Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

[서평]「Algorithms on Strings, Trees, and Sequences」

Digital technology works so well because at the heart of digital representation, there are only a few basic components. Nowadays, these basic components usually are binary digits, bits for short, called binary because they are designed to stand for only two different values, conveniently called zero and one. A stored or transmitted bit's state can deteriorate quite badly before a processing device will be mistaken in deciding which of the two possible values is the original.

Binary Arithmetic

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.

Constructions for permutation codes in powerline communications

For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.

/pdf/a-survey-on-the-computational-complexity-of-coloring-graphs-290lv8e457.pdf

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

The problem of computing the chromatic number of a P5-free graph (a graph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P5-free graph admits a k-coloring, and finding one, if it does.

/pdf/deciding-k-colorability-of-p-5-free-graphs-in-polynomial-21ee0ci08n.pdf

Deciding k -Colorability of P 5 -Free Graphs in Polynomial Time

Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials over GF(2)

A k-ary necklace is an equivalence class of k-ary strings under rotation. A necklace of xed density is a necklace where the number of zeros is xed. We present a fast, simple, recursive algorithm for generating (i.e., listing) xed-densit y k-ary necklaces or aperiodic necklaces. The algorithm is optimal in the sense that it runs in time proportional to the number of necklaces produced.

http://www.cis.uoguelph.ca/%7Esawada/papers/fix.pdf

An efficient algorithm for generating necklaces with fixed density

A k-ary necklace is an equivalence class of k-ary strings under rotation A necklace of fixed density is a necklace where the number of zeros is fixed We present a fast, simple, recursive algorithm for generating (ie, listing) fixed-density k-ary necklaces or aperiodic necklaces The algorithm is optimal in the sense that it runs in time proportional to the number of necklaces produced

An Efficient Algorithm for Generating Necklaces with Fixed Density

http://www.cis.uoguelph.ca/~sawada/papers/new_construction_binary.pdf

Joe Sawada

Papers

Deciding k -Colorability of P 5 -Free Graphs in Polynomial Time

Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials over GF(2)

An efficient algorithm for generating necklaces with fixed density

An Efficient Algorithm for Generating Necklaces with Fixed Density

A surprisingly simple de Bruijn sequence construction