1. Review of Chapter 9 張啟中

2. Inheritance Hierarchy Min PQ Max PQ Min Heap Mergeable Min PQ DeapMin-Max Min-LeftistMin-SkewMinFHeap MinBHeap DEPQMax Heap Symmetric Max Data Structures

3. Double-Ended Priority Queue A double-ended priority queue is a data structure that supports the following operations:  inserting an element with an arbitrary key  deleting an element with the largest key  deleting an element with the smallest key There are two kinds of DEPQ  Min-Max Heap  Deap

4. Min-Max Heaps Definition A min-max heap is a complete binary tree such that if it is not empty,  Each element has a data member called key.  Alternating levels of this tree are min levels and max levels, respectively.  The root is on a min level.  Let x be any node in a min-max heap. If x is on a min (max) level then the element in x has the minimum (maximum) key from among all elements in the subtree with root x. A node on a min (max) level is called a min (max) node.

5. Example 7 30 4550 9 3020 70 10 12 15 40 min max

6. Insertion of a Min-Max Heap Step 1  將欲新增的元素插入 Min-Max Heap 最後一個節點 Step 2  將新增節點與其父節點做比較，若父節點位於 Min (Max) Level ，且新增節點小於 ( 大於 ) 父節點，則交換二者的位置。 Step 3  若交換後新增節點的位於 Min (Max) Level ，則依序往上與 各 Min (Max) Level 的節點比較，若新增節點較小 ( 大 ) ，則二 者交換。 Step 4  重複 Step 3 ，直至不能再交換或到 root 為止。

7. Insertion of a Min-Max Heap 7 30 4550 9 3020 70 10 12 15 40 min max 5 O(logn)

8. Insertion of a Min-Max Heap 7 30 4550 9 3020 70 10 12 15 40 min max 80

9. Deletion of The Min Element Step 1  刪除 root Step 2  將最後一個節點刪除，並重新插入 Min-Max Heap 的 root

10. Deletion of The Min Element Step 3  Case 1 The root has no children. In this case x is to be inserted into the root.  Case 2 The root has at least one child. Find the smallest key from the children or grandchildren of root. Assume node k has the smallest key. x is inserted node.  x.key <= h[k].key  Insert x to the root  x.key > h[k].key and k is child of the root  Interchange x and k  x.key > h[k].key and k is grandchild of the root  (1) h[k] is moved to the root. (2) Let p is parent of node k. If x.key > h[p].key then h[p] and x are to interchanged. Repeat Step 3 with root k.

11. Deletion of The Min Element 7 30 4550 9 3020 70 10 12 15 40 min max O(logn)

12. Deaps Definition A deap is a complete binary tree that is either empty or satisfies the following properties  The root contains no element.  The left subtree is a min heap.  The right subtree is a max heap.  If the right subtree is not empty, then let i be any node in the left subtree. Let j be the corresponding node in the right subtree. If such a j does not exist, then let j be the node in the right subtree that corresponds to the parent of i. The key in node i is less than or equal to that of j.

13. Deap 10 1519 8 930 5 2540 45 20 Min Heap Max Heap

14. Insertion Into a Deap Step 1  將新增的元素插入 Deap 的最後面一個節點 Step 2  與新增節點相對位置的節點比較大小，若該新增節 點的位於 Max (Min) Heap ，則新增節點必須大於 相對節點，否則必須交換。 Step 3  若新增節點位於 Min (Max) Heap ，則依 Min (Max) Heap 的新增方式進行。

15. Insertion Into a Deap 10 1519 8 930 5 2540 45 20 4 j i

16. Insertion Into a Deap 10 1519 8 930 5 2540 45 20 30 j i

17. Deletion of Min Element Step 1  刪除 Min Heap 的 root Step 2  比較 Min Heap 中 root 的兩個 child ，將較小的值 移到 root ，直到 leaf 為止。 Step 3  將最後一個節點刪除，插入空出的 Leaf ，並依照 Deap 新增的方式操作

18. Deletion of Min Element 10 1519 8 930 5 2540 45 20 j i