Improved algorithms for graph four-connectivity
Journal of Computer and System Sciences
Fully dynamic algorithms for 2-edge connectivity
SIAM Journal on Computing
Some combinatorial properties of Sturmian words
Theoretical Computer Science
Sturmian words, Lyndon words and trees
Theoretical Computer Science
Handbook of formal languages, vol. 1
Sturmian words: structure, combinatorics, and their arithmetics
Theoretical Computer Science - Special issue: formal language theory
On the combinatorics of finite words
Theoretical Computer Science
Fine and Wilf's theorem for three periods and a generalization of Sturmian words
Theoretical Computer Science
Partial words and a theorem of Fine and Wilf
Theoretical Computer Science
Partial words and a theorem of Fine and Wilf revisited
Theoretical Computer Science
Computing vertex connectivity: new bounds from old techniques
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
On spaced seeds for similarity search
Discrete Applied Mathematics
Designing seeds for similarity search in genomic DNA
Journal of Computer and System Sciences - Special issue on bioinformatics II
Generalised fine and Wilf's theorem for arbitrary number of periods
Theoretical Computer Science - Combinatorics on words
Graph Theory With Applications
Graph Theory With Applications
Optimal spaced seeds for faster approximate string matching
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Hardness of optimal spaced seed design
CPM'05 Proceedings of the 16th annual conference on Combinatorial Pattern Matching
DNA'04 Proceedings of the 10th international conference on DNA computing
Periodicity properties on partial words
Information and Computation
The three-squares lemma for partial words with one hole
Theoretical Computer Science
Periods in partial words: an algorithm
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Periods in partial words: An algorithm
Journal of Discrete Algorithms
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The problem of computing periods in words, or finite sequences of symbols from a finite alphabet, has important applications in several areas including data compression, string searching and pattern matching algorithms. The notion of period of a word is central in combinatorics on words. There are many fundamental results on periods of words. Among them is the well known and basic periodicity result of Fine and Wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period. More precisely, any word with length at least p+q-gcd(p,q) having periods p and q has also period the greatest common divisor of p and q, gcd(p,q). Moreover, the bound p+q-gcd(p,q) is optimal since counterexamples can be provided for words of smaller length. Partial words, or finite sequences that may contain a number of ''do not know'' symbols or holes, appear in natural ways in several areas of current interest such as molecular biology, data communication, DNA computing, etc. Any long enough partial word with h holes and having periods p,q has also period gcd(p,q). In this paper, we give closed formulas for the optimal bounds L(h,p,q) in the case where p=2 and also in the case where q is large. In addition, we give upper bounds when q is small and h=3,4,5,6 or 7. No closed formulas for L(h,p,q) were known except for the cases where h=0,1 or 2. Our proofs are based on connectivity in graphs associated with partial words. A World Wide Web server interface has been established at www.uncg.edu/mat/research/finewilf3 for automated use of a program which given a number of holes h and two periods p and q, computes the optimal bound L(h,p,q) and an optimal word for that bound (a partial word u with h holes of length L(h,p,q)-1 is optimal if p and q are periods of u but gcd(p,q) is not a period of u).