Paths Of Equal Length Research Articles

Oddlab: fault-tolerant aware load-balancing framework for data center networks

Data center networks (DCNs) act as critical infrastructures for emerging technologies. In general, a DCN involves a multi-rooted tree with various shortest paths of equal length from end to end. The DCN fabric must be maintained and monitored to guarantee high availability and better QoS. Traditional traffic engineering (TE) methods frequently reroute large flows based on the shortest and least-congested paths to maintain high service availability. This procedure results in a weak link utilization with frequent packet reordering. Moreover, DCN link failures are typical problems. State-of-the-art approaches address such challenges by modifying the network components (switches or hosts) to discover and avoid broken connections. This study proposes Oddlab (Odds labels), a novel deployable TE method to guarantee the QoS of multi-rooted data center (DC) traffic in symmetric and asymmetric modes. Oddlab creatively builds a heuristic model for efficient flow scheduling and faulty link detection by exclusively using the gathered statistics from the DCN data plane, such as residual bandwidth and the number of installed elephant flows. Besides, the proposed method is implemented in an SDN-based DCN without altering the network components. Our findings indicate that Oddlab can minimize the flow completion time, maximize bisection bandwidth, improve network utilization, and recognize faulty links with sufficient accuracy to improve DC productivity.

Ant foraging path use responds to different types of risk and their encounter probabilities

Ant colonies are likely able to access food locations by multiple paths that can vary predictably in length or mortality risk. To favor one path over another requires ants to perceive, communicate and act upon important differences in length and risk between paths. Here, we present replicate Linepithema humile colonies with four equal-length paths to a sugar source. The paths vary in probabilities of encountering a risk cue that range from 0 to 100%. The risk cues were either live workers of an aggressive competitor (Liometopum occidentale: LO) or only formic acid (FA), a defensive chemical common in many formicine species. Both the probabilities of encounter and type of cue affected path preferences. Although across both cues the zero-risk path was most used, the use was not exclusive and patterns of response to the cues differed significantly. More ants were on all the paths when the cue was LO rather than FA, leading to a higher success rate at finding food. Worker numbers on paths with LO did decline over time as consistent with recognizing neighbors to be ‘dear enemies’ that are a reduced threat to the colony. Similarly, changes in path usage suggested that the FA cue also became viewed as less threatening over time when competitors were never simultaneously present. The results are consistent with L. humile exhibiting the behavioral plasticity and communicative ability needed to categorize and predict risk to efficiently collect food while defending against other ant species.

Throttling processes equivalent to full throttling on trees

Consider a discrete-time process on a graph G where a set B of initial vertices are chosen to be colored blue (the remainder being white) and then a time step consists of every currently blue vertex forcing all of its neighbors to become blue; this process stops when every vertex of the graph is blue, and the process is called full forcing. The full throttling number of G is then defined to be the minimum sum of the cardinality of B and the number of time steps needed to complete the forcing process. On trees, the full throttling number is equivalent to the throttling numbers of several other graph processes, such as positive-semidefinite zero forcing, the game of cops and robbers, and the distance domination number (alternately, the k-radius) of a graph. For all of these, it is known that maximum possible throttling number for a tree on n vertices is somewhere between 1.4502n and 142n, with the former exhibited by a family of spiders. After introducing some new ideas and methods for working with throttling on trees, this paper determines the exact full throttling number of all balanced spiders (trees with equal-length paths extending from a center vertex), and proves that their full throttling numbers are bounded above by that of paths of the same order n, which are known to have full throttling number ⌈2n−12⌉.

Computing Kemeny's constant for a barbell graph

In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.

Computing Kemeny's constant for a barbell graph

In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.

SimRank*: effective and scalable pairwise similarity search based on graph topology

Given a graph, how can we quantify similarity between two nodes in an effective and scalable way? SimRank is an attractive measure of pairwise similarity based on graph topologies. Its underpinning philosophy that “two nodes are similar if they are pointed to (have incoming edges) from similar nodes” can be regarded as an aggregation of similarities based on incoming paths. Despite its popularity in various applications (e.g., web search and social networks), SimRank has an undesirable trait, i.e., “zero-similarity”: it accommodates only the paths of equal length from a common “center” node, whereas a large portion of other paths are fully ignored. In this paper, we propose an effective and scalable similarity model, SimRank*, to remedy this problem. (1) We first provide a sufficient and necessary condition of the “zero-similarity” problem that exists in Jeh and Widom’s SimRank model, Li et al. ’s SimRank model, Random Walk with Restart (RWR), and ASCOS++. (2) We next present our treatment, SimRank*, which can resolve this issue while inheriting the merit of the simple SimRank philosophy. (3) We reduce the series form of SimRank* to a closed form, which looks simpler than SimRank but which enriches semantics without suffering from increased computational overhead. This leads to an iterative form of SimRank*, which requires O(Knm) time and O(n^2) memory for computing all (n^2) pairs of similarities on a graph of n nodes and m edges for K iterations. (4) To improve the computational time of SimRank* further, we leverage a novel clustering strategy via edge concentration. Due to its NP-hardness, we devise an efficient heuristic to speed up all-pairs SimRank* computation to O(Kn{tilde{m}}) time, where {tilde{m}} is generally much smaller than m. (5) To scale SimRank* on billion-edge graphs, we propose two memory-efficient single-source algorithms, i.e., ss-gSR* for geometric SimRank*, and ss-eSR* for exponential SimRank*, which can retrieve similarities between all n nodes and a given query on an as-needed basis. This significantly reduces the O(n^2) memory of all-pairs search to either O(Kn + {tilde{m}}) for geometric SimRank*, or O(n + {tilde{m}}) for exponential SimRank*, without any loss of accuracy, where {tilde{m}} ll n^2. (6) We also compare SimRank* with another remedy of SimRank that adds self-loops on each node and demonstrate that SimRank* is more effective. (7) Using real and synthetic datasets, we empirically verify the richer semantics of SimRank*, and validate its high computational efficiency and scalability on large graphs with billions of edges.

Same length, different shapes: ants collectively choose a straight foraging path over a bent one.

In socials insects, exploration is fundamental for the discovery of food resources and determines decision-making. We investigated how the interplay between the physical characteristics of the paths leading to food sources and the way it impacts the behaviour of individual ants affects their collective decisions. Colonies of different sizes of Lasius niger had access to two equal food sources through two paths of equal length but of different geometries: one was straight between the nest and the food source, and the other involved an abrupt change of direction at the midway point (135°). Both food sources were discovered simultaneously, but the food source at the end of the straight path was preferentially exploited by ants. Based on experimental and theoretical results, we show that a significantly shorter duration of nestbound travel on the straight path, which rapidly leads to a stronger pheromone trail, is at the origin of this preference.

Robust transport over networks.

We consider transportation over a strongly connected, directed graph. The scheduling amounts to selecting transition probabilities for a discrete-time Markov evolution which is designed to be consistent with initial and final marginal constraints on mass transport. We address the situation where initially the mass is concentrated on certain nodes and needs to be transported in a certain time period to another set of nodes, possibly disjoint from the first. The random evolution is selected to be closest to a prior measure on paths in the relative entropy sense-such a construction is known as a Schrödinger bridge between the two given marginals. It may be viewed as an atypical stochastic control problem where the control consists in suitably modifying the prior transition mechanism. The prior can be chosen to incorporate constraints and costs for traversing specific edges of the graph, but it can also be selected to allocate equal probability to all paths of equal length connecting any two nodes (i.e., a uniform distribution on paths). This latter choice for prior transitions relies on the so-called Ruelle-Bowen random walker and gives rise to scheduling that tends to utilize all paths as uniformly as the topology allows. Thus, this Ruelle-Bowen law (𝔐RB) taken as prior, leads to a transportation plan that tends to lessen congestion and ensures a level of robustness. We also show that the distribution 𝔐RB on paths, which attains the maximum entropy rate for the random walker given by the topological entropy, can itself be obtained as the time-homogeneous solution of a maximum entropy problem for measures on paths (also a Schrödinger bridge problem, albeit with prior that is not a probability measure). Finally we show that the paradigm of Schrödinger bridges as a mechanism for scheduling transport on networks can be adapted to graphs that are not strongly connected, as well as to weighted graphs. In the latter case, our approach may be used to design a transportation plan which effectively compromises between robustness and other criteria such as cost. Indeed, we explicitly provide a robust transportation plan which assigns maximum probability to minimum cost paths and therefore compares favourably with Optimal Mass Transportation strategies.

Shortest path in a multiply-connected domain having curved boundaries

Computing the shortest path, overcoming obstacles in a plane, is a well-known geometric problem. However, widely assumed obstacles are polygonal in nature. Very few papers have focused on curved obstacles, and in particular, for curved multiply-connected domains (domains having holes). Given a set of parametric curves forming a multiply-connected domain (MCD), with one closed curve acting as an outer boundary and several non-intersecting inner curves (loops) as representing holes and two distinct points S and E lying on the outer boundary, this paper provides an algorithm to find the shortest interior path (SIP) between the two points in the domain. SIP consists of portions of curves along with straight line segments that are tangential to the curve. The algorithm initially computes point–curve tangents (PCTs) and bitangents (BTs) using their respective constraints. A generalized region lemma is proposed, which is then employed to remove PCTs/BTs that will not contribute to the potential path and subsequently aiding in removing a few inner loops. The algorithm is designed to explore all potential paths. Merging of paths is also proposed to avoid redundant computations. A final SIP is chosen from potential paths using the length of each path. The algorithm also has the potential to give all paths of equal lengths contributing to shortest paths (within a tolerance level). As the algorithm computes all the potential paths on the fly, there is no need to employ a visibility graph to compute the shortest path. Curves are represented using non-uniform rational B-splines (NURBS). The algorithm uses the curves as such and does not discretize into point-sets or polygons. Extensions to handle curves with C1 discontinuities and S and E not on the outer boundary have also been described. Results of the implementation are provided and complexity of the algorithm is also discussed. This paper follows up the one presented for simply-connected domains (SCDs) in Bharath Ram and Ramanathan (2011) [4].

Coherent backscattering enhancement in cavities: The simple-shape cavity revisited

Coherent backscattering enhancement (CBE) was introduced in disordered, random or chaotic media. In this study, our attention is focused on simple parallelepipedic cavities, as contrary to the widespread opinion, CBE can also be observed for a pure-tone source in a one-dimensional (1-D) cavity. This approach is of two-fold interest. First, analytical calculus predicts a dimensional dependence of the CBE according to a R = 3 2 d law, with d = 1,2,3 as the dimensionality of the cavity; these computations have not yet been compared to experiments. Secondly, a ray approach in simple-shape cavities confirms the modal theory and opens new ballistic interpretations, for which each multiply reverberated path is not necessarily associated with one single reciprocal counterpart. The number of associated equal length paths ranges from zero to four in rectangular cavities.

Trail formation in ants. A generalized Polya urn process

Faced with a choice of paths, an ant chooses a path with a higher concentration of pheromone. Subsequently, it drops pheromone on the path chosen. The reinforcement of the pheromone-following behavior favors the selection of an initially discovered path as the preferred path. This may cause a long path to emerge as the preferred path, were it discovered earlier than a shorter path. However, the shortness of the shorter path offsets some of the pheromone accumulated on the initially discovered longer path. In this paper, we model the trail formation behavior as a generalized Polya urn process. For k equal length paths, we give the distribution of pheromone at any time and highlight its sole dependence on the initial pheromone concentrations on paths. Additionally, we propose a method to incorporate the lengths of paths in the urn process and derive how the pheromone distribution alters on its inclusion. Analytically, we show that it is possible, under certain conditions, to reverse the initial bias that may be present in favor of paths that were discovered prior to the discovery of more efficient (shorter) paths. This addresses the Plasticity-Stability dilemma for ants, by laying out the conditions under which the system will remain stable or become plastic and change the path. Finally, we validate our analysis and results using simulations.

Using Time-Reversal Symmetry for Sensitive Incoherent Matter-Wave Sagnac Interferometry

We present a theory of the transmission of guided matter-waves through Sagnac interferometers. Interferometer configurations with only one input and one output port have a property similar to the phase rigidity observed in the transmission through Aharonov-Bohm interferometers in coherent mesoscopic electronics. This property enables their operation with incoherent matter-wave sources. High rotation sensitivity is predicted for high finesse configurations.

Differential Geometric Path Planning of Multiple UAVs

Safe and simultaneous arrival of constant speed, constant altitude unmanned air vehicles (UAVs) on target is solved by design of paths of equal lengths. The starting point for our solution is the well-known Dubins path, which is composed of circular arc and line segments, thus requiring only one simple maneuver—constant rate turn. An explicit bound can be imposed on the rate during the design and the resulting paths are the minimum time solution of the problem. However, transition between arc and line segments entails discontinuous changes in lateral acceleration (latax), making this approach impractical for real fixed wing UAVs. Therefore, we replace the Dubins solution with a novel one, based on quintic Pythagorean hodograph (PH) curves, whose latax demand is continuous. The PH paths are designed to have lengths close to the lengths of the Dubins paths to stay close to the minimum time solution. To derive the PH paths, the Dubins solution is first interpreted in terms of differential geometry of curves using the path length and curvature as the key parameters. The curvature of a Dubins path is a piecewise constant and discontinuous function of its path length, which is a differential geometric expression of the discontinuous latax demand involved in transitions between the arc and the line segment. By contrast, the curvature of the PH path is a fifth order polynomial of its path length. This is not only continuous but also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and make the PH solution close to the minimum time one. The solution involves the design of paths meeting the curvature constraint and is followed by producing multiple paths of equal length by increasing the lengths of the shorter paths to match the longest one. The safety constraint, intercollision avoidance is achieved by satisfying two conditions: minimum separation distance and nonintersection at equal distance. The offset curves of the PH path are used to design a safety region along each path.

PATH PLANNING OF MULTIPLE UAVS WITH CLOTHOID CURVES IN TWO DIMENSIONS

This paper presents a design of the shortest path in the plane (2D) formed by clothoid arcs with straight line segments employing the principles of differential geometry. This path is used in path planning of multiple UAVs solving simultaneous arrival on target. The simultaneous arrival is achieved by the design of paths of equal length. The flight paths satisfy (i) constraint on maximum curvature bound and (ii) safety constraints to avoid collisions. The first constraint is met by designing flyable paths and the latter is met by varying curvature of the clothoid arc segments. The paths of equal length is generated by increasing the lengths of the shorter paths to equal to the longest one.

Spanning multiaccess channel hypercube computer interconnection

A distributed computer system based on a hypercube topology is proposed, where multiaccess channels spanning all dimensional axes provide processor interconnection. The multiprocessor system can be built at relatively low cost by combining the recent advances in high-capacity channels available through fiber optics, and demand-assignment multiple-access protocols, creating a highly fault-tolerant system with good average packet delay and average distance characteristics, at a significantly reduced cost. A totally distributed system has been achieved in which the packets are self-routing, able to take advantage of the multiple paths of equal length between two modes. The proposed structure gives a highly fault tolerant system, degrading gracefully in the presence of failures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Noncoherent Microholography Using A Holographic Optical Element As A Beamsplitter

A transmission holographic microscope is described. The technique consists of using a suitable holographic beamsplitter associated with the condenser of the microscope. An imaging system ensuring optical paths of equal length from the beamsplitter to the recording plane allows the use of this technique in nonmonochromatic light (spectral or arc lamps).

Joint statistics of rain attenuation on terrestrial and earth‐space propagation paths

Radar reflectivity data from the summer of 1975 were used to simulate the rain attenuation at 13 GHz on a network consisting of five earth terminals and six interconnecting terrestrial links. As rain echoes passed over the complex of terrestrial and earth‐space propagation paths, the attenuation over each of the eleven paths was calculated as a function of time with a resolution of a few minutes. Thirty‐four hours of data from twelve days were analyzed statistically to determine attenuation distributions for all paths in the network, separately and in various combinations. The five earth‐space paths were parallel, all at 18.5° elevation and 121.5° azimuth. The average attenuation distribution for these paths was essentially the same as that reported by Hodge for a network of eleven terminals located 35 km to the south of our network. Moreover, an analysis of the diversity gain achieved by using pairs of earth‐space paths revealed a dependence on terminal separation distance that was not significantly different from that of Hodge. A study of the attenuation distributions for the six terrestrial links of the network showed primarily a dependence on path length. For paths of equal length, however, moderate values of attenuation tended to occur less frequently on paths oriented perpendicular to the average direction of storm motion. Analyzed from the viewpoint of route diversity, data for the terrestrial links showed that the frequency of outages over a 40 km path could be reduced by an order of magnitude by adding a repeater along the route. Earth stations, defined as consisting of a main and an alternate earth terminal plus the terrestrial link between them, were found to have a reliability determined primarily by joint outages on the two earth‐space paths. Exceptions were cases with (1) baselines longer than about 30 km, (2) a low reference fade margin on the terrestrial link, (3) a terrestrial path leaving the main terminal in the same direction as the satellite azimuth. In these cases outages on the terrestrial link could further limit the overall system reliability.

Pairs of One Dimensional Random Walk Paths

We first show a way of constructing a path of length $2n$ from a pair of paths of length $n$ by means of which one may arrive at many results on pairs of paths of length $n$, simply by examining properties of paths of length $2n$. Secondly, for two random walk paths of length $n, A$ and $B$, with vertical coordinates $A(i)$ and $B(i)$ respectively, at times $i = 0,1,\cdots, n$, and such that for some $m A(m) > B(m)$ but $A(i) = B(i)$ when $i < m$, we define $d_{A,B}(i) = \frac{1}{2}(A(i) - B(i))$. For obvious reasons $A(i) - B(i)$ is always even, which incidentally, implies that the intersection of two paths are points with integral coordinates. We find that $d_{A,B}$ can be graphed against time by a three-valued random walk path, i.e. a path which may have horizontal steps. Questions about the pair consisting of $A$ and $B$ may then be answered by observing the path described by $d_{A,B}$. Results in the theory of three-valued random walk paths can thus be translated into results about pairs of random walk paths of equal length.

Reasoning in the Hooded Rat

A problem situation was devised to test Deutsch's (1956) theory of reasoning. In this situation various problems can be set the rat and these problems admit of various equally rational solutions. Predictions from the theory were made, not only that the animals would solve the problems, but also which particular solution they would adopt. These predictions are confirmed. The rats were trained to run to two goal-boxes, one near, the other more distant. The near one had two paths of equal length leading to it, the more distant goal only one. After training, the animals chose the near goal-box on the first trial of the day. On the test trials the near goal-box was left empty, or the chosen path blocked, or a mesh placed across the goal-box, with the food accessible via the other short path.

X-ray photography of zero-order reciprocal net planes of a crystal

Summary A method is described for obtaining an image of the zero-order reciprocal net plane without any distortion of the picture. This may be important, as all rays reflected by the crystal travel to the film along paths of equal length. Apart from the fact, that they reach the film at a constant angle, which can be adjusted so as to suit any special demands, the intermediate distances between the reciprocal net points on the photography can also be varied. These features render the method useful for the comparison of intensities. For research work on fibres it may also be of some use.