Cây nhị phân tìm kiếm là 1 cấu trúc dữ liệu quen thuộc với chúng ta. Có rất nhiều nghiên cứu và các thuật toán xoay quanh cấu trúc dữ liệu này. Trong talk này, xin giới thiệu một kỹ thuật giúp tối ưu cây nhị phân tìm kiếm dựa trên tần suất tìm kiếm, qua đó giúp giảm chi phí tìm kiếm xuống mức thấp nhất. - Speaker: Phong Vu - Software Engineer

1. Optimal Binary Search
Tree
Le Phong Vu

2. About me
▶ Le Phong Vu
▶ Member of Grokking
▶ Data structure and caching system
▶ Email: lephongvu90@gmail.com
2

3. Agenda
▶ Problem: Build a dictionary online with fast lookup
▶ Approach to apply OBST in dictionary application
▶ Use Dynamic programming to find OBST
▶ Optimize solution to reduce time create OBST
▶ Application of OBST
3

4. Problem: Build a dictionary
▶ Given an English world, you want to search its meaning quickly
4

5. Build a dictionary with BST
apple
hatch
mango
pizzajump
Balanced binary search tree
Worst-case O(log n) time per query
Total cost: 2000100
Word Total
search
Depth of
Node
Cost
apple 1.000.000 2 1000000 * 2
hatch 100 1 100 * 1
jump,
mango,
pizza
0 * 0
5

6. Re-structure BST
apple
hatch
…
Re-structure BST, move “apple” node
to root
Total cost: 1000200
Word Total search Depth of
Node
Cost
apple 1.000.000 1 1.000.000 * 1
hatch 100 2 100 * 2
jump,
mango,
pizza
0 * 0
OBTIMAL BINARY
SEARCH TREE
6

7. Approach 7
Build a
balanced
BST
Collect
search
information
Calculate
probability
search
Calculate a
Optimal BST
Change
current BST
by new
Optimal BST

8. Key and probability
▶
word key probability
apple
hatch
jump
mango
8

9. Dummy Key
Meaning Symbol
d0
d1
d2
di
d0 d1
d2 d3 d5d4
apple
hatch
jump
mango
pizza< apple (apple, hatch)
9

10. Example 10
word key total search probability
< apple 5
apple 10
(apple, hatch) … …
hatch … …
jump … …
mango … …
Total search = 100

11. Search time
▶
11

12. Optimal Binary Search Tree
▶ Given the set probabilities p and q, can we construct a
binary search tree whose expected search time is minimized?
▶ This tree is called an Optimal Binary Search Tree
12

13. Dynamic Programming
▶ Solves problems by combining the solutions to sub-problems.
13
Figure 1: Dynamic Programming
https://itnext.io/dynamic-programming-vs-divid
e-and-conquer-2fea680becbe

14. Step 1: The structure of OBST
▶ If an OBST has a sub-tree T, then this sub-tree T must be optimal BST.
14
Optimal
BST
Optimal
BST
Optimal
BST

15. Step 2: Deciding root of OBST
▶
15

16. Step 2: Deciding Root of OBST
▶
16
kr
Contribute
e [i, r-1]
Contribute
e [i, r-1]
Contribute
W[i,j]

17. Step 3: Bottom-up approach
▶
17

18.
18

19. Optimize solution
▶
19

20. Application
▶ Huffman Tree
▶ Optimal routing lookup
20

21. Reference
▶ Le Phong Vu, “Dùng kỹ thuật quy hoạch động giải bài toán tối ưu
cây nhị phân tìm kiếm”
https://engineering.grokking.org/optimal-binary-search-tree/
▶ D. Knuth, “Optimum binary search trees”, Acta Informatica
www.inrg.csie.ntu.edu.tw/algorithm2014/presentation/Knuth71.pdf
▶ Thomas H. Cormen, “Introduction to algorithm”, MIT Press, 3rd
Edition
▶ D. Knuth, “The art of computer programming”, volume 3
▶ Kurt Mehlhorn, “Nearly Optimal Binary Search Tree”
https://people.mpi-inf.mpg.de/~mehlhorn/ftp/mehlhorn3.pdf
21

22. Question
22

23. Example 23
An example from book:
Introduction to Algorithm, 3rd
edition

24. Example 24

25. Pseudocode 25