An Optimal Algorithm for the Maximum-Density Segment Problem

We address a fundamental problem arising from analysis of biomolecular sequences. The input consists of two numbers $w_{\min}$ and $w_{\max}$ and a sequence $S$ of $n$ number pairs $(a_i,w_i)$ with $w_i>0$. Let {\em segment} $S(i,j)$ of $S$ be the consecutive subsequence of $S$ between indices $i$ and $j$. The {\em density} of $S(i,j)$ is $d(i,j)=(a_i+a_{i+1}+...+a_j)/(w_i+w_{i+1}+...+w_j)$. The {\em maximum-density segment problem} is to find a maximum-density segment over all segments $S(i,j)$ with $w_{\min}\leq w_i+w_{i+1}+...+w_j \leq w_{\max}$. The best previously known algorithm for the problem, due to Goldwasser, Kao, and Lu, runs in $O(n\log(w_{\max}-w_{\min}+1))$ time. In the present paper, we solve the problem in O(n) time. Our approach bypasses the complicated {\em right-skew decomposition}, introduced by Lin, Jiang, and Chao. As a result, our algorithm has the capability to process the input sequence in an online manner, which is an important feature for dealing with genome-scale sequences. Moreover, for a type of input sequences $S$ representable in $O(m)$ space, we show how to exploit the sparsity of $S$ and solve the maximum-density segment problem for $S$ in $O(m)$ time.Comment: 15 pages, 12 figures, an early version of this paper was presented at 11th Annual European Symposium on Algorithms (ESA 2003), Budapest, Hungary, September 15-20, 200

Algorithms for the Problems of Length-Constrained Heaviest Segments

We present algorithms for length-constrained maximum sum segment and maximum density segment problems, in particular, and the problem of finding length-constrained heaviest segments, in general, for a sequence of real numbers. Given a sequence of n real numbers and two real parameters L and U (L <= U), the maximum sum segment problem is to find a consecutive subsequence, called a segment, of length at least L and at most U such that the sum of the numbers in the subsequence is maximum. The maximum density segment problem is to find a segment of length at least L and at most U such that the density of the numbers in the subsequence is the maximum. For the first problem with non-uniform width there is an algorithm with time and space complexities in O(n). We present an algorithm with time complexity in O(n) and space complexity in O(U). For the second problem with non-uniform width there is a combinatorial solution with time complexity in O(n) and space complexity in O(U). We present a simple geometric algorithm with the same time and space complexities. We extend our algorithms to respectively solve the length-constrained k maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the length-constrained $k$ maximum density segments problem in O(n min{k, U-L}) time and O(U+k) space. We present extensions of our algorithms to find all the length-constrained segments having user specified sum and density in O(n+m) and O(nlog (U-L)+m) times respectively, where m is the number of output. Previously, there was no known algorithm with non-trivial result for these problems. We indicate the extensions of our algorithms to higher dimensions. All the algorithms can be extended in a straight forward way to solve the problems with non-uniform width and non-uniform weight.Comment: 21 pages, 12 figure

Linear-Time Algorithms for Computing Maximum-Density Sequence Segments with Bioinformatics Applications

We study an abstract optimization problem arising from biomolecular sequence analysis. For a sequence A of pairs (a_i,w_i) for i = 1,..,n and w_i>0, a segment A(i,j) is a consecutive subsequence of A starting with index i and ending with index j. The width of A(i,j) is w(i,j) = sum_{i <= k <= j} w_k, and the density is (sum_{i<= k <= j} a_k)/ w(i,j). The maximum-density segment problem takes A and two values L and U as input and asks for a segment of A with the largest possible density among those of width at least L and at most U. When U is unbounded, we provide a relatively simple, O(n)-time algorithm, improving upon the O(n \log L)-time algorithm by Lin, Jiang and Chao. When both L and U are specified, there are no previous nontrivial results. We solve the problem in O(n) time if w_i=1 for all i, and more generally in O(n+n\log(U-L+1)) time when w_i>=1 for all i.Comment: 23 pages, 13 figures. A significant portion of these results appeared under the title, "Fast Algorithms for Finding Maximum-Density Segments of a Sequence with Applications to Bioinformatics," in Proceedings of the Second Workshop on Algorithms in Bioinformatics (WABI), volume 2452 of Lecture Notes in Computer Science (Springer-Verlag, Berlin), R. Guigo and D. Gusfield editors, 2002, pp. 157--17

Compaction of Quasi One-Dimensional Elastoplastic Materials

Insight in the crumpling or compaction of one-dimensional objects is of great importance for understanding biopolymer packaging and designing innovative technological devices. By compacting various types of wires in rigid confinements and characterizing the morphology of the resulting crumpled structures, here we report how friction, plasticity, and torsion enhance disorder, leading to a transition from coiled to folded morphologies. In the latter case, where folding dominates the crumpling process, we find that reducing the relative wire thickness counter-intuitively causes the maximum packing density to decrease. The segment-size distribution gradually becomes more asymmetric during compaction, reflecting an increase of spatial correlations. We introduce a self-avoiding random walk model and verify that the cumulative injected wire length follows a universal dependence on segment size, allowing for the prediction of the efficiency of compaction as a function of material properties, container size, and injection force.Comment: 7 pages, 6 figure

Coupled nonparametric shape priors for segmentation of multiple basal ganglia structures

This paper presents a new method for multiple structure segmentation, using a maximum a posteriori (MAP) estimation framework, based on prior shape densities involving nonparametric multivariate kernel density estimation of multiple shapes. Our method is motivated by the observation that neighboring or coupling structures in medical images generate configurations and co-dependencies which could potentially aid in segmentation if properly exploited. Our technique allows simultaneous segmentation of multiple objects, where highly contrasted, easy-to-segment structures can help improve the segmentation of weakly contrasted objects. We demonstrate the effectiveness of our method on both synthetic images and real magnetic resonance images (MRI) for segmentation of basal ganglia structures

High performance shape memory polyurethane synthesized with high molecular weight polyol as the soft segment

Shape memory polyurethanes (SMPUs) are typically synthesized using polyols of low molecular weight (MW~2,000 g/mol) as it is believed that the high density of cross-links in these low molecular weight polyols are essential for high mechanical strength and good shape memory effect. In this study, polyethylene glycol (PEG-6000) with MW ~6000 g/mol as the soft segment and diisocyanate as the hard segment were used to synthesize SMPUs, and the results were compared with the SMPUs with polycaprolactone PCL-2000. The study revealed that although the PEG-6000-based SMPUs have lower maximum elongations at break (425%) and recovery stresses than those of PCL-based SMPUs, they have much better recovery ratios (up to 98%) and shape fixity (up to 95%), hence better shape memory effect. Furthermore, PEG-based SMPUs showed a much shorter actuation time of < 10 s for up to 90% shape recovery compared to typical actuation times of tens of seconds to a few minutes for common SMPUs, demonstrated their great potential for applications in microsystems and other engineering components

STUDI HUBUNGAN ANTARA VOLUME, KECEPATAN DAN KEPADATAN PADA RUAS JALAN SLAMET RIYADI SAMARINDA

The movement of traffic flows on a road segment and the ability of a road segment to accommodate the flow of traffic need to be considered, and it will relate to quality and quantity of a transportation system. Volume (V), Speed (S), and Density (D) are the three main parameters that greatly affect the operational characteristics of the traffic flow. This relation between volume, speed and density can be used as a guide to determine the mathematical value of road segment for ideal conditions..............This study used comparative analysis with 3 (three) models, there are Greenshields, Greenberg and Underwood. Primary data is looking for on Slamet Riyadi Street Samarinda by survey of traffic volume and travel time on Monday, Wednesday, Saturday and Sunday from 07:00 to 09:00, at 11:00 to 13:00, and at 16:00 to 18:00 with intervals of 5 minutes every roads..............Based on the analysis and statistical test of the determinant value (R2), for the North uses Underwood relationship model with the determinant coefficient value (R2) = 0,525 which is gives the best accuracy rate of maximum speed is 21.115 km / hour, maximum density is 12,460 pcu/ km, and maximum volume is 263,101 pcu / hour. For the south uses underwood relationship model with the determinant coefficient value (R2) = 0.424 from the maximum speed is 19.707 km / hour, the maximum density is 14.293 pcu /km, and the maximum volume is 281,679 pcu / hour

Spectral densities and frequencies in the power spectrum of higher order repeat alpha satellite in human DNA molecule

Fast Fourier transform was applied to the central segment of a fully sequenced genomic segment from the centromeric region in human chromosome 7 (GenBank/AC017075.8, 193277 bp), which is characterized by alpha satellite higher order repeats (HOR). Frequencies and spectral densities were computed for all prominent peaks in the Fourier spectrum. We have additionally introduced a peak to noise ratio as effective spectral density in order to account for frequency variations of the noise level. We have shown that a very good description of computed Fourier frequencies can be obtained by using the multiple formula with the fundamental frequency corresponding to the 2734-bp HOR sequence. The peak at f(16) corresponds to the 171-bp monomer. Above the frequency f(16), the most pronounced peaks are mostly at multiples of f(16) (monomer-multiples). The lowest sixteen monomer-multiples kf(16) are locally dominant in spectral densities. The first monomer-multiple that is not locally dominant in spectral density is at k = 17. Above k = 27, the maximum of spectral density is systematically shifted to several neighboring higher frequency multiples. On the basis of the Fourier spectrum, the 171-bp monomer unit was subdivided into three approximately 57-bp subrepeats, which were further subdivided into 12-bp, 14-bp and 17-bp basic subrepeats