Search results for "combinatoric"
showing 10 items of 1776 documents
Polyomino coloring and complex numbers
2008
AbstractUsually polyominoes are represented as subsets of the lattice Z2. In this paper we study a representation of polyominoes by Gaussian integers. Polyomino {(x1,y1),(x2,y2),…,(xs,ys)}⊂Z2 is represented by the set {(x1+iy1),(x2+iy2),…,(xs+iys)}⊂Z[i]. Then we consider functions of type f:P→G from the set P of all polyominoes to an abelian group G, given by f(x,y)≡(x+iy)m(modv), where v is prime in Z[i],1≤m<N(v) (N(v) is the norm of v). Using the arithmetic of the ring Z[i] we find necessary and sufficient conditions for such a function to be a coloring map.
Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants
2020
We provide a quantitative lower bound to the Cheeger constant of a set $\Omega$ in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show that these quantitative estimates are sharp.
Strong chromatic index of products of graphs
2007
Graphs and Algorithms
Exact results for accepting probabilities of quantum automata
2001
One of the properties of Kondacs-Watrous model of quantum finite automata (QFA) is that the probability of the correct answer for a QFA cannot be amplified arbitrarily. In this paper, we determine the maximum probabilities achieved by QFAs for several languages. In particular, we show that any language that is not recognized by an RFA (reversible finite automaton) can be recognized by a QFA with probability at most 0.7726...
Word assembly through minimal forbidden words
2006
AbstractWe give a linear-time algorithm to reconstruct a finite word w over a finite alphabet A of constant size starting from a finite set of factors of w verifying a suitable hypothesis. We use combinatorics techniques based on the minimal forbidden words, which have been introduced in previous papers. This improves a previous algorithm which worked under the assumption of stronger hypothesis.
Local Normal Forms for First-Order Logic with Applications to Games and Automata
1999
Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata mode…
The pruning-grafting lattice of binary trees
2008
AbstractWe introduce a new lattice structure Bn on binary trees of size n. We exhibit efficient algorithms for computing meet and join of two binary trees and give several properties of this lattice. More precisely, we prove that the length of a longest (resp. shortest) path between 0 and 1 in Bn equals to the Eulerian numbers 2n−(n+1) (resp. (n−1)2) and that the number of coverings is (2nn−1). Finally, we exhibit a matching in a constructive way. Then we propose some open problems about this new structure.
From Nerode's congruence to Suffix Automata with mismatches
2009
AbstractIn this paper we focus on the minimal deterministic finite automaton Sk that recognizes the set of suffixes of a word w up to k errors. As first result we give a characterization of the Nerode’s right-invariant congruence that is associated with Sk. This result generalizes the classical characterization described in [A. Blumer, J. Blumer, D. Haussler, A. Ehrenfeucht, M. Chen, J. Seiferas, The smallest automaton recognizing the subwords of a text, Theoretical Computer Science, 40, 1985, 31–55]. As second result we present an algorithm that makes use of Sk to accept in an efficient way the language of all suffixes of w up to k errors in every window of size r of a text, where r is the…
Motif patterns in 2D
2008
AbstractMotif patterns consisting of sequences of intermixed solid and don’t-care characters have been introduced and studied in connection with pattern discovery problems of computational biology and other domains. In order to alleviate the exponential growth of such motifs, notions of maximal saturation and irredundancy have been formulated, whereby more or less compact subsets of the set of all motifs can be extracted, that are capable of expressing all others by suitable combinations. In this paper, we introduce the notion of maximal irredundant motifs in a two-dimensional array and develop initial properties and a combinatorial argument that poses a linear bound on the total number of …
On the exhaustive generation of k-convex polyominoes
2017
The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates, given an integer k, all k-convex polyominoes of area n in constant amortized time, using space O(n). Furthermore, by applying few changes, we are able to generate all convex polyominoes whose degree of convexity is exactly k.