Joe Sawada

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

Many applications call for exhaustive lists of strings subject to various constraints, such as inequivalence under group actions. A k-ary necklace is an equivalence class of k-ary strings under rotation (the cyclic group). A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algorithms for generating (i.e., listing) all necklaces and binary unlabeled necklaces. These algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. The algorithm for generating necklaces can be used as the basis for efficiently generating many other equivalence classes of strings under rotation and has been applied to generating bracelets, fixed density necklaces, and chord diagrams. As another application, we describe the implementation of a fast algorithm for listing all degree n irreducible and primitive polynomials over GF(2).

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 (i.e., 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.

A Certifying Algorithm for 3-Colorability of P5-Free Graphs

We provide a certifying algorithm for the problem of deciding whether a P 5-free graph is 3-colorable by showing there are exactly six finite graphs that are P 5-free and not 3-colorable and minimal with respect to this property.

Generating Bracelets in Constant Amortized Time

A bracelet is the lexicographically smallest element in an equivalence class of strings under string rotation and reversal. We present a fast, simple, recursive algorithm for generating (i.e., listing) k-ary bracelets. Using simple bounding techniques, we prove that the algorithm is optimal in the sense that the running time is proportional to the number of bracelets produced. This is an improvement by a factor of n (where n is the length of the bracelets being generated) over the fastest, previously known algorithm to generate bracelets.

A fast algorithm to generate necklaces with fixed content

We develop a fast algorithm for listing all necklaces with fixed content. By fixed content, we mean the number of occurrences of each alphabet symbol is fixed. Initially, we construct a simple but inefficient algorithm by making some basic modifications to a recursive necklace generation algorithm. We then improve it by using two classic combinatorial optimization techniques. An analysis using straight forward bounding techniques is used to prove that the algorithm runs in constant amortized time.

De Bruijn Sequences for Fixed-Weight Binary Strings

De Bruijn sequences are circular strings of length

The Number of Irreducible Polynomials and Lyndon Words with Given Trace

2^n

Binary bubble languages and cool-lex order

whose length n substrings are the binary strings of length n. Our focus is on creating circular strings of length

Constructions of k -critical P 5 -free graphs

\binom{n}{w}

Generating Necklaces and Strings with Forbidden Substrings

for the binary strings of length n with weight (number of 1s) equal to w. In this case, each fixed-weight string can be encoded by its first