Number Of Shortcuts Research Articles

On Computing the k -Shortcut Fréchet Distance

The Fréchet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min–max formulation that considers all orientation-preserving bijective mappings between the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fréchet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterised version of this problem, where the number of shortcuts is bounded by a parameter \(k\) . The corresponding decision problem can be stated as follows: Given two polygonal curves \(T\) and \(B\) of at most \(n\) vertices, a parameter \(k\) and a distance threshold \(\delta\) , is it possible to introduce \(k\) shortcuts along \(B\) such that the Fréchet distance of the resulting curve and the curve \(T\) is at most \(\delta\) ? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (1) there exists a decision algorithm with running time in \(\mathcal{O}(kn^{2k+2}\log n)\) ; (2) assuming the exponential-time hypothesis (ETH), there exists no algorithm with running time bounded by \(n^{o(k)}\) . In contrast, we also show that efficient approximate decider algorithms are possible, even when \(k\) is large. We present a \((3+\varepsilon)\) -approximate decider algorithm with running time in \(\mathcal{O}(kn^{2}\log^{2}n)\) for fixed \(\varepsilon\) . In addition, we can show that, if \(k\) is a constant and the two curves are \(c\) -packed for some constant \(c\) , then the approximate decider algorithm runs in near-linear time.

Fragility and robustness of self-sustained oscillations in an epidemiological model on small-world networks.

We investigate the susceptible-infected-recovered-susceptible epidemic model, typical of mathematical epidemiology, with the diversity of the durations of infection and recovery of the individuals on small-world networks. Infection spreads from infected to healthy nodes, whose infection and recovery periods denoted by τI and τR, respectively, are either fixed or uniformly distributed around a specified mean. Whenever τI and τR are narrowly distributed around their mean values, the epidemic prevalence in the stationary state is found to reach its maximal level in the typical small-world region. This non-monotonic behavior of the final epidemic prevalence is thought to be similar to the efficient navigation in small worlds with cost minimization. Besides, pronounced oscillatory behavior of the fraction of infected nodes emerges when the number of shortcuts on the underlying network become sufficiently large. Remarkably, we find that the synchronized oscillation of infection incidences is quite fragile to the variability of the two characteristic time scales τI and τR. Specifically, even in the limit of a random network (where the amplest oscillations are expected to arise for fixed τI and τR), increasing the variability of the duration of the infectious period and/or that of the refractory period will push the system to change from a self-sustained oscillation to a fixed point with negligible fluctuations in the steady state. Interestingly, negative correlation between τI and τR can give rise to the robustness of the self-sustained oscillatory phenomenon. Our findings thus highlight the pivotal role of, apart from the external seasonal driving force and demographic stochasticity, the intrinsic characteristic of the system itself in understanding the cycle of outbreaks of recurrent epidemics.

Jaywalking Your Dog: Computing the Fréchet Distance with Shortcuts

The similarity of two polygonal curves can be measured using the Fréchet distance. We introduce the notion of a more robust Fréchet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural approach for handling noise, in particular batched outliers. We compute a $(3+\varepsilon)$-approximation to the minimum Fréchet distance over all possible such shortcuts, in near linear time, if the curve is $c$-packed and the number of shortcuts is either small or unbounded. To facilitate the new algorithm we develop several new tools: (a) a data structure for preprocessing a curve (not necessarily $c$-packed) that supports $(1+\varepsilon)$-approximate Fréchet distance queries between a subcurve (of the original curve) and a line segment; (b) a near linear time algorithm that computes a permutation of the vertices of a curve, such that any prefix of $2k-1$ vertices of this permutation forms an optimal approximation (up to a constant factor) to the original curve compared to any polygonal curve with $k$ vertices, for any $k > 0$; and (c) a data structure for preprocessing a curve that supports approximate Fréchet distance queries between a subcurve and query polygonal curve. The query time depends quadratically on the complexity of the query curve and only (roughly) logarithmically on the complexity of the original curve. To our knowledge, these are the first data structures to support these kind of queries efficiently.

Boundary Perturbation Analysis of Complex Networks

In order to clarify the relation between network structure and dynamical phenomena in networks, we investigate the first passage and first return times of a random walker and the response to a boundary perturbation in complex networks. We introduce an efficient calculation method for the first passage time distribution (FPTD) and first return time distribution (FRTD) by introducing a perfectly absorbing node. Using this method, we calculate the FPTD and FRTD in small-world networks and study the relation between these distributions and the characteristics of network structures. We find that the characteristics of the FPTD and FRTD depend on the number of shortcuts in the network, and that the mean first passage time is not determined by the shortest path length alone. We also derive the relation between the FPTD and the complex admittance in boundary perturbation experiments, and show that the Cole–Cole diagram represents characteristic behavior depending on complex networks. The characteristics of the ne...

Optimal topology for parallel discrete-event simulations

The effect of shortcuts on the task completion landscape in parallel discrete-event simulation (PDES) is investigated. The morphology of the task completion landscape in PDES is known to be described well by the Langevin-type equation for nonequillibrium interface growth phenomena, such as the Kardar-Parisi-Zhang equation. From the numerical simulations, we find that the root-mean-squared fluctuation of task completion landscape, W(t,N), scales as W(t→∞,N)~N when the number of shortcuts, ℓ, is finite. Here N is the number of nodes. This behavior can be understood from the mean-field type argument with effective defects when ℓ is finite. We also study the behavior of W(t,N) when ℓ increases as N increases and provide a criterion to design an optimal topology to achieve a better synchronizability in PDES.

Transition of Firing Patterns in a Complex Neural Network

We study the effects of random long-range connections (shortcuts) on the firing patterns in a network composed of Hindmarsh-Rose neurons. It is found that the system can achieve the transition of neural firing patterns from the lower period state to the higher one, when the number of shortcuts in the neural network is greater than a threshold, indicating that the nervous system may make the optimal response to the change of stimulation by a corresponding adjustment of the shortcuts. Then we discuss the transition degree of firing patterns of neural network and its critical characteristics for different external stimulation current. Furthermore, the influences of coupling strength on such transition behavior of neural firing patterns are also considered. Our results may be useful in comprehending the real mechanism in neural coding and information transmission in neurobiological systems.

Kinetics of diffusion-limited aggregation-annihilation processes on small-world networks

Kinetics of diffusion-limitied aggregation-annihilation processes on NW small-world networks is investigated by Monte Carlo simulation. In the system, if two clusters of the same species meet at the same node, they will aggregate and form a large one; while if two clusters of different species meet at the same node, they will annihilate each other. Simulation results show that, if the value of p (a parameter that quantifies the number of shortcuts) is large or small enough, the concentration of clusters c(t) and the concentration of particles g(t) follow power laws at large times, i.e.c(t)∝t-α and g(t)∝t－β. Moreover, the relation between the exponents α and β is found to satisfy α=2β. However, if p is of medium value, the concentration of clusters and the concentration of particles do not follow the power laws exactly. Our simulation results agree with the reported theoretical analysis very well.

A MODEL OF GRAPH COLORING DYNAMICS WITH ATTENTION WAVES AND STRATEGIC WAITING

Recently, Kearns et al. [Kearns, M., Suri, S. and Montfort, N., An experimental study of the coloring problem on human subject networks, Science313 (2006) 824–827] studied the topology dependence of graph coloring dynamics. In their empirical study, the authors analyze, how a network of human subjects acting as autonomous agents performs in solving a conflict-avoidance task (the graph coloring problem) for different network architectures. A surprising result was that the run-time of the empirical dynamics decreases with the number of shortcuts in a Watts–Strogatz small-world graph. In a simulation of the dynamics based on randomly selecting color conflicts for update, they observe a strong increase of the run-time with the number of shortcuts. Here, we propose classes of strategies, which are capable of explaining the decrease in run-time with an increasing number of shortcuts. We show that the agent's strategy, the graph topology, and the complexity of the problem (essentially given by the graph's chromatic number) interact nontrivially yielding unexpected insights into the problem-solving capacity of organizational structures.

Greedy pathlengths and small world graphs

We use matrix analysis to study a cycle plus random, uniform shortcuts—the classic small world model. For such graphs, in addition to the usual edge and vertex information there is an underlying metric that determines distance between vertices. The metric induces a natural greedy algorithm for navigating between vertices and we use this to define a pathlength. This pathlength definition, which is implicit in [J. Kleinberg, The small-world phenomenon: an algorithmic perspective, in: Proceedings of the 32nd ACM Symposium on Theory of Computing, 2000] is entirely appropriate in many message passing contexts. Using a Markov chain formulation, we set up a linear system to determine the expected greedy pathlengths and then use techniques from numerical analysis to find a continuum limit. This gives an asymptotically correct expression for the expected greedy pathlength in the limit of large network size: both the leading term and a sharp estimate of the remainder are produced. The results quantify how the greedy pathlength drops as the number of shortcuts is increased. Further, they allow us to measure the amount by which the greedy pathlength, which is based on local information, exceeds the traditional pathlength, which requires knowledge of the whole network. The analysis allows for either O(1) shortcuts per node or O(1) shortcuts per network. In both cases we find that the greedy algorithm fails to exploit fully the existence of short paths.

Thin Watts-Strogatz networks

A modified version of the Watts-Strogatz (WS) network model is proposed, in which the number of shortcuts scales with the network size N as Nalpha, with alpha < 1. In these networks, the ratio of the number of shortcuts to the network size approaches zero as N --> infinity, whereas in the original WS model, this ratio is constant. We call such networks "thin Watts-Strogatz networks." We show that even though the fraction of shortcuts becomes vanishingly small for large networks, they still cause a kind of small-world effect, in the sense that the length L of the network increases sublinearly with the size. We develop a mean-field theory for these networks, which predicts that the length scales as N1-alpha ln N for large N. We also study how a search using only local information works in thin WS networks. We find that the search performance is enhanced compared to the regular network, and we predict that the search time tau scales as N1-alpha/2. These theoretical results are tested using numerical simulations. We comment on the possible relevance of thin WS networks for the design of high-performance low-cost communication networks.

Planning open spaces for wildlife 2: modeling and verifying focal species habitat

In the face of human population growth that is transforming the Earth, scientists, land managers, and planners are working to prevent, mitigate, and reverse the consequent loss of species, ecosystems, and landscapes. Because of the need to act quickly with incomplete data, a number of shortcuts have been developed that rely on identifying key species for planning efforts. By developing conservation plans for a small set of carefully selected focal species, planners hope to create a protective umbrella for a wider array of species and functional landscapes. In an earlier paper, we described an approach for selecting a set of focal species. In this paper, we report a process for the rapid identification and verification of potential habitat for a focal species. Using the barred owl as an example, we present the process for a suburbanizing region of North Carolina, USA. The barred owl was selected to represent bottomland hardwood and forested wetland landscapes in the region. Using a geographic information system (GIS), we assembled data layers from readily available remotely sensed, conventional survey, and physiographic data to create a model of barred owl habitat. Barred owls occupy bottomland hardwood forests, which we identified using land cover, soils, and wetlands data. We eliminated from consideration bottomland forest habitat within 100 m of a road and within 60 m of open vegetative cover. Patches of the remaining bottomland forest larger than 86 ha in size were considered large enough to meet all barred owl habitat needs. Simple presence/absence surveys detected barred owls in approximately 65% of patches identified by our model as suitable habitat. We tested the barred owl’s suitability as an umbrella for bottomland forest species using an existing database of rare and outstanding elements of natural diversity. Umbrella coverage for barred owl habitat (bottomland forest patches≥86 ha) varied with taxa from 0% for invertebrate species to 75% for vertebrate species. However, umbrella coverage for all bottomland forest, including patches <86 ha, was at or near 100% for all taxa. The relatively simple modeling and verification processes we used can be carried out with a minimal amount of data and time, making it an attractive tool in situations where time and resources are in short supply.

The Trevor-Roper Trap or the Imperialism of History. An Essay

This is, as the title makes clear, an essay; that is to say, a genre in which it is considered legitimate for the author to put forward his own more or less (in this case rather more) subjective viewpoints. As such it contains quite a number of short cuts and mouthfuls. I have also deemed it necessary, for the sake of the logic of the argumentation, to make occasional and rather long de-tours via a number of obvious, and at times downright elementary, points. My excuse is that the genre virtually requires it. And my hope is that the following pages will provide at least some food for thought.Back in the early 1960s the distinguished Professor Hugh Trevor-Roper of Oxford University proclaimed, as every Africanist probably knows, that at least precolonial Black Africa had no history. He must have meant what he said, for he repeated his contention in 1969 by putting the label “unhistoric” on the African continent; the whole of the African continent that is, including Ethiopia, Egypt, and the Maghrib.On the face of it there is little reason why we should bother with this type of point of view now in the 1990s. After all, the avalanche of articles and books on African history—including several multi-volume General Histories—which have been published since the 1960s, in a sense bear testimony to the absurdity of Trevor-Roper's position.And yet, for all that, I am not quite certain that the malaise engendered by Trevor-Roper and his like has been entirely dissipated. After all, Trevor-Roper remains a frequently-quoted historian. But more to the point, there is often in my opinion a rather embarrassing insistence in the specialist Africanist litera¬ture on the “extraordinary complexity and dynamism” of Black Africa's past; an insistence not infrequently coupled with the urge, apparently never appeased, to put to rest the myth of Primitive Africa. There is also an equally embarrassing insistence on behalf of many Africanists to pin the label “state” on even the tiniest of polities in precolonial Africa, thus obscuring the appar¬ent fact that perhaps a majority of Africans in the precolonial era lived in so-called “acephalous” societies.

Shortcuts I’ve Already Found with the PRISM Service and PASSPORT Software

Suggests a number of shortcuts to help with workflow when using PASSPORT and PRISM. Describes the following functions: stacking commands with the semicolon, function key definition, use of “rep” function key definition, additional function key files, COPY and PASTE, authority file displays, and specific cursor placement.