# Spring 2021 — 30540 Computer Science 2

This course introduces the beautiful and powerful techniques of theoretical computer science for the design of algorithms, the analysis of algorithms, and the study of the computational complexity of problems.

### General information

Instructors: Luca Trevisan, Fabrizio Iozzi

Lectures:

• Tuesdays 14:00-16:30 Room N20
• Thursdays 8:30-10:00 Room N11
• Fridays 14:00-16:30 Room N06

Office hours:

• Luca Trevisan: Thursdays 15:00-16:00 (appointment required)

• S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani. Algorithms
It is easy to find online a pdf of the Dasgupta et al. book. As far as we know, these pdf copies are circulating with the blessing of the authors.

Additional lecture notes will be distributed when the content of the lectures deviates from the book.

### Lecture plan

• 2021-02-04: Introduction
Contents: What this course is about. Review of the topics and goals of the course. Preview of some highlights. Mathematical preliminaries: proving things by induction, bounding summations, big-Oh and little-oh notations, basic probability
Instructor: Iozzi
Suggested Exercises:
• 2021-02-05: Divide-and-conquer I
Contents: Integer multiplication in O(n^{log_2 3}) time, analysis of mergesort
Instructor: Iozzi
Suggested Exercises:
• 2021-02-09: Divide-and-conquer II
Contents: Median in linear time, Master Theorem
Instructor: Iozzi
Suggested Exercises:
• 2021-02-11: Graph Decomposition I
Contents: Graphs, representations of graphs, directed and undirected graphs, paths, connected components, DFS for connected components in undirected graphs
Instructor: Iozzi
Suggested Exercises:
• 2021-02-12: Graph Decomposition II
Contents: Directed acyclic graphs, linearization, DFS finds linearization
Readings: Section 3.3 and Notes on topological sort
Instructor: Trevisan
Suggested Exercises:
• 2021-02-16: Graph Decomposition III
Contents: Finding strongly connected components by running two DFSs
Instructor: Trevisan
Suggested Exercises:
• 2021-02-18: Shortest Paths I
Contents: BFS and shortest paths in unweighted graphs
Instructor: Trevisan
Suggested Exercises:
• 2021-02-19: Shortest Paths II
Contents: Dijkstra's algorithm
Readings: Sections 4.3 and 4.4, notes
Instructor: Trevisan
Suggested Exercises:
• 2021-02-23: Shortest Paths III
Contents: Continuation of Dijkstra's algorithm, priority queue implementation as sorted lists and as heaps
Instructor: Trevisan
Suggested Exercises:
• 2021-02-25: Lab I
Contents: Discuss solutions to exercises
Instructor: Iozzi
Suggested Exercises:
• 2021-02-26: Greedy Algorithms I
Contents: Minimum spanning tree problem, Kruskal and Prim's algorithm, proofs of correctness
Instructor: Trevisan
Suggested Exercises:
• 2021-03-02: Greedy Algorithms II
Contents: Correctness of Kruskal and Prim's algorithm, continued. Union-find data structure
Instructor: Trevisan
Suggested Exercises:
• 2021-03-04: Greedy Algorithms III
Contents: Continuation of previous lecture
Instructor: Trevisan
Suggested Exercises:
• 2021-03-05: Greedy Algorithms IV
Contents: Prefix codes, Huffman coding
Instructor: Trevisan
Suggested Exercises:
• 2021-03-09: Greedy Algorithms V
Contents: Huffman coding, entropy, shortest paths in DAGs
Instructor: Trevisan
Suggested Exercises:
• 2021-03-11: Dynamic Programming I (moved to 03-24)
Contents: Shortest paths with negative weights. All-pairs shortest paths. Rethinking Dijkstra's algorithm
Instructor: Trevisan
Suggested Exercises:
• 2021-03-12: Lab II
Contents: Discuss solutions to exercises and implementation of dynamic programming algorithms
Instructor: Iozzi
Suggested Exercises:
• 2021-03-25: Dynamic Programming II
Contents: Edit distance, knapsack
Instructor: Trevisan
Suggested Exercises:
• 2021-03-26: Flows I
Contents: Flows in networks, residual network, Ford-Fulkerson
Instructor: Trevisan
Suggested Exercises:
• 2021-03-30: Flows II
Contents: Max flow min cut theorem, analysis of Edmonds-Karp, reduction to other problems
Instructor: Trevisan
Suggested Exercises:
• 2021-04-01: Linear Programming I
Contents: Introduction, geometry of solutions, weak duality
Instructor: Iozzi
Suggested Exercises:
• 2021-04-02: Linear Programming II
Contents: The simplex algorithm
Instructor: Iozzi
Suggested Exercises:
• 2021-04-13: Linear Progamming III
Contents: Problems modelled by linear programming: flows, zero-sum games
Instructor: Iozzi
Suggested Exercises:
• 2021-04-15: Lab III
Contents: More on the simplex algorithm
Instructor: Iozzi
Suggested Exercises:
• 2021-04-16: NP-completeness I
Contents: Combinatorial explosion, search problems, P, NP, reductions, NP-completeness
Instructor: Trevisan
Suggested Exercises:
• 2021-04-20: NP-completeness II
Contents: NP-completeness of SAT, Independent Set, Clique