Implement Data Structures with Python Fast using Sparse Matrix. kung fu computer science, geometric complexity theory

1. I want to become a
programming God
How to implement any data structures fast in Python?
Java
C++
Python
Matlab
Programming Languages… Convex Optimization
Competitive Programming
Data Structures
1
2 3
4 5
𝑀 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
M(final)=𝛿>0( 𝑖=1
∞
𝑀
𝑖
)
Transitive Closure
Algorithm

2. Sparse Matrix
• Any data structures can be represented in dense or sparse matrix
• Data structures are
• List
• Queue
• Tree
• Graph
• Hash Table
• …
• Sparse Matrix can be easily created in Python

3. Sparse Matrix in Python
# Python program to multpliply two
# csc matrices using multiply()
# Import required libraries
import numpy as np
from scipy.sparse import csc_matrix
# Create first csc matrix A
row_A = np.array([0, 0, 1, 2 ])
col_A = np.array([0, 1, 0, 1])
data_A = np.array([4, 3, 8, 9])
cscMatrix_A = csc_matrix((data_A,
(row_A, col_A)),
shape = (3, 3))
# print first csc matrix
print("first csc matrix: n",
cscMatrix_A.toarray())
# Create second csc matrix B
row_B = np.array([0, 1, 1, 2 ])
col_B = np.array([0, 0, 1, 0])
data_B = np.array([7, 2, 5, 1])
cscMatrix_B = csc_matrix((data_B, (row_B,
col_B)),
shape = (3, 3))
# print second csc matrix
print("second csc matrix:n",
cscMatrix_B.toarray())
# Multiply these matrices
sparseMatrix_AB =
cscMatrix_A.multiply(cscMatrix_B)
# print resultant matrix
print("Product Sparse Matrix:n",
sparseMatrix_AB.toarray())

4. Matlab or Python?
• When I started writing codes in Matlab, I felt in love with it
• It can do any array operations using indexing syntax, matrix and vector
• It eliminates the for loop in array operations
• Later I found out that Numpy library in Python can do the same
• And even more with more functions in Numpy
• Forget about old school Java programming language where data
structures are implemented in Objects with methods to operate on
the data structures

5. List and Queue
• List can be represented in a vectors
• Or it can be represented in the sparse matrix M
• mij is an element of matrix M
• mij is 1 if item j of list is connected to item i
• Item i is the next item of item j
• Queue is represented in the same approach

6. Tree and Graph
• Tree and Graph can be represented by a set of directional edges that
connects the items (or vertices) together
• The objects of the items are stored in a vector
• Example:
1
2 3
4 5
𝑀 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
M is a sparse Matrix
Item 1: John
Item 2: Peter
Item 3: Eric
Item 4: Ken
Item 5: Bob
𝑣 =
′𝐽𝑜ℎ𝑛′
′𝑃𝑒𝑡𝑒𝑟′
𝐸𝑟𝑖𝑐′
′𝐾𝑒𝑛′
′𝐵𝑜𝑏′
Vector containing object

7. • Navigating through a tree can be represented by matrix multiplication
• 𝑀𝑎 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
1
0
0
0
0
=
0
1
1
0
0
Immediate children of item 1
Level 1 items of tree
1
2 3
4 5
Level 1

8. • Navigating through a tree can be represented by matrix multiplication
• 𝑀𝑀𝑎 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
1
0
0
0
0
=
0
0
0
1
1
Immediate children of item 1
Level 2 items of tree
1
2 3
4 5
Level 2

9. • Navigating through a tree can be represented by matrix multiplication
• 𝑀𝑀𝑎 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
1
0
0
0
0
=
0
0
0
0
0
Immediate children of item 1
Level 3 items of tree
No items at level 3
1
2 3
4 5
Level 3

10. • For a graph, the transitive closure is simply the repeated application
of matrix multiplication
• 𝑀 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
1
2 3
4 5

11. • 1 step of transitive closure
• 𝑀𝑀 + 𝑀 =
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
+
0 0 0 0 0
1 0 0 0 0
1
0
0
0
0
0
0 0 0
1 0 0
1 0 0
• =
0 0 0 0 0
1 0 0 0 0
1
1
1
0
0
0
0 0 0
1 0 0
1 0 0
1
2 3
4 5

12. • 2 step of transitive closure
• 𝑀𝑀𝑀 + 𝑀𝑀 + 𝑀
• =
0 0 0 0 0
1 0 0 0 0
1
1
1
0
0
0
0 0 0
1 0 0
1 0 0
1
2 3
4 5

13. • Transitive closure algorithm in matrix representation is simply
• M(final)=𝛿>0( 𝑖=1
∞
𝑀𝑖)
• Can be written as M(final)=𝛿>0( 𝑀 + 𝐼 𝑖 − 𝐼), where I is an identity matrix
• No need to compute until infinity, only need to compute until the matrix converges
• It means matrix M raised to the power of infinity (M multiplied to itself infinite times)
• And the final matrix after multiplication is thresholded (𝛿>0()) element by element,
• Each element is threshold to 1 if it is greater than 0, else it is threshold to 0
• In the example, M(final) =
0 0 0 0 0
1 0 0 0 0
1
1
1
0
0
0
0 0 0
1 0 0
1 0 0

14. Algebraic Representation of Algorithm
• I believe a super programming
will represent an algorithm in
higher order abstract form
(algebraic form)
• It is much more cleaner and
easier to understand
• The algorithm can be analyzed
symbolically
• The properties of the algorithm
can be derived algebraically
Begin
copy the adjacency matrix into another matrix named
transMat
for any vertex k in the graph, do
for each vertex i in the graph, do
for each vertex j in the graph, do
transMat[i, j] := transMat[i, j] OR (transMat[i, k]) AND
transMat[k, j])
done
done
done
Display the transMat
End
If represented in pseudo codes, it will look very messy.
The algebraic matrix notation of the transitive closure algorithm
looks better
Transitive Closure Algorithm:

15. Convex Optimization representation of
Algorithms
• I like the convex optimization algorithm, because it is represented so
succinctly in algebraic form. And it is so beautiful
• Minimize f(x)
• Such that g(x)=0
• h(x)>=0
• If all algorithms can be represented in this format, it will be so beautiful
• I did some formulation to represent algorithms in convex optimization format
• https://www.slideshare.net/SingKuangTan/discrete-markov-random-field-relaxation
• https://vixra.org/abs/2112.0151

16. Competitive Programming
• I hope my sparse matrix approach to implement data structure will
help to speed up programming in competition
• Hope that my convex optimization representation of algorithm will also help
• I have not take part in programming competition before
• Would like to if given a chance
• I have did so much programming before and studied my different algorithms…

17. Links to my papers
● https://vixra.org/author/sing_kuang_tan
● Link to my NP vs P paper
● And Discrete Markov Random Field relaxation paper

18. About Me
● My job uses Machine Learning to solve problems
○ Like my posts or slides in LinkedIn, Twitter or Slideshare
○ Follow me on LinkedIn
■ https://www.linkedin.com/in/sing-kuang-tan-b189279/
○ Follow me on Twitter
■ https://twitter.com/Tan_Sing_Kuang
○ Send me comments through these links
● Look at my Slideshare slides
○ https://www.slideshare.net/SingKuangTan
■ NP vs P Proof using Discrete Finite Automata
■ Use Inductive or Deductive Logic to solve NP vs P?
■ Kung Fu Computer Science, Clique Problem: Step by Step
■ Beyond Shannon, Sipser and Razborov; Solve Clique Problem like an Electronic Engineer
■ A weird Soviet method to partially solve the Perebor Problems
■ 8 trends in Hang Seng Index
■ 4 types of Mathematical Proofs
■ How I prove NP vs P
○ Follow me on Slideshare

19. Share my links
● I am a Small Person with Big Dreams
○ Please help me to repost my links to other platforms so that I can spread my ideas to the rest of the world
● 我人小，但因梦想而伟大。
○ 请帮我的文件链接传发到其他平台，让我的思想能传遍天下。
● Comments? Send to singkuangtan@gmail.com
● Link to my paper NP vs P paper
○ https://www.slideshare.net/SingKuangTan/brief-np-vspexplain-249524831
○ Prove Np not equal P using Markov Random Field and Boolean Algebra Simplification
○ https://vixra.org/abs/2105.0181
○ Other link
■ https://www.slideshare.net/SingKuangTan