Following are the courses that I typically teach. I have tried to add notes for each course - though they may not always be as self-contained as a comprehensive set of lecture notes. These notes are simply meant for students to use as a quick refresher.
Part 1 Introduction to Problems, Algorithms and Computation, Basic Abstract Data Types (list or dynamic multi set, stack, queue) and Data Structures (array, linked list)
Part 2 Basic Algorithm Design & Analysis: RAM model of computation, notion of efficiency, asymptotic analysis, sorting arrays using iterative algorithms (selection sort, insertion sort) and recursive algorithms (merge sort, quick sort)
Part 3 Priority Queue, Expressions & Dictionary: Heap (for priority queue), hash table (for dictionary), binary search tree (for both dictionary & priority queue), expression trees (for arithmetic expressions)
Part 4 Graphs & Algorithms on Graphs: Representing graphs using list and matrix, depth-first search, topological orderings, strongly connected components, breadth-first search and shortest paths
Part 1 Algorithms with numbers - arithmetic on integers, modular arithmetic, GCD (Euclid’s algorithm). Factoring and primality testing. Iterative and recursive algorithms - multiplication and division algorithms, Towers of Hanoi. Divide and Conquer - multiplication. Solving recurrences using induction, recursion tree, Master Theorem. Estimates of factorials and binomials.
Part 2 Sorting (comparison model and exchange model) and searching lower bounds (sorted and unsorted lists). Find Max, Find Max-Min, Find Second Max upper and lower bounds.
Part 3 Graph algorithms using greedy, dynamic programming and iterative improvement - single source shortest path (in uniform weighted (di)graphs, arbitrary weighted DAGs, non-negative weights digraphs, arbitrary weighted digraphs), all pairs shortest path, minimum spanning trees, maximum flow, maximum bipartite matching.
Part 4 Theory of NP-completeness - polynomial-time reductions, classes NP, co-NP and P, NP-complete problems.
Part 1 Logic [Notes Part 1] [Notes Part 2]
Logical formulas - propositional and predicate. Introduction to proofs.
Part 2 Set Theory [Notes Part 1] [Notes Part 2]
Sets, relations, equivalences and partial orders. Logic of sets. Well ordering principle. Induction. Recursive definitions. Infinite Sets: Cantor’s theorem, diagonalization argument, halting problem.
Part 3 Graph Theory [Notes]
Graphs - directed and undirected. Walks, paths, cycles, trees, forests. Cliques and independent sets. Graph Isomorphism. Bipartite graphs and matchings.
Part 4 Combinatorics [Notes]
Product rule, division rule, counting subsets, sequences with repetitions - binomial and multinomial theorems. Pigeonhole principle. Inclusion-exclusion principle. Combinatorial proofs. Recurrence relations.
Part 5 Discrete Probability [Notes]
Events and probability spaces. Conditional probability and law of total probability. Independence. Random variables, distribution functions and expectations. Linearity of expectation. Markov's inequality and Chebyshev's inequality. Law of large numbers. An introduction to probabilistic method and randomized algorithms.
Basic Toolkit: branching, kernelization, iterative compression, LP techniques, crown decomposition, sunflower lemma
Randomized Techniques: color coding, chromatic coding, random separation, derandomization
Algebraic Techniques: principle of inclusion-exclusion, polynomial identity testing, representative sets
Treewidth: dynamic programming over tree decomposition, computing/approximating treewidth
Hardness: fixed-parameter intractability, kernelization lower bounds, ETH-based lower bounds
Maximum Bipartite Matching: Maximum matching, alternating paths, augmenting paths, Berge's algorithm for maximum bipartite matching, stable matchings, Gale-Shapley algorithm for stable marriage problem
Maximum Flow: Maximum flow problem, iterative improvement idea, residual network, augmenting paths, Ford-Fulkerson algorithm, max flow min cut theorem, integrality of flows, Edmonds-Karp algorithm, application of maximum flow to maximum bipartite matching, team elimination in tournaments, image segmentation, LP formulation for maximum flow and minimum cut
Matroids: Definition, examples, algorithmic characterization of matroids, 2-matroid intersection, applications of 2-matroid intersection to maximum bipartite matching, minimum weight arborescences, color-constrained spanning trees, NP-hardness of 3-matroid intersection, matroid partitioning, matroid pairity
Algebraic algorithms: Maximum bipartite matching using Edmonds matrix, maximum matching in general graphs using Tutte's matrix, isolation lemma for parallelization, perfect matching with k red edges
Graph theory
Basic definitions, terminology, modelling using graphs
Basic counting results on graphs
Important results on trees
Matching and Vertex Cover in bipartite graphs - Hall's theorem and Kőnig's theorem
Eulerian graphs - definition and characterization
Hamiltonian graphs, Dirac's theorem and Ore's theorem
Connectivity - vertex connectivity, edge connectivity, Whitney's inequality, Menger's theorems
Coloring - vertex coloring, edge coloring, relation to maximum degree, Vizing's theorem
Planar graphs - definition, Euler's formula, 5-coloring of planar graphs
Combinatorics
Counting sequences, subsets, functions, compositions and set/integer partitions
Binomial coefficients and Binomial theorem
Inclusion-exclusion principle, recurrence relations, pigeonhole principle
Ordinary and exponential generating functions
Selected Advanced Topics
Introduction to extremal graph theory - Mantel's theorem, Turan's theorem
Introduction to Ramsey theory - Ramsey numbers: upper bound, lower bound, recurrence
Introduction: Network Flows, Ford-Fulkerson Method, Edmonds-Karp Algorithm, Linear Programming (LP), LP Formulation for Flows, LP Duality, Weak Duality Theorem, Complementary Slackness, Farkas’ Lemma
Linear Programming based Optimization: Minimum Spanning Trees, Shortest Paths, Totally Unimodular Matrices from Bipartite Graphs, Maximum Matching and Minimum Vertex Cover on Bipartite Graphs
Polyhedral Techniques: Polyhedra and Polytopes of Linear Programs, The Matching and Perfect Matching Polytopes for General Graphs, The Independent Set Polytope, Polyhedral Characterization of Bipartite Graphs and Perfect Graphs
Advanced Topics: Semidefinite programming based algorithms and matroid based algorithms