ["cycle start"] }
+              - { value: 4, state: visited }
+              - { value: 5, state: active, annotations: ["→3"] }
+            pointers:
+              slow: 2
+              fast: 4
+        variables: {}
+        codeHighlight:
+          startLine: 5
+          endLine: 6
+
+      - id: step-4
+        description: "slow moves to 4, fast loops back to 4 (via 5→3→4). They meet!"
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited }
+              - { value: 2, state: visited }
+              - { value: 3, state: visited, annotations: ["cycle start"] }
+              - { value: 4, state: found, annotations: ["meeting point"] }
+              - { value: 5, state: visited, annotations: ["→3"] }
+            pointers:
+              "slow, fast": 3
+        variables:
+          cycle_detected: true
+        codeHighlight:
+          startLine: 8
+          endLine: 9

backend/data/patterns/greedy.yaml

@@ -1,6 +1,8 @@
 name: Greedy
 slug: greedy
 difficulty_level: 3
+pattern_type: technique
+display_order: 12
 
 description: >
   Make locally optimal choices at each step, hoping to find a global optimum.

backend/data/patterns/heap.yaml

@@ -1,6 +1,8 @@
 name: Heap / Priority Queue
 slug: heap
 difficulty_level: 3
+pattern_type: data_structure
+display_order: 16
 
 description: >
   A data structure that efficiently maintains the minimum or maximum element,

backend/data/patterns/intervals.yaml

@@ -1,6 +1,8 @@
 name: Overlapping Intervals
 slug: intervals
 difficulty_level: 2
+pattern_type: technique
+display_order: 13
 
 description: >
   Process and manipulate intervals (ranges) that may share common regions.

backend/data/patterns/linkedlist-reversal.yaml

@@ -1,6 +1,8 @@
 name: LinkedList In-Place Reversal
 slug: linkedlist-reversal
 difficulty_level: 2
+pattern_type: technique
+display_order: 14
 
 description: >
   Reverse linked list nodes in-place by manipulating pointers without allocating
@@ -277,3 +279,130 @@ related_patterns:
   - two-pointers
 
 prerequisite_patterns: []
+
+visualization_examples:
+  - id: reverse-linked-list
+    title: Reverse Entire Linked List
+    input:
+      list: [1, 2, 3, 4]
+    code: |
+      def reverse_list(head):
+          prev = None
+          curr = head
+
+          while curr:
+              next_node = curr.next
+              curr.next = prev
+              prev = curr
+              curr = next_node
+
+          return prev
+    steps:
+      - id: step-1
+        description: "Initialize prev=None, curr=head (node 1)."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: active }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+              - { value: 4, state: default }
+            pointers:
+              curr: 0
+        variables:
+          prev: "null"
+        codeHighlight:
+          startLine: 2
+          endLine: 3
+
+      - id: step-2
+        description: "Save next=2, reverse link: 1→null. Move pointers forward."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited, annotations: ["→null"] }
+              - { value: 2, state: active }
+              - { value: 3, state: default }
+              - { value: 4, state: default }
+            pointers:
+              prev: 0
+              curr: 1
+        variables:
+          next_node: 2
+        codeHighlight:
+          startLine: 6
+          endLine: 9
+
+      - id: step-3
+        description: "Save next=3, reverse link: 2→1. Move pointers forward."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited, annotations: ["→null"] }
+              - { value: 2, state: visited, annotations: ["→1"] }
+              - { value: 3, state: active }
+              - { value: 4, state: default }
+            pointers:
+              prev: 1
+              curr: 2
+        variables:
+          next_node: 3
+        codeHighlight:
+          startLine: 6
+          endLine: 9
+
+      - id: step-4
+        description: "Save next=4, reverse link: 3→2. Move pointers forward."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited, annotations: ["→null"] }
+              - { value: 2, state: visited, annotations: ["→1"] }
+              - { value: 3, state: visited, annotations: ["→2"] }
+              - { value: 4, state: active }
+            pointers:
+              prev: 2
+              curr: 3
+        variables:
+          next_node: 4
+        codeHighlight:
+          startLine: 6
+          endLine: 9
+
+      - id: step-5
+        description: "Save next=null, reverse link: 4→3. curr becomes null, loop ends."
+        structures:
+          list:
+            type: linkedlist
+            nodes:
+              - { value: 1, state: visited, annotations: ["→null"] }
+              - { value: 2, state: visited, annotations: ["→1"] }
+              - { value: 3, state: visited, annotations: ["→2"] }
+              - { value: 4, state: found, annotations: ["→3", "new head"] }
+            pointers:
+              prev: 3
+        variables:
+          next_node: "null"
+        codeHighlight:
+          startLine: 11
+          endLine: 11
+
+      - id: step-6
+        description: "Return prev (node 4). Reversed list: 4→3→2→1→null"
+        structures:
+          reversed:
+            type: linkedlist
+            nodes:
+              - { value: 4, state: found, annotations: ["head"] }
+              - { value: 3, state: default }
+              - { value: 2, state: default }
+              - { value: 1, state: default }
+        variables:
+          result: "4→3→2→1→null"
+        codeHighlight:
+          startLine: 11
+          endLine: 11

backend/data/patterns/matrix-traversal.yaml

@@ -1,6 +1,8 @@
 name: Matrix Traversal
 slug: matrix-traversal
 difficulty_level: 3
+pattern_type: algorithm
+display_order: 9
 
 description: >
   Navigate 2D grids systematically using DFS, BFS, or directional iteration.

backend/data/patterns/monotonic-stack.yaml

@@ -1,6 +1,8 @@
 name: Monotonic Stack
 slug: monotonic-stack
 difficulty_level: 3
+pattern_type: data_structure
+display_order: 15
 
 description: >
   Maintain a stack where elements are always in sorted order (either increasing or
@@ -267,3 +269,245 @@ related_patterns:
   - sliding-window
 
 prerequisite_patterns: []
+
+visualization_examples:
+  - id: largest-rectangle-histogram
+    title: "Largest Rectangle in Histogram"
+    input:
+      heights: [2, 1, 5, 6, 2, 3]
+    code: |
+      def largest_rectangle(heights):
+          stack = []  # (index, height)
+          max_area = 0
+
+          for i, h in enumerate(heights):
+              start = i
+              while stack and stack[-1][1] > h:
+                  idx, height = stack.pop()
+                  max_area = max(max_area, height * (i - idx))
+                  start = idx
+              stack.append((start, h))
+
+          for idx, height in stack:
+              max_area = max(max_area, height * (len(heights) - idx))
+
+          return max_area
+    steps:
+      - id: step-1
+        description: "Goal: Find the largest rectangle that fits in this histogram. Each bar's height is shown, and we want to find the biggest rectangular area."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: default }
+              - { value: 1, state: default }
+              - { value: 5, state: default }
+              - { value: 6, state: default }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+        variables:
+          question: "What's the largest rectangle?"
+          max_area: 0
+
+      - id: step-2
+        description: "We scan left to right. Bar 0 (height 2) could be the shortest bar in some rectangle. Push it to stack. Format: h:height @start_index."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: active }
+              - { value: 1, state: default }
+              - { value: 5, state: default }
+              - { value: 6, state: default }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+            pointers:
+              i: 0
+          stack (h:height @start):
+            type: stack
+            values:
+              - { value: "h:2 @0", state: active }
+        variables:
+          max_area: 0
+
+      - id: step-3
+        description: "Bar 1 (height 1) is SHORTER than bar 0 (height 2). Bar 0's rectangle can't extend further right! Pop it and calculate: width=1, area=2x1=2."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: comparing }
+              - { value: 1, state: active }
+              - { value: 5, state: default }
+              - { value: 6, state: default }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+            pointers:
+              i: 1
+            rectangle:
+              startIndex: 0
+              endIndex: 0
+              height: 2
+              state: comparing
+              label: "Area: 2"
+          stack (h:height @start):
+            type: stack
+            values: []
+        variables:
+          max_area: 2
+          calculation: "height(2) × width(1) = 2"
+
+      - id: step-4
+        description: "Now push bar 1 (height 1). Since bar 0 was popped, bar 1's rectangle can extend back to index 0. Stack stores (0, 1)."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: visited }
+              - { value: 1, state: active }
+              - { value: 5, state: default }
+              - { value: 6, state: default }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+            pointers:
+              i: 1
+            maxArea: 2
+          stack (h:height @start):
+            type: stack
+            values:
+              - { value: "h:1 @0", state: active }
+        variables:
+          max_area: 2
+          insight: "Height 1 can extend back to index 0!"
+
+      - id: step-5
+        description: "Bar 2 (height 5) is taller than bar 1. Taller bars might extend further right, so push it. Bar 3 (height 6) is also taller — push it too."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: visited }
+              - { value: 1, state: default }
+              - { value: 5, state: active }
+              - { value: 6, state: active }
+              - { value: 2, state: default }
+              - { value: 3, state: default }
+            pointers:
+              i: 3
+            maxArea: 2
+          stack (h:height @start):
+            type: stack
+            values:
+              - { value: "h:6 @3", state: active }
+              - { value: "h:5 @2", state: active }
+              - { value: "h:1 @0", state: default }
+        variables:
+          max_area: 2
+          stack_note: "Stack keeps increasing heights (bottom to top)"
+
+      - id: step-6
+        description: "Bar 4 (height 2) is SHORTER than bar 3 (height 6). Pop bar 3: width=1, area=6x1=6. Still shorter than bar 2 (height 5), pop it: width=2, area=5x2=10!"
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: visited }
+              - { value: 1, state: default }
+              - { value: 5, state: comparing }
+              - { value: 6, state: comparing }
+              - { value: 2, state: active }
+              - { value: 3, state: default }
+            pointers:
+              i: 4
+            rectangle:
+              startIndex: 2
+              endIndex: 3
+              height: 5
+              state: found
+              label: "Area: 10"
+            maxArea: 10
+          stack (h:height @start):
+            type: stack
+            values:
+              - { value: "h:1 @0", state: default }
+        variables:
+          max_area: 10
+          calculation: "height(5) × width(2) = 10"
+
+      - id: step-7
+        description: "Push bar 4 at index 2 (it extends back to where bar 2 was). Bar 5 (height 3) is taller, so push it too."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: visited }
+              - { value: 1, state: default }
+              - { value: 5, state: visited }
+              - { value: 6, state: visited }
+              - { value: 2, state: active }
+              - { value: 3, state: active }
+            pointers:
+              i: 5
+            maxArea: 10
+          stack (h:height @start):
+            type: stack
+            values:
+              - { value: "h:3 @5", state: active }
+              - { value: "h:2 @2", state: active }
+              - { value: "h:1 @0", state: default }
+        variables:
+          max_area: 10
+          note: "h:2 starts @2 because it extends back"
+
+      - id: step-8
+        description: "Done scanning! Process remaining stack: bar 5 (height 3) extends from index 5 to end (width 1), area=3. Bar 4 (height 2) extends from 2 to end (width 4), area=8."
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: visited }
+              - { value: 1, state: visited }
+              - { value: 5, state: visited }
+              - { value: 6, state: visited }
+              - { value: 2, state: comparing }
+              - { value: 3, state: comparing }
+            rectangle:
+              startIndex: 2
+              endIndex: 5
+              height: 2
+              state: comparing
+              label: "Area: 8"
+            maxArea: 10
+          stack (h:height @start):
+            type: stack
+            values:
+              - { value: "h:1 @0", state: default }
+        variables:
+          max_area: 10
+          remaining: "Still processing stack..."
+
+      - id: step-9
+        description: "Bar 1 (height 1) extends from index 0 to end (width 6), area=6. The largest rectangle has area 10 — the one spanning bars 2-3 with height 5!"
+        structures:
+          heights:
+            type: histogram
+            bars:
+              - { value: 2, state: visited }
+              - { value: 1, state: visited }
+              - { value: 5, state: found }
+              - { value: 6, state: found }
+              - { value: 2, state: visited }
+              - { value: 3, state: visited }
+            rectangle:
+              startIndex: 2
+              endIndex: 3
+              height: 5
+              state: found
+              label: "Max: 10"
+            maxArea: 10
+          stack (h:height @start):
+            type: stack
+            values: []
+        variables:
+          max_area: 10
+          answer: "Largest rectangle = 10"

backend/data/patterns/prefix-sum.yaml

@@ -1,6 +1,8 @@
 name: Prefix Sum
 slug: prefix-sum
 difficulty_level: 2
+pattern_type: technique
+display_order: 5
 
 description: >
   Precompute cumulative sums to answer range sum queries in O(1) time. This

backend/data/patterns/sliding-window.yaml

@@ -1,6 +1,8 @@
 name: Sliding Window
 slug: sliding-window
 difficulty_level: 2
+pattern_type: algorithm
+display_order: 2
 
 description: >
   Maintain a window of elements that slides through the data, tracking a
@@ -192,3 +194,129 @@ related_patterns:
 
 prerequisite_patterns:
   - two-pointers
+
+visualization_examples:
+  - id: max-sum-subarray
+    title: Maximum Sum Subarray of Size K
+    input:
+      nums: [2, 1, 5, 1, 3, 2]
+      k: 3
+    code: |
+      def max_sum_subarray(nums, k):
+          window_sum = sum(nums[:k])
+          max_sum = window_sum
+
+          for i in range(k, len(nums)):
+              window_sum += nums[i] - nums[i - k]
+              max_sum = max(max_sum, window_sum)
+
+          return max_sum
+    steps:
+      - id: step-1
+        description: "Initialize window with first k=3 elements: [2, 1, 5]. Sum = 8"
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 2, state: active, annotations: ["window"] }
+              - { value: 1, state: active, annotations: ["window"] }
+              - { value: 5, state: active, annotations: ["window"] }
+              - { value: 1, state: default }
+              - { value: 3, state: default }
+              - { value: 2, state: default }
+            pointers:
+              left: 0
+              right: 2
+        variables:
+          window_sum: 8
+          max_sum: 8
+        codeHighlight:
+          startLine: 2
+          endLine: 3
+
+      - id: step-2
+        description: "Slide window: remove 2, add 1. New window [1, 5, 1]. Sum = 7"
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 2, state: visited }
+              - { value: 1, state: active, annotations: ["window"] }
+              - { value: 5, state: active, annotations: ["window"] }
+              - { value: 1, state: active, annotations: ["window"] }
+              - { value: 3, state: default }
+              - { value: 2, state: default }
+            pointers:
+              left: 1
+              right: 3
+        variables:
+          window_sum: 7
+          max_sum: 8
+          i: 3
+        codeHighlight:
+          startLine: 5
+          endLine: 7
+
+      - id: step-3
+        description: "Slide window: remove 1, add 3. New window [5, 1, 3]. Sum = 9. New max!"
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 2, state: visited }
+              - { value: 1, state: visited }
+              - { value: 5, state: found, annotations: ["window"] }
+              - { value: 1, state: found, annotations: ["window"] }
+              - { value: 3, state: found, annotations: ["window"] }
+              - { value: 2, state: default }
+            pointers:
+              left: 2
+              right: 4
+        variables:
+          window_sum: 9
+          max_sum: 9
+          i: 4
+        codeHighlight:
+          startLine: 5
+          endLine: 7
+
+      - id: step-4
+        description: "Slide window: remove 5, add 2. New window [1, 3, 2]. Sum = 6"
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 2, state: visited }
+              - { value: 1, state: visited }
+              - { value: 5, state: visited }
+              - { value: 1, state: active, annotations: ["window"] }
+              - { value: 3, state: active, annotations: ["window"] }
+              - { value: 2, state: active, annotations: ["window"] }
+            pointers:
+              left: 3
+              right: 5
+        variables:
+          window_sum: 6
+          max_sum: 9
+          i: 5
+        codeHighlight:
+          startLine: 5
+          endLine: 7
+
+      - id: step-5
+        description: "Loop complete. Maximum sum found is 9 (window [5, 1, 3])"
+        structures:
+          nums:
+            type: array
+            values:
+              - { value: 2, state: visited }
+              - { value: 1, state: visited }
+              - { value: 5, state: found }
+              - { value: 1, state: found }
+              - { value: 3, state: found }
+              - { value: 2, state: visited }
+        variables:
+          max_sum: 9
+        codeHighlight:
+          startLine: 9
+          endLine: 9

backend/data/patterns/topological-sort.yaml

@@ -0,0 +1,104 @@
+name: Topological Sort
+slug: topological-sort
+difficulty_level: 3
+pattern_type: algorithm
+display_order: 22
+
+description: >
+  Order vertices in a directed acyclic graph (DAG) such that for every
+  edge u -> v, vertex u comes before v in the ordering.
+
+when_to_use: |
+  - Course prerequisites / task dependencies
+  - Build order / compilation order
+  - Detecting cycles in directed graphs
+  - Any problem with "do A before B" constraints
+
+metaphor: |
+  Imagine getting dressed: you must put on underwear before pants, socks
+  before shoes. Topological sort finds a valid order that respects all
+  "must come before" rules. If there's a cycle (shirt requires jacket,
+  jacket requires shirt), no valid order exists.
+
+core_concept: |
+  Two main approaches:
+
+  **Kahn's Algorithm (BFS):** Start with nodes having no incoming edges
+  (in-degree 0). Process them, remove their edges, repeat. If all nodes
+  processed, valid order exists.
+
+  **DFS-based:** Do DFS, add nodes to result when backtracking (post-order).
+  Reverse at end. Cycle exists if we revisit a node in current path.
+
+code_template: |
+  from collections import deque
+
+  def topological_sort_bfs(n: int, edges: list[tuple[int, int]]) -> list[int]:
+      """Kahn's algorithm - returns empty list if cycle exists."""
+      # Build adjacency list and in-degree count
+      graph = [[] for _ in range(n)]
+      in_degree = [0] * n
+
+      for u, v in edges:  # u -> v (u must come before v)
+          graph[u].append(v)
+          in_degree[v] += 1
+
+      # Start with nodes having no prerequisites
+      queue = deque([i for i in range(n) if in_degree[i] == 0])
+      result = []
+
+      while queue:
+          node = queue.popleft()
+          result.append(node)
+
+          for neighbor in graph[node]:
+              in_degree[neighbor] -= 1
+              if in_degree[neighbor] == 0:
+                  queue.append(neighbor)
+
+      # If we processed all nodes, valid topological order exists
+      return result if len(result) == n else []
+
+recognition_signals:
+  - "prerequisites"
+  - "dependencies"
+  - "ordering tasks"
+  - "course schedule"
+  - "build order"
+  - "detect cycle in directed graph"
+  - "do X before Y"
+
+common_mistakes:
+  - title: Forgetting Cycle Detection
+    description: |
+      Assuming input is always a valid DAG. Must check if all nodes were
+      processed (BFS) or if back-edge exists (DFS).
+    fix: |
+      BFS: Check `len(result) == n`. DFS: Track "visiting" state separately
+      from "visited."
+
+  - title: Wrong Edge Direction
+    description: |
+      Confusing "A depends on B" vs "A must come before B." These are
+      opposite edge directions.
+    fix: |
+      Clarify: If A depends on B, edge is B -> A (B comes before A).
+
+variations:
+  - name: Course Schedule (cycle detection)
+    description: Return true/false if valid ordering exists
+    example: "course-schedule"
+  - name: Course Schedule II (find order)
+    description: Return the actual ordering
+    example: "course-schedule-ii"
+  - name: Alien Dictionary
+    description: Infer ordering from sorted alien words
+    example: "alien-dictionary"
+
+related_patterns:
+  - bfs
+  - dfs
+
+prerequisite_patterns:
+  - bfs
+  - dfs

backend/data/patterns/tree-traversal.yaml

@@ -1,6 +1,8 @@
 name: Binary Tree Traversal
 slug: tree-traversal
 difficulty_level: 2
+pattern_type: algorithm
+display_order: 8
 
 description: >
   Visit all nodes in a binary tree in specific orders: preorder (root-left-right),

backend/data/patterns/trie.yaml

@@ -1,6 +1,8 @@
 name: Trie
 slug: trie
 difficulty_level: 3
+pattern_type: data_structure
+display_order: 17
 
 description: >
   A tree-like data structure for efficient string prefix operations. Each node

backend/data/patterns/two-pointers.yaml

@@ -1,6 +1,8 @@
 name: Two Pointers
 slug: two-pointers
 difficulty_level: 2
+pattern_type: algorithm
+display_order: 1
 
 description: >
   Use two pointers to traverse data from different positions, often moving

backend/data/patterns/union-find.yaml

@@ -1,6 +1,8 @@
 name: Union Find
 slug: union-find
 difficulty_level: 3
+pattern_type: data_structure
+display_order: 18
 
 description: >
   Track disjoint sets with efficient union and find operations. Union-Find