CSCI-GA.3520-001

Fall 2024

CS Dept, New York University

Honors Analysis of Algorithms

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General Information and Announcements.

Final exam is scheduled on Fri, Dec 13, 1:30-5:30PM, Room 101 WWH. You must arrive 20 mins early, i.e. by 1:10PM. It is a closed book exam.

The 2023 final exam is HERE.

The class will meet in person and will be taught on the whiteboard. It will not be recorded, but the video recordings of the class from Fall 2020 will be made available. All course notes are available (see below). All homeworks are available too (see below, there could be some changes, deadlines TBA).

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Administrative Information

Lectures: T Th 6:20-7:35 PM (CIWW 102)

Instructor: Subhash Khot, Off-416 WWH, Ph: 212-998-4859. Office hours: W 9:30-11:30

Grader: Haitian Jiang, haitian.jiang @ nyu

Class assistant: Utkarsh Parkhi up2043 @ nyu

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Course Description

This course is intended to cover the topics needed for the departmental comprehensive exam in Algorithms. Unlike in the past, the course and the final exam will *not* include topics from the theory of computation. The goal of the course, in addition to covering the topics listed below, is to improve your algorithmic problem solving skills.

Topics:

• Recurrence equations and divide & conquer
• Linear time algorithms (radix sort, lexicographic sort, selection)
• Hashing
• Balanced trees and their augmentation
• Amortized analysis (Fibonnacci heaps, Union-Find)
• Algorithms on graphs (DFS, path problems, MST; using reductions for algorithm design)
• Dynamic programming
• Randomization
• NP-completeness

Text: (Not necessarily required. The lectures should be self-contained).

• Algorithm Design, by Kleinberg and Tardos.

This book may be useful: Introduction to Algorithms (Second Edition), by Cormen, Leiserson, Rivest, and Stein.

Grading: 50% problem sets, 50% final exam.

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Problems Sets:

Practice Problems (Problems added to the top of the list).

Problems from past PhD Algorithms Exam

Problems from past MS Algorithms Core Exam (easy but may be good as practice problems)

Towards the preparation for the final exam, the main thing to do is practice, practice and practice! In addition to the problems from the past PhD/MS exams and homework problems, you can also work through problems in the [KT] textbook. Also, you may not want to wait till the relevant topics are covered in class (which might be too late, especially for the topics to be covered towards the end).

Homeworks: I highly recommend hand-written solutions.

Deadlines to be announced as we go along.

Homework 1 Solutions

Homework 2 Solutions

Homework 3 Solutions

Homework 4 Solutions

Homework 5 Solutions

Homework 6 Solutions

Homework 6 could have some broken links. These refer to 2008, 2009 exams from here.

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Lectures (Rough plan)

Class Notes: 1 2 DC DC+1 G G+1 D D+1 B B+1 B+2 B+3 B+4 MF R R+1 R+2 NP NP+1 NP+2 NP+3 SC

Date	Topics covered	Source
	Introduction; Basic data structures: arrays, linked lists, merging sorted lists	KT: Chapter 1,2
	Heapsort, Divide and conquer: mergesort, recurrence relations	KT: Chapter 5
	Divide and conquer: finding median, quicksort	KT: Chapter 5
	Divide and conquer: counting inversions, sorting lower bound, radix sort	KT: Chapter 5
	Divide and conquer: fast Fourier transform, polynomial multiplication	KT: Chapter 5
	Greedy algorithms: interval scheduling, interval partitioning	KT: Chapter 4
	Greedy algorithms: minimum spanning tree	KT: Chapter 4
	Dynamic programming: Subset-sum with bounded integers, matrix chain multiplication, longest common subsequence	KT: Chapter 6
	Dynamic programming: weighted interval scheduling, shortest paths, Maximum independent sets in trees	KT: Chapter 6
	Amortized analysis: stack, binary counter, binomial heaps	CLRS: Chapter 17,19,20
	Amortized analysis: Fibonacci heaps	CLRS: Chapter 17,19,20
	Binary search trees, 2-3 Trees, Breadth first search	KT: Chapter 3
	Acyclic graphs, topological sort, strongly connected graphs, strongly connected components, depth first search	KT: Chapter 3
	Dijkstra: shortest path, Max-flow	CLRS: Chapter 24
	Max-flow: Max-flow = Min-cut, Ford-Fulkerson algorithm
	Max-flow: Application to Hall Theorem Randomized algorithms: Basics of probability
	Randomized algorithms: Union bound, Ramsey Numbers (application of probabilistic method)
	Randomized algorithms: Independence, Contention resolution
	Randomized algorithms: Expectation, Randomized Quick-Sort, MAX-CUT
	Randomized algorithms: Hashing
	Turing machines, running time, P, non-determinism, NP
	NP-completeness: reductions
	Cook-Levin Theorem