ALGORITHMS

General description

The course presents the main design methodoligies (incremental, recursion, dynamic programming, greedy algorithms) and the techniques for their complexity analysis, both in the worst and average cases.

Training objectives

Knowledge and understanding: students will acquire knowldege relative to the main methodologies for the design of efficient algorithms (incremental, recursion, dynamic programming, greedy algorithms), as well as the techniques for their computational analysis, both in the worst and in the average cases.
Applying knowledge and understanding: students will acquire the ability to solve problems of low difficulty, requiring the design and analysis of elementary algorithmic solutions.
Making judgements: students will be able to evaluate the quality of an algorithmic solution in terms of efficiency and reuse.
Communication skills: students will acquire the necessary communication skills and expressive ability in communicate problems regarding the algorithmic studies, also to non-expert interlocutors.
Learning skills: students will have the ability to adapt the knowledge acquired also in new contexts and to advance his/her knowledge through the consultation of specialist sources in the algorithmic field.

DETAILED SYLLABUS

Introduction
Computational problems and algorithms: the sorting problem
Algorithms as technology
Incremental methodology: Insertion-Sort (correctness and complexity)
Divide-and-Conquer methodology: Merge-Sort (complexity)
Asymptotic notations and relationships among them
Standard notations and common functions.

Recurrences
The substitution method
The iterative method and the recursion tree
The master theorem

Sorting and order statistics
Heaps and their construction
Priority queues
Quicksort and its randomized version
Analysis of Quicksort in the worst- and average case
Lower bounds for sorting
Sorting in linear time: Counting-Sort, Radix-Sort, Bucket-Sort
Medians and order statistics

Hashing
Hash table
Hash functions (division method, multiplication method, universal hashing)
Open addressing

Red-black trees
Rotations, insertions, deletions
Complexity analysis

Elements of dynamic programming
Optimal substructure, overlapping subproblems, reconstructing an optimal solution
Some case studies: assembly line scheduling, matrix-chain multiplication, longest common subsequence, editing distance

Elements of the greedy strategy
Greedy-choice property, optimal substructure
Some case studies: the activity-selection problem, Huffman codes

Elementary graph algorithms
Breadth-first search
Depth-first search (edges classification)
Topological sorting
Strongly connected components

T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein. Introduction to algorithms (Third Edition), The MIT Press, Cambridge - Massachusetts, 2009.