feat(viz): heap pattern with kth largest

frontend/src/lib/visualizations/types.ts

@@ -89,6 +89,10 @@ export interface DataState {
   // Grid support (for DP visualizations)
   grids?: GridState[];
   gridPointers?: GridPointerState[];
+  // Decision tree support (for Backtracking)
+  decisionTrees?: DecisionTreeState[];
+  // Heap support
+  heaps?: HeapState[];
 }
 
 /** Single step in the visualization */
@@ -250,6 +254,30 @@ export interface QueueState {
   label?: string;
 }
 
+// ============================================
+// Decision Tree Types (for Backtracking)
+// ============================================
+
+/** State of a decision tree node for backtracking */
+export interface DecisionNodeState {
+  id: string;
+  value: string; // Current subset as string: "[]", "[1]", "[1,2]"
+  state: 'normal' | 'exploring' | 'complete' | 'backtracking' | 'current';
+  left: string | null; // Include branch
+  right: string | null; // Exclude branch
+  decision?: string; // "Include 2?" label
+  depth: number; // Recursion depth (0-indexed)
+}
+
+/** Complete decision tree state */
+export interface DecisionTreeState {
+  id: string;
+  nodes: DecisionNodeState[];
+  rootId: string;
+  label?: string;
+  currentPath?: string[]; // IDs of nodes in current exploration path
+}
+
 // ============================================
 // Grid Types (for DP visualizations)
 // ============================================
@@ -280,3 +308,24 @@ export interface GridPointerState {
   col: number;
   color: 'current' | 'dependency' | 'result' | 'default';
 }
+
+// ============================================
+// Heap Types
+// ============================================
+
+/** State of a heap node */
+export interface HeapNodeState {
+  id: string;
+  value: number;
+  index: number;  // Position in heap array (0-indexed)
+  state: 'normal' | 'comparing' | 'swapping' | 'settled' | 'highlighted' | 'removing';
+}
+
+/** Complete heap state */
+export interface HeapState {
+  id: string;
+  elements: HeapNodeState[];
+  label?: string;
+  heapType: 'min' | 'max';
+  maxSize?: number;  // For bounded heaps (like top-k)
+}