Partial match queries in relaxed K-dt trees

The study of partial match queries on random hierarchical multidimensional data structures dates back to Ph. Flajolet and C. Puech’s 1986 seminal paper on partial match retrieval. It was not until recently that fixed (as opposed to random) partial match queries were studied for random relaxed K-d trees, random standard K-d trees, and random 2-dimensional quad trees. Based on those results it seemed natural to classify the general form of the cost of fixed partial match queries into two families: that of either random hierarchical structures or perfectly balanced structures, as conjectured by Duch, Lau and Martínez (On the Cost of Fixed Partial Queries in K-d trees Algorithmica, 75(4):684–723, 2016). Here we show that the conjecture just mentioned does not hold by introducing relaxed K-dt trees and providing the average-case analysis for random partial match queries as well as some advances on the average-case analysis for fixed partial match queries on them. In fact this cost –for fixed partial match queries– does not follow the conjectured forms.Peer ReviewedPostprint (author's final draft

Quiet Planting in the Locked Constraint Satisfaction Problems

We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles. The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to be crucial. Our main result is the location of the hard region in the planted ensemble. In a part of that hard region instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio

S-TREE: Self-Organizing Trees for Data Clustering and Online Vector Quantization

This paper introduces S-TREE (Self-Organizing Tree), a family of models that use unsupervised learning to construct hierarchical representations of data and online tree-structured vector quantizers. The S-TREE1 model, which features a new tree-building algorithm, can be implemented with various cost functions. An alternative implementation, S-TREE2, which uses a new double-path search procedure, is also developed. S-TREE2 implements an online procedure that approximates an optimal (unstructured) clustering solution while imposing a tree-structure constraint. The performance of the S-TREE algorithms is illustrated with data clustering and vector quantization examples, including a Gauss-Markov source benchmark and an image compression application. S-TREE performance on these tasks is compared with the standard tree-structured vector quantizer (TSVQ) and the generalized Lloyd algorithm (GLA). The image reconstruction quality with S-TREE2 approaches that of GLA while taking less than 10% of computer time. S-TREE1 and S-TREE2 also compare favorably with the standard TSVQ in both the time needed to create the codebook and the quality of image reconstruction.Office of Naval Research (N00014-95-10409, N00014-95-0G57

Testing robustness of relative complexity measure method constructing robust phylogenetic trees for Galanthus L. Using the relative complexity measure

Background: Most phylogeny analysis methods based on molecular sequences use multiple alignment where the quality of the alignment, which is dependent on the alignment parameters, determines the accuracy of the resulting trees. Different parameter combinations chosen for the multiple alignment may result in different phylogenies. A new non-alignment based approach, Relative Complexity Measure (RCM), has been introduced to tackle this problem and proven to work in fungi and mitochondrial DNA. Result: In this work, we present an application of the RCM method to reconstruct robust phylogenetic trees using sequence data for genus Galanthus obtained from different regions in Turkey. Phylogenies have been analyzed using nuclear and chloroplast DNA sequences. Results showed that, the tree obtained from nuclear ribosomal RNA gene sequences was more robust, while the tree obtained from the chloroplast DNA showed a higher degree of variation. Conclusions: Phylogenies generated by Relative Complexity Measure were found to be robust and results of RCM were more reliable than the compared techniques. Particularly, to overcome MSA-based problems, RCM seems to be a reasonable way and a good alternative to MSA-based phylogenetic analysis. We believe our method will become a mainstream phylogeny construction method especially for the highly variable sequence families where the accuracy of the MSA heavily depends on the alignment parameters