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Group Phil♂sophy

Binary Search Tree (BST)

12 5 3 6 8 4 15 7 9 17 16

A binary search tree

Binary Search Tree

• Construct binary search tree
• Input[2, 4, 5, 1, 3]

2 4 1 5 3

• O(logN)

• What if the input is [1, 2, 3, 4, 5] ?

1 2 3 4 5

• A linked list.
• Only O(N)

Some Self-Balancing Binary Search Trees

• AVL(Adelson-Velskii & Landis Tree)
• Red Black Tree
• Size Balanced Tree
• Treap

Treap

a randomized

binary search tree

A tree or a heap?

• Key [4, 2, 1, 3, 5]
• Priority [15, 11, 13, 18, 19]

2 3 4 1 5

11 19 18 15 13

Rotate

4 2 5 1 3 2 1 4 3 5

right rotate left rotate

Rotate

4 2 5 1 3 2 1 4 3 5

right rotate 30 10 50 20 40 10 20 30 50 40

Insert

Lets build a treap together!

2 1 5 3 6

10 18 20 40 30

4

15

Insert

Key = 4 Priority = 15

Left rotate 3 Right rotate 5

Search

Just as you do in the BST

2 1 4 3 5

10 18 15 20 30

6

40

Delete

Just do it!

2 1 6 4 7

10 15 12 15 13

8

30

5

19

3

40

Rotate then do it

2 1 6 4 7

10 15 12 15 13

8

30

5

19

3

40

Find node 6 Right rotate 6 Left rotate 6 Delete 6

Deeper understanding of treap

• DEF. mean that i is a proper ancestor of k.
• If i is proper ancestor of k,
• Otherwise

k i  k i   1 k i  

2 3 4 1 5

5 3 4 1 1 3 4 1 3 2 1 5 2 1 4 2            

Input array is [1, 2, 3, 4, 5] Input priority is [0.3, 0.1, 0.8, 0.5, 0.9] 0.1 0.9 0.8 0.5 0.3

• The depth of node k:

 

 





 

A i

k i k depth

   

   

 

 

   

A i A i

k i Pr k i E E E

• DEF. X(i,k) denotes set of nodes which key value is between i and k.
• For input array [2, 3, 1, 5, 4]
• X(2,5)={2,3,4,5}
• X(1,3)={1,2,3}

• Lemma. For all , i has the smallest priority in X(i,k)

k i 

iff 1 k i  

 

 

                    k i if 1 k i 1 k i if k i if 1 i k 1 1 i k k i k i Pr

   

 

 

2 n ln 2 2 1 k n ln k ln 1 H 1 H j 1 j 1 1 k i 1 1 i k 1 1 k i 1 k i Pr k depth E

1 k n k 1 k n 2 i k 2 j n 1 k i 1 k 1 i k i A i A i

                       

            

     

• depth(k) = O(logN) in expectation.
• search(k) takes time O(depth(k)) = O(logN) in expectation.
• insert(k) takes time O(depth(k))=O(logN) in expectation
• …

• Lemma.

k is a node in treap, then

   

 

 

N log O k depth max E 

Why we are using a treap?

• Easy to understand
• Easy to use
• Well reliability
• Average depth O(logN)

Thank you

Q & A