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Optimal Binary Search
Tree
Le Phong Vu
About me
▶ Le Phong Vu
▶ Member of Grokking
▶ Data structure and caching system
▶ Email: lephongvu90@gmail.com
2
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
Problem: Build a dictionary
▶ Given an English world, you want to search its meaning quickly
4
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
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
Approach 7
Build a
balanced
BST
Collect
search
information
Calculate
probability
search
Calculate a
Optimal BST
Change
current BST
by new
Optimal BST
Key and probability
▶  
word key probability
apple
hatch
jump
mango
8
Dummy Key
Meaning Symbol
d0
d1
d2
di
d0 d1
d2 d3 d5d4
apple
hatch
jump
mango
pizza< apple (apple, hatch)
9
Example 10
word key total search probability
< apple 5
apple 10
(apple, hatch) … …
hatch … …
jump … …
mango … …
Total search = 100
Search time
▶  
11
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
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
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
Step 2: Deciding root of OBST
▶  
15
Step 2: Deciding Root of OBST
▶  
16
kr
Contribute
e [i, r-1]
Contribute
e [i, r-1]
Contribute
W[i,j]
Step 3: Bottom-up approach
▶  
17
 
18
Optimize solution
▶  
19
Application
▶ Huffman Tree
▶ Optimal routing lookup
20
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
Question
22
Example 23
An example from book:
Introduction to Algorithm, 3rd
edition
Example 24
Pseudocode 25

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