The.Algorithm.Design.Manual.Springer-Verlag.1998

Combinatorial Problems
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Combinatorial Problems
In this section, we consider several classic algorithmic problems of a purely combinatorial nature. These
include sorting and permutation generation, both of which were among the first nonnumerical problems
arising on electronic computers. Sorting, searching, and selection can all be classified in terms of
operations on a partial order of keys. Sorting can be viewed as identifying or imposing the total order on
the keys, while searching and selection involve identifying specific keys based on their position in the
total order.
The rest of this section deals with other combinatorial objects, such as permutations, partitions, subsets,
calendars, and schedules. We are particularly interested in algorithms that rank and unrank
combinatorial objects, i.e. that map each distinct object to and from a unique integer. Once we have rank
and unrank operations, many other tasks become simple, such as generating random objects (pick a
random number and unrank) or listing all objects in order (iterate from 1 to n and unrank).
We conclude with the problem of generating graphs. Graph algorithms are more fully presented in
subsequent sections.
Books on general combinatorial algorithms, in this restricted sense, include:
● Nijenhuis and Wilf [NW78] - This book specializes in algorithms for constructing basic
combinatorial objects such as permutations, subsets, and partitions. Such algorithms are often
very short but hard to locate and usually are surprisingly subtle. Fortran programs for all of the
algorithms are provided, as well as a discussion of the theory behind each of them. See Section
for details.
● Ruskey [Rus97] - On its completion, this manuscript in preparation will become the standard
reference on generating combinatorial objects. A preview is available via the WWW at
http://www-csc.uvic.ca/home/fruskey/cgi-bin/html/main.html .
● Knuth [Knu73a, Knu73b] - The standard reference on searching and sorting, with significant
material on combinatorial objects such as permutations.
● Reingold, Nievergelt, Deo [RND77] - A comprehensive algorithms text with a particularly
thorough treatment of combinatorial generation and search.
● Stanton and White [SW86] - An undergraduate combinatorics text with algorithms for generating
permutations, subsets, and set partitions. It contains relevant programs in Pascal.
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