Algorithm Of Linear Time Complexity Research Articles

Efficient computations with counting functions on free groups and free monoids

We present efficient algorithms to decide whether two given counting functions on nonabelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on nonabelian free groups are cohomologous. We work in the multi-tape Turing machine model with nonconstant-time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, that is, it is the same as in addition of rational numbers (implemented using the Harvey-van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions. Bibliography: 20 titles.

Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment.

Point cloud (PC)-a collection of discrete geometric samples of a 3D object's surface-is typically large, which entails expensive subsequent operations. Thus, PC sub-sampling is of practical importance. Previous model-based sub-sampling schemes are ad-hoc in design and do not preserve the overall shape sufficiently well, while previous data-driven schemes are trained for specific pre-determined input PC sizes and sub-sampling rates and thus do not generalize well. Leveraging advances in graph sampling, we propose a fast PC sub-sampling algorithm of linear time complexity that chooses a 3D point subset while minimizing a global reconstruction error. Specifically, to articulate a sampling objective, we first assume a super-resolution (SR) method based on feature graph Laplacian regularization (FGLR) that reconstructs the original high-res PC, given points chosen by a sampling matrix H. We prove that minimizing a worst-case SR reconstruction error is equivalent to maximizing the smallest eigenvalue λmin of matrix HT H+ μL, where L is a symmetric, positive semi-definite matrix derived from a neighborhood graph connecting the 3D points. To arrive at a fast algorithm, instead of maximizing λmin, we maximize a lower bound λ-min(HT H+ μL) via selection of H-this translates to a graph sampling problem for a signed graph G with self-loops specified by graph Laplacian L. We tackle this general graph sampling problem in three steps. First, we approximate G with a balanced graph GB specified by Laplacian LB. Second, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform Lp = SLB S-1, so that all Gershgorin disc left-ends of Lp are aligned exactly at λmin(LB). Finally, we choose samples on GB using a previous graph sampling algorithm to maximize λ-min(HT H+ μLp) in linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in reconstruction quality.

Large-Scale Robust Beamforming via $\ell _{\infty }$ -Minimization

In this paper, linearly constrained and robust $\ell _{\infty }$ -norm beamforming techniques are proposed for non-Gaussian signals. A conventional approach for $\ell _{\infty }$ -minimization needs to solve a linear programming (LP) or second-order cone programming (SOCP). However, this strategy is computationally prohibitive for “big data” because the existing algorithms for LP or SOCP, such as simplex method or interior point method, can only solve small- or medium-scale problems. In this paper, the alternating direction method of multipliers (ADMM) is devised for large-scale $\ell _{\infty }$ -beamforming problems, where the core subproblems can be formulated concisely as a linearly or second-order cone constrained least squares and the proximity operator of the $\ell _{\infty }$ -norm in each iteration. Remarkably, a linear-time complexity algorithm is devised that efficiently computes the $\ell _{\infty }$ -norm proximity operator. Simulation results verify the high efficiency of the ADMM and the superiority of the $\ell _{\infty }$ -norm beamforming techniques over several representative beamformers, indicating that its performance can approach the optimal upper bound.

FatRegion: A Fast Adaptive Tree-Structured Region Extraction Approach

Coherent image regions can be used as good features for many computer vision tasks, such as object tracking, segmentation, and recognition. Most of previous region extraction methods, however, are not suitable for online applications because of their either heavy computations or unsatisfactory results. We propose a seed-based region growing and merging approach to generate simultaneously coherent and discriminative image regions. We present a quadtree-based seed initialization algorithm to adaptively place seeds into different image areas and then grow them into regions by a color- and edge-guided growing procedure. To merge these regions in different levels, we propose to use the generalized boundary strength to measure the quality of region merging result. In addition, we present a region merging algorithm of linear time complexity to perform efficient and effective region merging. Overall, our new approach simultaneously holds these advantages: 1) it is extremely fast with linear complexity in both time and space, which takes less than 50 ms to process an HVGA image; 2) it can give a direct control of the region number and well adapt to image regions with various sizes and shapes; and 3) it provides a tree-structured representation of the regions and thus can model the image from multiple scales. We evaluate the proposed approach on the standard benchmarks with extensive comparisons with the state-of-the-art methods. The experimental results demonstrate its good comprehensive performances. Example applications using the extracted regions as features for online object tracking and multiclass object segmentation also exhibit its potential for many computer vision tasks.

Adaptive thresholding with inverted triangular area for real-time detection of the heart rate from photoplethysmogram traces on a smartphone.

Photoplethysmogram (PPG) signals acquired by smartphone cameras are weaker than those acquired by dedicated pulse oximeters. Furthermore, the signals have lower sampling rates, have notches in the waveform and are more severely affected by baseline drift, leading to specific morphological characteristics. This paper introduces a new feature, the inverted triangular area, to address these specific characteristics. The new feature enables real-time adaptive waveform detection using an algorithm of linear time complexity. It can also recognize notches in the waveform and it is inherently robust to baseline drift. An implementation of the algorithm on Android is available for free download. We collected data from 24 volunteers and compared our algorithm in peak detection with two competing algorithms designed for PPG signals, Incremental-Merge Segmentation (IMS) and Adaptive Thresholding (ADT). A sensitivity of 98.0% and a positive predictive value of 98.8% were obtained, which were 7.7% higher than the IMS algorithm in sensitivity, and 8.3% higher than the ADT algorithm in positive predictive value. The experimental results confirmed the applicability of the proposed method.

A novel method for performance evaluation of text chunking

Evaluation of text chunking is revisited. The proposed method tries to analyze the errors made by a chunker and formulates an evaluation strategy that brings out the strength and weakness of a chunker in a better way than the existing precision, recall and F score based methods or their variants do. A tree-matching based algorithm of linear time complexity is designed, analyzed, and illustrated by giving examples. Correctness of the algorithm is checked by using a chunker and a set of test sentences.

Algorithms for connected component labeling based on quadtrees

AbstractAn algorithm of linear time complexity is presented to label connected components of a binary image by a quadtree. For a given node, the search for all adjacent nodes is carried out in O(1) (i.e., constant time complexity for the worst case) using our formerly presented algorithm in (Aizawa et al., 3rd International Symposium on Communications, Control, and Signal Processing,2008, 505–510), whereas it explores all possible adjacencies for each node in a usual way. Then during the process of tree formulation in the search, all equivalent relations of labels are stored as lists. Time complexity of the algorithm is O(B+W) for the worst case and its auxiliary space is no more than O(B), where B and W correspond to the number of leaf nodes in a quadtree representing black and white quadrants, respectively. Empirical tests of the algorithm are employed in comparison with another linear time connected component labeling algorithm based on top‐down quadtree traversal algorithm (Samet, IEEE Trans Pattern Anal Mach Intell PAMI‐7 (1985), 94–98), as well as traditional row‐by‐row scanning algorithm using linear time Union‐Find (Fiorio and Gustedt, Theor Comput Sci 154 (1996), 165–181). Our algorithm has shown the best performance in large images. © 2009 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 19, 158–166, 2009.

Iterative Converging Algorithms for Computing Bounds on Durations of Activities in Pert and Pert-Like Models

Since its invention in 1958, Program Evaluation and Review Technique (PERT) has been widely used during the planning, design, and implementation of projects. Pert models the activities of a project as a single source-single sink directed acyclic graph where nodes represent events (end or beginning of activities) and arcs activities. The maximum amount by which an activity can be delayed without delaying the overall project is called the slack. Critical tasks have zero slack whereas all noncritical tasks have positive slacks. Pert is a valuable tool in the management of large projects since it allows to compute the slack of each activity of the project. Such information may be crucial in avoiding cost overruns that would be caused by delays to critical activities and/or excessive delays to noncritical activities. What Pert fails to provide is how one should go about distributing remaining slack on noncritical activities while taking into consideration properties of the activities as well as precedence relationships among them, so as to have reasonable upper bounds on duration of all activities, critical or noncritical. In this paper we propose several algorithms for the distribution of slack on non-critical activities. We show that if one desires to distribute the remaining slack proportionally to the initially assigned activity durations then the problem is in P, and propose an algorithm of linear time complexity. However if one desires to use distribution functions other than the initial durations of activities, then the problem of slack distribution becomes NP-complete. Finding the maximal bounds corresponding to zero-slack solution at the sink requires iterative application of exponential algorithm. For that case we introduce an approximation algorithm of linear time complexity on each iteration. The algorithm iteratively increases bounds on durations of activities and converges to the zero-slack solution on all paths from the source node to the sink node in the Pert-like graph. The algorithms described in this paper were successfully applied to solving timing bounds problems in VLSI design.

An Efficient Algorithm for Topologically Correct Segmentation of the Cortical Sheet in Anatomical MR Volumes

Polygon-mesh representations of the cortices of individual subjects are of anatomical interest, aid visualization of functional imaging data and provide important constraints for their statistical analysis. Due to noise and partial volume sampling, however, conventional segmentation methods rarely yield a voxel object whose outer boundary represents the folded cortical sheet without topological errors. These errors, called handles, have particularly deleterious effects when the polygon mesh constructed from the segmented voxel representation is inflated or flattened. So far handles had to be removed by cumbersome manual editing, or the computationally more expensive method of reconstruction by morphing had to be used, incorporating the a priori constraint of simple topology into the polygon-mesh model. Here we describe a linear time complexity algorithm that automatically detects and removes handles in presegmentations of the cortex obtained by conventional methods. The algorithm's modifications reflect the true structure of the cortical sheet. The core component of our method is a region growing process that starts deep inside the object, is prioritized by the distance-to-surface of the voxels considered for inclusion and is selftouching-sensitive, i.e., voxels whose inclusion would add a handle are never included. The result is a binary voxel object identical to the initial object except for “cuts” located in the thinnest part of each handle. By applying the same method to the inverse object, an alternative set of solutions is determined, correcting the errors by addition instead of deletion of voxels. For each handle separately, the solution more consistent with the intensities of the original anatomical MR scan is chosen. The accuracy of the resulting polygon-mesh reconstructions has been validated by visual inspection, by quantitative comparison to an expert's manual corrections, and by crossvalidation between reconstructions from different scans of the same subject's cortex.

An optimal algorithm for solving the 1-median problem on weighted 4-cactus graphs

The median problem has been extensively studied in the last three decades. On general graphs, the problem can be solved in O( n 3), where n is the number of vertices in a graph. For tree networks, however, more efficient algorithms can be devised to find the medians in O( n). In this paper, we study weighted 4-cactus (W4C) graphs which provide a less restrictive network structure than trees and show that median problem can be solved just as efficiently on W4C graphs as on trees. Many examples are provided demonstrating the rudiments of our algorithm and a linear time complexity algorithm is developed.

A linear algorithm to decompose inheritance graphs into modules

Inheritance graphs of object-oriented languages can be decomposed into independent subgraphs, or modules, which are inheritance graphs themselves. This paper is devoted to the study of efficient algorithms of decomposition into modules similar to substitution decomposition algorithms. For 2-connected inheritance graphs we search for maximal modules (under inclusion), whereas we show that the significant modules of non-2-connected graphs are the biconnected components. For the 2-connected case, a method based upon a property of greedy linear extensions is proposed. However, in both cases a linear-time complexity algorithm is provided to decompose into modules. Furthermore, the algorithm can be inserted into the inheritance mechanism (such as in CLOS [14]).

New algorithm for testing random access memories

A linear time complexity algorithm to test RAMs is presented. The algorithm requires only 7n read/write operations and provides an extensive fault coverage. The main advantage of the proposed scheme is that a small test time can be achieved.

Kürzeste Streckenzüge im Raum

A classical problem of geometry is the following: given a convex polygon in the plane, find an inscribed polygon of shortest circumference. In this paper we generalize this problem to arbitrary polygonal paths in space and consider two cases: in the “open” case the wanted path of shortest length can have different start and end point, whereas in the “closed” case these two points must coincide. We show that finding such shortest paths can be reduced to finding a shortest path in a planar “channel”. The latter problem can be solved by an algorithm of linear-time complexity in the open as well in the closed case. Finally, we deal with constrained problems where the wanted path has to fulfill additional properties; in particular, if it has to pass straight through a further point, we show that the length of such a constrained polygonal path is a strictly convex function of some angle α, and we derive an algorithm for determining such constrained polygonal paths efficiently.