COMP 3804:  Design and Analysis of Algorithms (Fall 2021)

Weekly-outline     Assignments


Instructor: Anil Maheshwari

Course Delivery:
The video lectures and notes are posted from the Winter 2021 offering of this course from Professor Michiel Smid (see https://cglab.ca/~michiel/3804.html). He has graciously agreed to share the contents with us. (Note: Except for a couple of initial videos, the rest are professionally recorded in a classroom setting at Carleton.) In addition to these, my office hours will be during the class time slot on Tuesday and Thursdays from 14:35 to 15:55. You are welcome to join live during the office hour. During the office hour(s), in addition to answering your questions, we may solve a few problems related to that weeks topic from the weekly practice problem set and/or assignments. Some relevant parts of the office hours will be recorded and posted in the Brightspace system.  Ample resources and relevant pointers will be provided during the course.

OFFICE HOURS: Tuesday and Thursday 14:35-15:55 Online over zoom (link provided in Brightspace). 

E-mail: anil@scs.carleton.ca, WWW:http://www.scs.carleton.ca/~anil


Course-TA: 

 
TAs will be mainly marking assignments, tests and exam.

Course Objectives:
An introduction to the design and analysis of algorithms. Topics include: divide-and-conquer, dynamic programming, linear programming, greedy algorithms, graph algorithms, NP-completeness.
(Also listed as MATH 3804.) Prerequisites: COMP 2002 or COMP 2402, COMP 2804, MATH 2007 and MATH 2108, or equivalents. Lectures three hours a week.

A bit of detailed Objectives and Contents are as follows:
Objectives:
Designing algorithms, their correctness and efficiency.
How to think about abstract computation - develop algorithmic intuition - how to attack a problem algorithmically.
Algorithmic techniques - learn key techniques (greedy, divide-and-conquer, dynamic programming, LP, ..) - which technique to use for a given problem. How to compare one algorithmic solution versus another for a problem
How to express the algorithmic solution - pseodo-code - steps - analyze - argue that it is correct - essentially how will you explain your algorithmic solutions to others?

Contents (tentatively):
Introduction: Whats an Algorithm, big-O notation, recurrence relation.
Divide and Conquer: Multiplying two n-bit integers, merge-sort, Matrix Multiplication, Median, ..
Graphs: Representation, Traversals - DFS & BFS, Shortest Paths (including A* heuristic).
Greedy Algorithms: Minimum Spanning Trees, Huffman Encodings, ...
Dynamic Programming: Edit Distance, Longest Common Sequence, ...
NP-Completeness: search problems, polynomial time reductions, ideas of Cooks Theorem.
Whats Next: Brief idea on how to cope with NP-Complete problems - introduction to approximation algorithms, exhaustive search and search heuristics.


Text-Book
DPV: Das, Papadimitriou and Vazirani, "Introduction to Algorithms", Mc Graw Hill 2008.
(See Papadimitriou's Websitepublisher's web-site)

There are numerous other textbooks on Algorithms and  Reference Materials

The following is the schedule for 4 Assignments, 2 Tests,  and the Final Exam:


Posted On
Due Date

% of Total Marks

BACKGROUND

Quiz 1

Quiz 2

Sept 9
Never

Make sure you know this material - this is minimum background required to understand the material in this course.


A1
PDF-File
LaTeX-File
Sep 9

Sep 24


10%

A2
PDF-File
LaTeX File
(Figures: dfs-undirected, grid-graph and tree)
Sep 23


Oct 8

10%
A3
PDF-File
LaTeX-File
Oct 22

Nov 12

10%
A4
PDF-File
LaTeX-File
Nov 18
Dec 2

10%
Test - 1
Oct 14
Oct 14
15% (14:35-15:55 PM on Thursday October 14th)

Test -2
Nov 25
Nov 25
15% (14:35-15:55 PM on Thursday November 25th)
Final Exam
Please check exam schedule


30%


To pass this course:
You must secure at least a total of 50% marks in the whole course.

Weekly Outline:
(Video Recordings for the relevant part of the office hours are posted in Brightspace)
Week #
Michiel's Video
Michiel's Notes
Practice Problem Set/ Assignment Solutions
during office hour.
Topics +
Additional pointers
0 (Sep 9)
 Introductory Remarks
Introduction


Naive and faster methods for
computing Fibonacci numbers,
O, Omega, Theta Notation

See Chapter 1 (Section 1.3) for more details
1 (Sep 14, 16)
Part 1(Merge-Sort Analysis)

Part 2(Multiplying Integers)

Part 3(Faster Multiplication)

Part 4 (Recursion Tree)
Divide-and-Conquer

Recursion Tree
Set - I



Set-II
Divide-and-Conquer Technique
Analyzing merge-sort, multiplication of
two n-bit numbers, matrix multiplication.

Additional notes on analyzing recurrences including
substitution method.


See Chapter 1 (Section 1.4 and 1.5) for more details

Also, see the following link for Master Method simulator:
https://www.nayuki.io/page/master-theorem-solver-javascript
2 (Sept 21, 23)
Deterministic Selection

Randomized Selection+ Heaps
Selection (deterministic)

Selection (randomized)

Heaps

Set-III
Group of 5 deterministic selection algorithm.
Randomized selection algorithm.
What are Heaps?
3 (Sept 28, 30)
Operations on Heaps

More on Heaps + Graphs
Introduction to Graphs and
Graph Exploration

Set-III continued +
Assignment 1 Solutions
Heap operations, heap sort.
Introduction to Graphs.
4 (Oct 5, 7)
Graph Representation- Explore

Depth-first-search

DAGs, Topological Sort, and DFS

Depth-first-search



Directed-graphs
Set-IV
Graph representation, Graph exploration,
tree and back edges,
depth-first search traversal,
finding connected components using explore and
cc-number

Is G bipartite?
Directed graphs.
Acyclic directed graphs (DAGs)
Topological Sorting
5 (Oct 12, 14)
DFS in directed graphs


TEST-1 (October 14th)

DFS in directed graphs
Assignment- 2 Solutions
DFS in directed graphs - forward, backward,
cross and back edges in the traversal.

Test takes place during the class time slot of 14:35-15:55 via Brightspace system on October 14th. The syllabus consists of all the material covered in Weeks 1-5 (inclusive), Assignments and the Problem Sets.  This includes all the lectures from Sept 9 - Oct 12th (inclusive).
6 (Oct 19, 22)
Shortest-paths in directed graphs

Correctness of Dijkstra's algorithm
shortest-paths


correctness
Test-1 Solutions

SSSP Examples

BFS+ Set-V
Shortest paths in DAGs

Dijsktra's single-source shortest path
algorithm for general directed graphs
Oct 25-29
FALL BREAK


NO OFFICE HOURS
7 (Nov 2, 4)
Union-Find DS + Kruskal's MST algorithm



Kruskal and Prim's MST algorithms
union-find data structure + MST

Prim's MST algorithms

Additional Notes on MSTs





NO OFFICE HOURS on November 4th
Union-find data structure
Minimum spanning tree algorithms - Kruskal and Prim's Algorithm.

Analysis of Kruskal's algorithm using union-find DS
8 (Nov 9, 11)
Prim's MST Analysis & 
Introduction to Dynamic Programming I

Dynamic Programming

Analysis of Prim's using priority queues
Revisiting shortest paths in DAGs, dynamic programming framework - structure of optimal solution, set-up recurrence relation, solve recurrence bottom-up.
Longest paths in DAGs
9 (Nov 16, 18)
Dynamic Programming II

Dynamic Programming III


Longest increasing sequence,  matrix-chain multiplication, longest-common sequence, all-pairs shortest paths - Floyd-Warshall algorithm. Longest paths in graphs - optimal substructure property?
10 (Nov 23, 25)
Decision Problems


Decision Problems using languages
Introduction to P and NP Complexity Classes

Formal definitions of decision problems, P and NP.

Complexity classes P and NP
Decision problems - SAT, Clique, Hamiltonian Cycle, TSP, Subset Sum


P subset NP,

Polynomial-time reductions.
11 (Nov 30, Dec 2)
Polynomial-time reductions

NP-Completeness
Polynomial-time reductions

NP-Complete Problems

Definition, Clique to Independent set, Clique to Vertex cover,  3SAT to Independent set.

NP-Hard and NP-Complete Problems. How to show a problem is NP-Complete?
12 (Dec 7, 9)
Hardness Reductions

Circuit-SAT is NP-Complete


More Hardness Reductions

Circuit-SAT problem

3SAT, Clique, 0-1 Integer Linear Programs are NP-Complete.

Sketch of the proof of Circuit-SAT is NP-Complete.


Undergraduate Academic Advisor:

The undergraduate advisor for the School of Computer Science is available in Room 5302C HP, by telephone at 520-2600, ext. 4364 or by email at undergraduate_advisor@scs.carleton.ca. The advisor can assist with information about prerequisites and preclusions, course substitutions/equivalencies, understanding your academic audit and the remaining requirements for graduation. The undergraduate advisor will also refer students to appropriate resources such as the Science Student Success Centre, Learning Support Services and the Writing Tutorial Services.
University Policies:

Student Academic Integrity Policy: Every student should be familiar with the Carleton University student academic integrity policy. A student found in violation of academic integrity standards may be awarded penalties which range from a reprimand to receiving a grade of F in the course or even being expelled from the program or University. Some examples of offences are: plagiarism and unauthorized co-operation or collaboration. Information on this policy may be found in the Undergraduate Calendar.

Plagiarism: As defined by Senate, "plagiarism is presenting, whether intentional or not, the ideas, expression of ideas or work of others as one's own". Such reported offences will be reviewed by the office of the Dean.

Unauthorized Co-operation or Collaboration:
We will follow the normal policies of Carleton University for providing academic accommodations as deemed necessary. Please contact the appropriate authorities well in advance so that you can be accommodated.

Equity Statements:
You may need special arrangements to meet your academic obligations during the term. For an accommodation request the processes are as follows:

Pregnancy obligation: write to me with any requests for academic accommodation during the first two weeks of class, or as soon as possible after the need for accommodation is known to exist. For more details visit the Equity Services website: http://www2.carleton.ca/equity/

Religious obligation: write to me with any requests for academic accommodation during the first two weeks of class, or as soon as possible after the need for accommodation is known to exist. For more details visit the Equity Services website: http://www2.carleton.ca/equity/

Academic Accommodations: The Paul Menton Centre for Students with Disabilities (PMC) provides services to students with Learning Disabilities (LD), psychiatric/mental health disabilities, Attention Deficit Hyperactivity Disorder (ADHD), Autism Spectrum Disorders (ASD), chronic medical conditions, and impairments in mobility, hearing, and vision. If you have a disability requiring academic accommodations in this course, please contact PMC at 613-520-6608 or pmc@carleton.ca for a formal evaluation. If you are already registered with the PMC, contact your PMC coordinator to send me your Letter of Accommodation at the beginning of the term, and no later than two weeks before the first in-class scheduled test or exam requiring accommodation (if applicable). Requests made within two weeks will be reviewed on a case-by-case basis. After requesting accommodation from PMC, meet with me to ensure accommodation arrangements are made. Please consult the PMC website (www.carleton.ca/pmc)  for the deadline to request accommodations for the formally-scheduled exam (if applicable).

Medical Certificate: The following is a link to the official medical certificate accepted by Carleton University for the deferral of final examinations or assignments in undergraduate courses. To access the form, please go to http://www1.carleton.ca/registrar/forms/


Announcements 
  1. No Office Hours on Thursday November 4th.
  2. We won't have any office hours during the Fall Break. But if you have questions etc., please send me an e-mail.
  3. Test 1 takes place on Thursday October 14th during the class time slot on Brightspace system.
  4. Assignment 1 has been marked. If you want any of the questions to be remarked, please send me an e-mail with some reasoning.
  5. Assignment 2 is posted. My recommendation will be to start early.
  6. Video Recordings for the relevant part of the office hours are posted in Brightspace
  7. Assignment 1 is posted
  8. We will likely send e-mail via Brightspace for important information. You may also check here.
  9. For NP-Completeness you may like to visit some of the following references:
    1. A List of NP-Complete Problems
    2. Some reductions illustrated nicely 1 2 3
    3. NP-Completeness Column
    4. Graph of NP-Complete Problems
    5. Puzzles are NP-Complete
    6. The Girl who proved P=NP
    7. Here are the lists of proofs that P=NP as well as proofs that P<>NP
    8. Here is the description from Clay Mathematical Institute
    9. Richard Karp Video

http://www.scs.carleton.ca/~maheshwa