Polynomial kernels for 3-leaf power graph modification problems

A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u,v) is an edge iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, Huffner and Niedermeier (2005).Comment: Submitte

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

Hierarchical testing designs for pattern recognition

We explore the theoretical foundations of a ``twenty questions'' approach to pattern recognition. The object of the analysis is the computational process itself rather than probability distributions (Bayesian inference) or decision boundaries (statistical learning). Our formulation is motivated by applications to scene interpretation in which there are a great many possible explanations for the data, one (``background'') is statistically dominant, and it is imperative to restrict intensive computation to genuinely ambiguous regions. The focus here is then on pattern filtering: Given a large set Y of possible patterns or explanations, narrow down the true one Y to a small (random) subset \hat Y\subsetY of ``detected'' patterns to be subjected to further, more intense, processing. To this end, we consider a family of hypothesis tests for Y\in A versus the nonspecific alternatives Y\in A^c. Each test has null type I error and the candidate sets A\subsetY are arranged in a hierarchy of nested partitions. These tests are then characterized by scope (|A|), power (or type II error) and algorithmic cost. We consider sequential testing strategies in which decisions are made iteratively, based on past outcomes, about which test to perform next and when to stop testing. The set \hat Y is then taken to be the set of patterns that have not been ruled out by the tests performed. The total cost of a strategy is the sum of the ``testing cost'' and the ``postprocessing cost'' (proportional to |\hat Y|) and the corresponding optimization problem is analyzed.Comment: Published at http://dx.doi.org/10.1214/009053605000000174 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

High-dimensional approximate nearest neighbor: k-d Generalized Randomized Forests

We propose a new data-structure, the generalized randomized kd forest, or kgeraf, for approximate nearest neighbor searching in high dimensions. In particular, we introduce new randomization techniques to specify a set of independently constructed trees where search is performed simultaneously, hence increasing accuracy. We omit backtracking, and we optimize distance computations, thus accelerating queries. We release public domain software geraf and we compare it to existing implementations of state-of-the-art methods including BBD-trees, Locality Sensitive Hashing, randomized kd forests, and product quantization. Experimental results indicate that our method would be the method of choice in dimensions around 1,000, and probably up to 10,000, and pointsets of cardinality up to a few hundred thousands or even one million; this range of inputs is encountered in many critical applications today. For instance, we handle a real dataset of $10^6$ images represented in 960 dimensions with a query time of less than $1$sec on average and 90\% responses being true nearest neighbors