feat(patterns): pattern taxonomy + is_optimal

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