Review of Chapter 9

1. Review of Chapter 9 張啟中

2. Inheritance Hierarchy Max PQ Min PQ Min Heap DEPQ Max Heap Mergeable Min PQ Deap Min-Max Symmetric Max Data Structures Min-Leftist Min-Skew MinFHeap MinBHeap

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 min 70 40 max 30 9 10 15 min 45 50 30 20 12 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 O(logn) 7 min 70 40 max 30 9 10 15 min 45 50 30 20 12 5 max

8. Insertion of a Min-Max Heap 7 min 70 40 max 30 9 10 15 min 45 50 30 20 12 80 max

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 O(logn) 7 min 70 40 max 30 9 10 15 min 45 50 30 20 12 max

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 5 45 10 8 25 40 20 15 19 9 30 Max Heap Min 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 5 45 10 8 25 40 20 15 19 9 30 4 j i

16. Insertion Into a Deap 5 45 10 8 25 40 20 15 19 9 30 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 5 45 10 8 25 40 j 20 15 19 9 30 i