Update 6.5.md

docs/Chap06/6.5.md

@@ -133,12 +133,43 @@ Both are not very efficient. Furthermore, if the priority can overflow or underf
 
 ```cpp
 HEAP-DELETE(A, i)
-    A[i] = A[A.heap-size]
+    if A[i] > A[h.heap-size]
+        A[i] = A[A.heap-size]
+        MAX-HEAPIFY(A, i)
+    else
+        HEAP-INCREASE-KEY(A, i, A[A.heap-size])
     A.heap-size = A.heap-size - 1
+```
+
+**Note:** The following algorithm is wrong. For example, given an array $A = [15, 7, 9, 1, 2, 3, 8]$ which is a max-heap, and if we delete $A[5] = 2$, then it will fail.
+
+```cpp
+HEAP-DELETE(A, i)
+    A[i] = A[A.heap-size]
     MAX-HEAPIFY(A, i)
+    A.heap-size = A.heap-size - 1
 ```
 
-We just move the last element of the heap to the deleated position and then call $\text{MAX-HEAPIFY}$ on it. This works, because the element is already smaller than its parent (because it was already under it on the heap), but might be larger than its children. $\text{MAX-HEAPIFY}$ restored the heap property.
+- before:
+
+    ```
+          15
+        /    \
+       7      9
+      / \    / \
+     1   2  3   8
+    ```
+
+- after (which is wrong since $8 > 7$ violates the max-heap property):
+
+    ```
+          15
+        /    \
+       7      9
+      / \    /
+     1   8  3
+    ```
+
 
 ## 6.5-9