The document provides guidelines for a presentation by two professors. It includes their names, the presenter's name, date and location of the presentation. It then discusses topics related to algorithms and computation such as deterministic and nondeterministic computation modes, complexity classes like TIME and SPACE, and the time complexity hierarchy theorem. It also mentions random number generators, pseudo-random number generators, and Monte Carlo methods for testing random numbers.

1. ‫ارائه‬ ‫عنوان‬
‫راهنما‬ ‫اساتید‬
:
‫اول‬ ‫استاد‬ ‫نام‬
‫دوم‬ ‫استاد‬ ‫نام‬
‫دهنده‬ ‫ارائه‬
:
‫باقی‬ ‫وحید‬
‫بهمن‬
۱۳۹۹
‫تهران‬ ‫دانشگاه‬
‫فنی‬ ‫های‬ ‫دانشکده‬ ‫پردیس‬
‫مهندسی‬ ‫علوم‬ ‫دانشکده‬
‫کامپیوتر‬ ‫مهندسی‬
–
‫محاسبات‬ ‫و‬ ‫ها‬ ‫الگوریتم‬

2. Department of Algorithms and Computation
School of Engineering Science
College of Engineering
University of Tehran
Winter 2019
Test Subject
Presentation by : Vahid Baghi

3. mode of computation
deterministic mode
Parameters for a Complexity Class
nondeterministic mode

4. Time is Tape Dependent
Theorem : Let 𝑡 𝑛 be a function , where 𝑡 𝑛 > 𝑛 .Then every 𝑡 𝑛 time multitape Turing
machine has an equivalent 𝑂(𝑡2
(𝑛)) time single-tape Turing machine.
Transition
Logic
⊔
1
0 ∞
⊔
#
0
0
1
#
0
1
1
#
1
0
∞
⊔
0
1
1 ∞
⊔
0
0
1
Transition
Logic
110

5. Proper Complexity Functions
Definition
There exists a TM M that outputs
exactly 𝑓 𝑛 symbols on input 1𝑛
and runs in time O(𝑓 𝑛 + 𝑛) and
space O(𝑓 𝑛 )
f is a proper complexity function if :
∀𝑛 ∶ 𝑓 𝑛 ≥ 𝑓(𝑛 − 1)
For Example : log 𝑛
, 𝑛 , 𝑛2 , …

6. A complexity class is a set of classes of decision problems (or languages) with the same
worst-case complexity
Complexity Classes
DTIME or TIME is the computational resource of
computation time for a deterministic Turing
machine
DTIME
DSPACE
DSPACE or SPACE is the computational
resource describing the resource of memory
space for a deterministic Turing machine

7. Time Complexity Hierarchy
Theorem : for any 𝑡 𝑛 > 0 , there exists a decidable language
L ∉ 𝐷𝑇𝐼𝑀𝐸(𝑡 𝑛 )
Define 𝐿 = 𝑖 𝑀𝑖 does not accept i within 𝑡 𝑖 time }
Question : is L ∈ 𝐷𝑇𝐼𝑀𝐸 𝑡 𝑛 ?
Proof :
Assume (towards contradiction) L ∈ 𝐷𝑇𝐼𝑀𝐸 𝑡 𝑛
∃ a fixed K ∈ 𝑁 such that Turing machine 𝑀𝐾 decides L within time bound 𝑡 𝑛

8. ‫تصادفی‬ ‫شبه‬ ‫و‬ ‫تصادفی‬ ‫اعداد‬ ‫مولد‬
‫تصادفی‬ ‫اعداد‬ ‫مولد‬
(
Random Number Generator
)
‫ای‬‫وسیله‬
‫تولید‬ ‫برای‬ ‫که‬ ‫است‬ ‫فیزیکی‬
‫ای‬‫دنباله‬
‫کار‬ ‫به‬ ‫ندارند‬ ‫خاصی‬ ‫الگوی‬ ‫که‬ ‫اعداد‬ ‫از‬
‫رو‬‫می‬
‫د‬
.
‫مثال‬ ‫برای‬
•
‫پرتاب‬
‫تاس‬
•
‫سکه‬ ‫پرتاب‬
•
‫شانس‬ ‫گردونه‬
‫تصادفی‬ ‫شبه‬ ‫اعداد‬ ‫مولد‬
(
Pseudo Random Number Generator
: )
‫که‬ ‫است‬ ‫الگوریتمی‬
‫ای‬‫دنباله‬
‫تول‬ ، ‫هستند‬ ‫تصادفی‬ ‫مناسبی‬ ‫تقریب‬ ‫با‬ ‫که‬ ‫را‬ ‫اعداد‬ ‫از‬
‫ید‬
‫کند‬‫می‬
.

9. ‫تصادفی‬ ‫اعداد‬ ‫های‬ ‫مولد‬ ‫تست‬
‫روش‬
‫مونت‬
‫کارلو‬
‫عدد‬ ‫تخمین‬ ‫برای‬
𝜋
𝐴𝑐𝑖𝑟𝑐𝑙𝑒
𝐴𝑠𝑞𝑢𝑎𝑟𝑒
=
𝜋𝑟2
(2𝑟)2 =
𝜋
4
𝜋 = 4 ∗
𝐴𝑐𝑖𝑟𝑐𝑙𝑒
𝐴𝑠𝑞𝑢𝑎𝑟𝑒
X
Y

10. ‫مقدمه‬
‫تحق‬ ‫پیشینه‬
‫یق‬
‫پیشنهادی‬ ‫روش‬
‫نتایج‬ ‫ارزیابی‬
‫گیری‬ ‫نتیجه‬
‫پیشنهادات‬
‫نمونه‬ ‫عنوان‬
1/n

11. ‫مقدمه‬
‫تحق‬ ‫پیشینه‬
‫یق‬
‫پیشنهادی‬ ‫روش‬
‫نتایج‬ ‫ارزیابی‬
‫گیری‬ ‫نتیجه‬
‫پیشنهادات‬
‫نمونه‬ ‫عنوان‬
‫عنوان‬ ‫زیر‬
۱
‫عنوان‬ ‫زیر‬
۲
‫نمونه‬ ‫فرمول‬
1/n
𝑠𝑡+1 = {𝑒𝑖, 𝑒𝑖 ∈ 𝑠𝑡, 𝑟 + 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑝i|𝑝i ∈ 𝑝, 1 ≤ 𝑖 ≤ 𝑟}

13. Big Picture

14. User ID Movie ID Rating Timestamp
1 40 3 881250929
2 32 4 886176814
1 10 3.5 881250939
3 20 5 874833878
1 20 2 881250949
2 50 3 886176824
4 11 4.5 879196566
User ID Movie ID Rating Timestamp
1 40 3 881250929
1 10 3.5 881250939
1 20 2 881250949
2 32 4 886176814
2 50 3 886176824
3 20 5 874833878
4 11 4.5 879196566
MovieLens Dataset