Department of Computer Science at the Technische Universität München Chair for Efficient Algorithms Postal address: 80290 München; Premises: Arcisstr.21, 80333 München

Core practical: Design of Algorithms (WS 2001/2002)

Background

Many problems that are encountered in practice can be solved efficiently if they are modeled as graph problems. For several decades researchers in mathematics and computer science have devised algorithms for the efficient solution of typical graph problems. It has been shown that many problems can be solved by very fast algorithms even though a straightforward implementation requires exponential running-times.

Maximum matching in a bipartite graph

So-called matching algorithms, for example, can be used to solve a variety of assignment problems. One such problem could be to assign lecturers to courses in a way that ensures that the maximum possible number of courses can be taught. If every lecturer teaches at most one course and every course is taught at most once, the problem is equivalent to the matching problem in a bipartite graph as depicted in the figure above. The top row of nodes represents lecturers, the bottom row represents courses. An edge between a lecturer node and a course node means that the lecturer is competent to teach the respective course. Now, a valid assignment of courses to lecturers corresponds to a matching in the graph. In the example above, at most six out of eight courses can be taught. The bold edges correspond to such an assignment. The algorithm by Hopcroft and Karp computes maximum matchings in bipartite graphs in time O(sqrt(n)*m), where n is the number of nodes and m is the number of edges of the graph.

Graph algorithms are interesting not only because of their usefulness for the solution of practical problems, but also because of the algorithmic techniques and the used data structures. Therefore, students who intend to take part in this practical course should be prepared to deal with complicated algorithms.

Contents

	depth-first-search, breadth-fist-search
	connected components
	minimum spanning trees
	shortest paths
	matchings
	network flow
	vertex-coloring planar graphs
	approximating the traveling salesman problem

In addition to these graph problems, several other problems originating from different application contexts and not directly related to graphs will be covered as well. Examples are:

	string matching
	computing the edit distance of strings

Many of these algorithms are presented in the courses "Efficient Algorithms and Data Structures I/II". The content of these lectures is useful for the practical but it is also possible to aquire the knowledge in parallel to the practical. A special two-hours-per-week discussion group is part of the practical and gives explanations regarding the implementation of the algorithms.

The goal of the practical course is to help students gain a better understanding of how sophisticated algorithms, graph algorithms in particular, work and to convince them of the benefit obtained by using high-level data structures.

Organization

LEDA is a class library supplying frequently used data structures such as stacks, queues, priority queues, lists and graphs. Using LEDA reduces substantially the programming effort necessary to implement sophisticated algorithms in C++ efficiently. In addition, the class GraphWin offered by LEDA is a flexible tool for visualization of graph algorithms.

The programming exercises can either be solved in our computer room (S2237), on the workstations in the HP-Halle (beware: compilation of LEDA programs takes quite long here!), or on any PC running Linux (the latter case requires to install LEDA first). Groups of three students solve the programming exercises together. We strongly advise the groups not to divide the work load and work on the exercises separately; instead, they should actually cooperate in solving the problems and implementing the algorithms. In the oral exam at the end of the semester, every student is expected to be knowledgable about the actual implementation of all solutions handed in by his/her group.

Students and coordinators meet once per week for a two-hour discussion group in which the coordinators present new exercises and explain the required algorithms and theoretical background. Students send their solutions by e-mail to algoprak@informatik.tu-muenchen.de or algoprak@in.tum.de. Comments on their solutions and grading information are sent to the students by e-mail by their respective supervisors as well. Problems encountered by the students can be discussed and questions can be answered either during the discussion group or with their supervisors.

Programs handed in by the students will be graded according to the following criteria:

1. the computed solutions must be correct
2. the implementation must be efficient, i.e., the given worst-case complexity must not be exceeded
3. the visualization of the algorithm should be at least acceptable
4. the source code must be comprehensible and well-structured

In order to get a course certificate, the following criteria must be satisfied:

• the group of three students must hand in satisfactory solutions to all exercises
• each student must pass the oral examination at the end of the semester

Literature:

	Cormen, Leiserson, Rivest. Introduction to Algorithms MIT Press, 1990
	Turau. Algorithmische Graphentheorie. Addison Wesley, 1996
	Gibbons. Algorithmic Graph Theory. Cambridge University Press, 1985
	Mehlhorn. Data Structures and algorithms 2: Graph algorithms and NP-Completeness, Springer-Verlag, 1984
	Papadimitriou, Steiglitz. Combinatorial Optimization: Algorithms and complexity. Prentice-Hall, 1982
	Sedgewick. Algorithms. Addison-Wesley, 1988
	Tarjan. Data Structures and Network Algorithms. SIAM, 1983
	Reingold, Nievergelt, Deo. Combinatorial Algorithms. Prentice-Hall, 1977
	Ahuja, Magnanti, Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.