Generalized Fibonacci cubes

AbstractGeneralized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Qd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f, the f-dimension of a graph is introduced. Several problems and conjectures are also listed

The degree sequence of Fibonacci and Lucas cubes

AbstractThe Fibonacci cube Γn is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γn and Λn is ∑i=0k(n−2ik−i)(i+1n−k−i+1) and ∑i=0k[2(i2i+k−n)(n−2i−1k−i)+(i−12i+k−n)(n−2ik−i)], respectively. Both results are obtained in two ways, since each of the approaches yields additional results on the degree sequences of these cubes. In particular, the number of vertices of high resp. low degree in Γn is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of Γn and Λn are easily computed

Advances and applications of automata on words and trees : abstracts collection

From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes

Abstract Let Γ n and Λ n be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λ n ) is bounded below by , where L n is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that and the exact values of ρ(Γ n ) and ρ(Λ n ) are obtained for n ≤ 10. It is also shown that Aut(Γ n ) Z 2

Analysing and Enhancing the Coarse Registration Pipeline

The current and continual development of sensors and imaging systems capable of acquiring three-dimensional data provides a novel form in which the world can be expressed and examined. The acquisition process, however, is often limited by imaging systems only being able to view a portion of a scene or object from a single pose at a given time. A full representation can still be produced by shifting the system and registering subsequent acquisitions together. While many solutions to the registration problem have been proposed, there is no quintessential approach appropriate for all situations. This dissertation aims to coarsely register range images or point-clouds of a priori unknown pose by matching their overlapping regions. Using spherical harmonics to correlate normals in a coarse registration pipeline has been shown previously to be an effective means for registering partially overlapping point-clouds. The advantage of normals is their translation invariance, which permits the rotation and translation to be decoupled and determined separately. Examining each step of this pipeline in depth allows its registration capability to be quantified and identifies aspects which can be enhanced to further improve registration performance. The pipeline consists of three primary steps: identifying the rotation using spherical harmonics, identifying the translation in the Fourier domain, and automatically verifying if alignment is correct. Having achieved coarse registration, a fine registration algorithm can be used to refine and complete the alignment. Major contributions to knowledge are provided by this dissertation at each step of the pipeline. Point-clouds with known ground-truth are used to examine the pipeline's capability, allowing its limitations to be determined; an analysis which has not been performed previously. This examination allowed modifications to individual components to be introduced and measured, establishing their provided benefit. The rotation step received the greatest attention as it is the primary weakness of the pipeline, especially as the nature of the overlap between point-clouds is unknown. Examining three schemes for binning normals found that equiangular binning, when appropriately normalised, only had a marginal decrease in accuracy with respect to the icosahedron and the introduced Fibonacci schemes. Overall, equiangular binning was the most appropriate due to its natural affinity for fast spherical-harmonic conversion. Weighting normals was found to provide the greatest benefit to registration performance. The introduction of a straightforward method of combining two different weighting schemes using the orthogonality of complex values increased correct alignments by approximately 80% with respect to the next best scheme; additionally, point-cloud pairs with overlap as low as 5% were able to be brought into correct alignment. Transform transitivity, one of two introduced verification strategies, correctly classified almost 100% of point-cloud pair registrations when there are sufficient correct alignments. The enhancements made to the coarse registration pipeline throughout this dissertation provide significant improvements to its performance. The result is a pipeline with state-of-the-art capabilities that allow it to register point-cloud with minimal overlap and correct for alignments that are classified as misaligned. Even with its exceptional performance, it is unlikely that this pipeline has yet reached its pinnacle, as the introduced enhancements have the potential for further development

A Minimal Periods Algorithm with Applications

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and describe in detail how to compute in O(k|w|)-time the minimal k-th power, with period of length larger than s, starting at each position in a word w for arbitrary exponent $k\geq2$ and integer $s\geq0$. We provide the complete proof of correctness of the algorithm, which is somehow not completely clear in Kosaraju's original paper. The algorithm can be used as a sub-routine to detect certain types of pseudo-patterns in words, which is our original intention to study the generalization.Comment: 14 page

A Novel Latin Square Image Cipher

In this paper, we introduce a symmetric-key Latin square image cipher (LSIC) for grayscale and color images. Our contributions to the image encryption community include 1) we develop new Latin square image encryption primitives including Latin Square Whitening, Latin Square S-box and Latin Square P-box ; 2) we provide a new way of integrating probabilistic encryption in image encryption by embedding random noise in the least significant image bit-plane; and 3) we construct LSIC with these Latin square image encryption primitives all on one keyed Latin square in a new loom-like substitution-permutation network. Consequently, the proposed LSIC achieve many desired properties of a secure cipher including a large key space, high key sensitivities, uniformly distributed ciphertext, excellent confusion and diffusion properties, semantically secure, and robustness against channel noise. Theoretical analysis show that the LSIC has good resistance to many attack models including brute-force attacks, ciphertext-only attacks, known-plaintext attacks and chosen-plaintext attacks. Experimental analysis under extensive simulation results using the complete USC-SIPI Miscellaneous image dataset demonstrate that LSIC outperforms or reach state of the art suggested by many peer algorithms. All these analysis and results demonstrate that the LSIC is very suitable for digital image encryption. Finally, we open source the LSIC MATLAB code under webpage https://sites.google.com/site/tuftsyuewu/source-code.Comment: 26 pages, 17 figures, and 7 table

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn