min heap, still need to reverse

lectures/24_priority_queues_II/README.md

@@ -116,6 +116,10 @@ organized heap data, but incur a O(n log n) cost. Why?
 - The following is the sort algorithm with a main function to test it; the code makes a min heap.
 
 ```cpp
+#include <iostream>
+#include <vector>
+#include <algorithm>
+
 /* The heapify function is designed to ensure that a subtree rooted at a given index i
  * in an array representation of a min heap maintains the heap property.
  * 
@@ -160,14 +164,18 @@ std::vector<int> sortArray(std::vector<int>& nums) {
         heapify(nums, n, i);
     }
 
-    // now the first one is the largest, swap it to the back
+    // now the first one is the smallest, swap it to the back
     // do this n-1 times.
-    for(int i=0; i<(n-1); i++){
+    for(int i=n-1; i>0; i--){
         // build the min heap again, with 0 being the root.
-        // but only consider n-1-i elements, as the others are already in the right place.
+        // but only consider i elements, as the others are already in the right place.
-        heapify(nums, n-1-i, 0);
+	std::swap(nums[0], nums[i]);     // move smallest to the end
+        heapify(nums, i, 0);
     }
 
+    // reverse to get ascending order
+    std::reverse(nums.begin(), nums.end());
+
     return nums;
 }
 
@@ -203,10 +211,10 @@ The above program prints the following:
 $ g++ heap_sort.cpp
 $ ./a.out
 Original array:
-42 12 13 65 98 45 97 85 76 90
+42 12 13 65 98 45 97 85 76 90 
 
 Sorted array:
-12 42 13 65 90 45 97 85 76 98
+12 13 42 45 65 76 85 90 97 98 
 ```
 
 ## 24.9 Summary Notes about Vector-Based Priority Queues

lectures/24_priority_queues_II/heap_sort.cpp

@@ -1,8 +1,9 @@
 #include <iostream>
 #include <vector>
+#include <algorithm>
 
 /* The heapify function is designed to ensure that a subtree rooted at a given index i 
- * in an array representation of a heap maintains the heap property. 
+ * in an array representation of a min heap maintains the heap property. 
  * While the function doesn't have an explicit base case like some recursive functions, 
  * it inherently terminates due to the following conditions:
  *  
@@ -11,19 +12,19 @@
  * In this scenario, the conditions left < n and right < n in the if statements evaluating the children will both be false, 
  * preventing further recursive calls.
  * 
- * Heap Property Satisfaction: If the node at index i is greater than or equal to its children (or if it has no children), 
+ * Heap Property Satisfaction: If the node at index i is less than or equal to its children (or if it has no children), 
- * the heap property is already satisfied. Consequently, the variable largest remains equal to i, 
+ * the heap property is already satisfied. Consequently, the variable smallest remains equal to i, 
- * and the condition largest != i evaluates to false. 
+ * and the condition smallest != i evaluates to false. 
  * This prevents the swap operation and the subsequent recursive call, leading to termination. 
  * In essence, the function will return when:
  * The node is a leaf node. 
- * The node's value is greater than or equal to its children's values, maintaining the heap property.
+ * The node's value is less than or equal to its children's values, maintaining the heap property.
  * These implicit conditions ensure that the recursion does not continue indefinitely.
  *
  * Why Not Just Heapify Once? A single call to heapify on the entire array 
  * wouldn't suffice because heapify is designed to correct violations of 
  * the heap property starting from a specific node, assuming its subtrees are already heaps. 
- * Initially, the array doesn't have this structure, so multiple calls are necessary to build the initial max-heap. 
+ * Initially, the array doesn't have this structure, so multiple calls are necessary to build the initial min-heap. 
  * Similarly, during the sorting phase, each extraction disrupts the heap structure, necessitating a call to heapify to restore order.
  * */
 
@@ -40,8 +41,8 @@ void heapify(std::vector<int>& nums, int n, int i){
         smallest = right;
     }
 
-    // after the above, smallest basically will either stay the same, or will be either left or right, depending on nums[left] is larger or nums[right] is larger. largest stays the same if it is already larger than its two children.
+    // after the above, smallest basically will either stay the same, or will be either left or right, depending on nums[left] is smaller or nums[right] is smaller. smallest stays the same if it is already smallest than its two children.
-    // if largest is changed, then we do need to swap.
+    // if smallest is changed, then we do need to swap.
     if(smallest != i){
         std::swap(nums[i], nums[smallest]);
         heapify(nums, n, smallest);
@@ -58,20 +59,22 @@ std::vector<int> sortArray(std::vector<int>& nums) {
     // This bottom-up approach guarantees that each subtree satisfies the heap property before moving to the next node.
     for(int i=n/2-1; i>=0; i--){
         // heapify the subtree whose root is at i
-        // i.e., build a max heap, with i being the root; and this heap contains nodes from i to n-1;
+        // i.e., build a min heap, with i being the root; and this heap contains nodes from i to n-1;
         heapify(nums, n, i);
     }
 
-    // now the first one is the largest, swap it to the back
+    // now the first one is the smallest, swap it to the back
     // do this n-1 times.
-    for(int i=0; i<(n-1); i++){
+    for(int i=n-1; i>0; i--){
-        // nums[0] is always the largest one
+        // build the min heap again, with 0 being the root.
-        std::swap(nums[0], nums[n-1-i]);
+        // but only consider i elements, as the others are already in the right place.
-        // build the max heap again, with 0 being the root.
+	std::swap(nums[0], nums[i]);     // move smallest to the end
-        // but only consider n-1-i elements, as the others are already in the right place.
+        heapify(nums, i, 0);
-        heapify(nums, n-1-i, 0);
     }
 
+    // reverse to get ascending order
+    std::reverse(nums.begin(), nums.end());
+
     return nums;
 }