Total Number Of Paths Research Articles

On Leech labelings of graphs and some related concepts

Let G=(V,E) be a graph and let f:E→{1,2,3,…} be an edge labeling of G. The path weight of a path P in G is the sum of the labels of the edges of P and is denoted by w(P). The path number of G, tp(G) is the total number of paths in a graph G. If the set of all path weights S in G with respect to the labeling f is {1,2,3,…,tp(G)}, then f is called a Leech labeling of G. A graph which admits a Leech labeling is called a Leech graph. Leech index is a parameter which evaluates how close a graph is towards being Leech. In this paper, the path number of the wheel graph Wn is obtained. We also determine a bound for the Leech index of Wn and a subclass of unicyclic graphs. A python program that gives all possible Leech labelings of a cycle Cn for n≥3, if it exists, is also provided.

Channel reconstruction for mmWave Massive MIMO systems based on Channel Path Map

For the needs of high data rate and massive device access in current fifth-generation (5G) mobile communication networks and future sixth-generation (6G) mobile communication networks, Millimeter-Wave (mmWave) Massive Multiple-Input Multiple-Output (MIMO) is a crucial technical support. However, the conventional pilot based channel estimation methods are exceedingly time-consuming and substantially degrade system performance when the number of users or antennas is high. This paper proposes a Path Clustering based Channel Reconstruction (PCCR) scheme for channel reconstruction in order to solve this issue. By leveraging channel information (such as angle of arrival, angle of departure, and path gain) at a number of geographical locations that are recorded in the Channel Path Map (CPM), we aim to reconstruct the mmWave channel of any locations of interests. To be specific, the CPM is used to extract the neighborhood propagation path data, and an autonomous determination algorithm is used to find the number of path clusters at the location of interest. The recorded propagation path information is processed by the Equivalent Base Station (EBS) scheme, and the path clustering is accomplished by using the AGglomerative NESting (AGENS) algorithm. In each path cluster, Inverse Distance Weighted (IDW) algorithm is developed to forecast the path information at the user location. With all path information at user location are forecast, the channel can be reconstructed. The anticipated overhead is substantially lower than that of conventional methods because the computational cost of the procedure only depends on the total number of paths at the nearby locations. Ray-tracing based simulations are carried out to demonstrate the performance of the proposed algorithm. The results show that, when applied to a CPM with a spatial location precision of 5 m, the Normalized Mean Squared Error (NMSE) of the proposed algorithm is nearly identical to that of Linear Minimum Mean Square Error (LMMSE) estimation. The actual achievable date rate can match the needs of upcoming wireless communications and is superior over the conventional LMMSE.

Linguistic and nonlinguistic evaluation of motion events in a path-focused language

AbstractThis study examines how properties of path (the trajectory of motion) and manner (how an action is performed) components of motion events are reflected in linguistic and nonlinguistic motion event conceptualization in a path-focused language, Turkish. In two experiments, we investigated how path and manner differed in salience (i.e., prominence) and ease of expression (EoE, i.e., effort of describing), and how these factors were related to lexicalization and similarity judgments of motion events. In Experiment 1, participants rated motion events based on path and manner salience and EoE and expressed path and manner in a written format. Results indicated that manner was rated as more salient and path as easier to express. Path salience and EoE were related to both types (i.e., number of different expressions) and the total number of paths and manners used. However, manner EoE but not salience was associated with only types and the total number of manners used. In Experiment 2, participants rated the similarity of motion event pairs created using the ratings in Experiment 1. We found that higher manner salience and EoE difference were associated with lower similarity ratings. These findings suggest that salience and EoE of path and manner are related to both linguistic and nonlinguistic aspects of motion event conceptualization.

Two extensions of Leech labeling to the class of all graphs

Let be a tree of order n and let be an injective edge labeling of T. The weight of a path P is the sum of the labels of the edges of P and is denoted by If the set of weights of the paths in T is then f is called a Leech labeling of T and a tree which admits a Leech labeling is called a Leech tree. In this paper, we present two extensions of Leech labeling to the class of all graphs, namely, Leech labeling and geodesic Leech labeling. For this purpose we introduce two parameters and where is the total number of paths in G and is the total number of geodesic paths in G. We determine the values of these parameters for some graph classes. We present several families of graphs which are Leech graphs, geodesic Leech graphs and also graphs which are not Leech graphs. A list of open problems is also included.

Homotopic [formula omitted]-oriented routing with few links and thick edges

We study the NP-hard optimization problem of finding non-crossing thick C-oriented paths that are homotopic to a set of input paths in an environment with C-oriented obstacles, with the goal to minimize the total number of links of the paths. We introduce a special type of C-oriented paths—smooth paths—and present a 2-approximation algorithm for smooth paths that runs in O(n3log⁡κ+kinlog⁡n+kout) time, where n is the total number of paths and obstacle vertices, kin and kout are the total complexities of the input and output paths, and κ=|C|. The algorithm also computes an O(κ)-approximation for general C-oriented paths. In particular we give a 2-approximation algorithm for rectilinear paths. Our algorithm not only approximates the minimum number of links, but also simultaneously minimizes the total length of the paths. As a related result we show that, given a set of (possibly crossing) C-oriented paths with a total of L links, non-crossing C-oriented paths homotopic to the input paths can require a total of Ω(Llog⁡κ) links.

Methods for removing links in a network to minimize the spread of infections

Minimizing the spread of infections is a challenging problem, and it is the subject matter in many different fields such as epidemiology and cyber-security. In this paper, we investigate link removal as an intervention strategy and study the relative effectiveness of different link removal methods in minimizing the spread of infections in a network. With that in mind, we develop four connectivity-based network interdiction models and formulate these models as mixed integer linear programs. The first model minimizes the number of connections between infected and susceptible nodes; the second the number of susceptible nodes having one or more connections with infected nodes; the third the total number of paths between infected and susceptible nodes; and the fourth the total weight of the paths between infected and susceptible nodes. We also propose heuristic algorithms to solve the models. The network interdiction models act as link removal methods, i.e., each return a solution consisting of a set of links to remove in the network. We compare the effectiveness of these four methods with the effectiveness of an existing link removal method [25], a method based on link betweenness centrality [18], and random link removal method. Our results show that complete isolation of susceptible nodes from infected nodes is the most effective method in reducing the average number of new infections (reduce occurrence) under most scenarios, and the relative effectiveness of the complete isolation method increases with transmission probability. In contrast, removing the highest probability transmission paths is the most effective method in increasing the average time to infect half of the susceptible nodes (reduce speed) under most scenarios, and the relative effectiveness of this method decreases with transmission probability.

On the hardness of network design for bottleneck routing games

In routing games, the selfish behavior of the players may lead to a degradation of the network performance at equilibrium. In more than a few cases however, the equilibrium performance can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braessʼs paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve their equilibrium performance by edge removal. In this work, we investigate the computational complexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network, i.e. the maximum latency of a used edge. We first show that bottleneck routing games do not suffer from Braessʼs paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA cannot be improved by edge removal, even if their PoA is as large as Ω(n0.121). This implies that the network design problem for linear bottleneck routing games is NP-hard to approximate within a factor of O(n0.121−ε), for any constant ε>0. The proof is based on a recursive construction of hard instances that carefully exploits the properties of bottleneck routing games, and may be of independent interest. On the positive side, we present an algorithm for finding a subnetwork that is almost optimal with respect to the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. We show that the running time is essentially determined by the total number of paths in the network, and is quasipolynomial when the number of paths is quasipolynomial.

A Note on Observables for Counting Trails and Paths in Graphs

We point out that the total number of trails and the total number of paths of given length, between two vertices of a simple undirected graph, are obtained as expectation values of specifically engineered quantum mechanical observables. Such observables are contextual with some background independent theories of gravity and emergent geometry. Thus, we point out yet another situation in which the mathematical formalism of a physical theory has some computational aspects involving intractable problems.

MIMO frequency-selective channels with multiple-frequency offsets: estimation and detection techniques

Channel gains and frequency offsets are estimated in a multiple input and multiple output (MIMO) scenario, allowing for more than one path between the transmit and the receive antennas, with each path possibly having a different frequency offset. Such a scenario may arise due to Doppler shift associated with various multipaths. Here, the general case is examined and an approximate maximum likelihood (AML) estimator is derived by exploiting the correlation property of the pilot signal. The AML estimator can be written as the sum of nTL one-dimensional maximisations, where nT is the number of transmit antennas and L is the total number of paths. The proposed estimator is validated by comparing the results with the Cramér–Rao lower bound (CRLB). Further, a minimum mean square error recursive (MMSER) equaliser is proposed for the considered scenario. Finally, the performance of the MMSER equaliser is compared with the equaliser not accounting for frequency offsets.

Blind reception of multicarrier ds-cdma using antenna arrays

An asynchronous, multicarrier direct-sequence code-division multiple-access (DS-CDMA) array receiver is proposed based upon blind, composite channel vector estimation. Due to the fact that the combined effect of all received paths is estimated, the total number of paths can be greater than the number of antennas in the receiver array. Furthermore, the proposed approach is not limited by multipath coherency. The receiver is classified as an interference cancelling zero-forcing receiver, and multicarrier operation in a frequency-selective channel means that its performance can exceed that of a single-carrier array DS-CDMA. The effectiveness of the proposed approach, even in the presence of strong interference, is demonstrated by computer simulation studies.

Distance-related indexes in the quantitative structure-property relationship modeling.

A comparative study of structure-boiling point modeling for a set of 180 acyclic and cyclic hydrocarbons (DS-180) and two of its subsets (one containing a selection of 76 acyclic and cyclic alkanes (DS-76), and the other containing 104 (DS-104) mono- and polycyclic butanes through octanes) using several known and novel distance-related indices is reported. The distance-related indices used were as follows: Wiener index, hyper-Wiener index, detour index, hyper-detour index, Harary index, Pasaréti index, Vérhalom index, Wiener-sum index, inverse Wiener-sum index and the product-form version of the Wiener index. Additional indices used were the total number of paths, the Hosoya Z index, the total walk count index, the number of carbon atoms, and the number of rings in the hydrocarbon. The best models for predicting the boiling points of 76, 104, and 180 acyclic and cyclic alkanes contain the natural logarithm of the cross-products of the Hosoya and detour index and of the Pasaréti index and the number of rings. This result extends earlier work by us and Rücker and Rücker on the use of the Wiener, detour, and Hosoya indices in modeling boiling points of alkanes and cycloalkanes. It also supports later work by Rücker and Rücker on the use of the descriptor combination for the same purpose.

Communication capabilities of product networks

Instead of providing separate solutions for each individual network, a unified theory is desirable to cover the study of a class of networks. Cartesian product graphs provide a common framework to investigate the performance of several individual networks. This paper addresses communication capabilities of product networks. Communication cost is generally characterized by the diameter, the average distance, the total number of paths, the traffic intensity, the saturation level, the queue length in each node, the communication delay and the network throughput. The diameter and average distance of product networks have been studied. However, no work has addressed the remaining measures for product networks. This paper presents a unified theory to evaluate the traffic intensity and the saturation level of product networks. We have theoretically computed the traffic intensity and the saturation level. Intensive simulations have been conducted to validate the analytical results and to compute the other measures for different workloads, different networks, and different network sizes. Examples of product networks that have been investigated are multidimensional meshes, multidimensional toruses, and r?ary n?cube networks. We have also shown that the structure (geometry) of a network is a primary factor for network high performance. For meshes and toruses, square networks present an optimal structure. While in case of an r?ary n?cubes, networks with higher radix outperform those with smaller radix. In particular, cross?cubes (4?ary n?cube) are shown to perform better than binary cubes.

Upper bounds on trellis complexity of lattices

Unlike block codes, n-dimensional lattices can have minimal trellis diagrams with an arbitrarily large number of states, branches, and paths. In particular, we show by a counterexample that there is no f(n), a function of n, such that all rational lattices of dimension n have a trellis with less than f(n) states. Nevertheless, using a theorem due to Hermite, we prove that every integral lattice /spl Lambda/ of dimension n has a trellis T, such that the total number of paths in T is upper-bounded by P(T)/spl les/n!(2//spl radic/3)/sup n2(n-1/2)/V(/spl Lambda/)/sup n-1/ where V(n) is the volume of /spl Lambda/. Furthermore, the number of states at time i in T is upper-bounded by |S/sub i/|/spl les/(2//spl radic/3)/sup i2(n-1)/V(/spl Lambda/)/sup 2i2/n/. Although these bounds are seldom tight, these are the first known general upper bounds on trellis complexity of lattices.

Resynthesis of Combinational Circuits for Path Count Reduction and for Path Delay Fault Testability

Path delay fault model is the most suitable model for detecting distributed manufacturing defects which can cause delay faults. However, the number of paths in a modern design can be extremely large and the path delay testability of many practical designs is very low. In this paper we show how to resynthesize a combinational circuit in order to reduce the total number of paths in the circuit. Our results show that it is possible to obtain circuits with a significant reduction in the number of paths while not increasing area and/or delay of the longest sensitizable path in the circuit. Research on path delay testing shows that in many circuits a large portion of paths does not have a test that can guarantee detection of a delay fault. The path delay testability of a circuit would increase if the number of such paths is reduced. We show that addition of a small number of test points into the circuit can help reducing the number of such paths in the given design.

Clustering based on multiple paths

This paper deals with a separation of strongly touching clusters using a concept of n-connectedness between pattern pairs, i.e., the number of paths between patterns. This new concept is similar to the concept of n-connectedness of graphs. Classification algorithms based on the number of independent paths and edge-disjoint paths are presented. It is shown that the latter has a transitivity. Finally, a simpler algorithm, i.e., a classification method based on the total number of paths, is presented. The proposed algorithm may be seen as an intermediate one between the ordinary connectedness algorithm and the maximal complete subgraph (MCS) algorithm.