Hop Length Research Articles

Optimizing MFCC parameters for the automatic detection of respiratory diseases

Voice signals originating from the respiratory tract are utilized as valuable acoustic biomarkers for the diagnosis and assessment of respiratory diseases. Among the employed acoustic features, Mel Frequency Cepstral Coefficients (MFCC) are widely used for automatic analysis, with MFCC extraction commonly relying on default parameters. However, no comprehensive study has systematically investigated the impact of MFCC extraction parameters on respiratory disease diagnosis. In this study, we address this gap by examining the effects of key parameters, namely the number of coefficients, frame length, and hop length between frames, on respiratory condition examination. Our investigation uses four datasets: the Cambridge COVID-19 Sound database, the Coswara dataset, the Saarbrücken Voice Disorders (SVD) database, and a TACTICAS dataset. The Support Vector Machine (SVM) is employed as the classifier, given its widespread adoption and efficacy. Our findings indicate that the accuracy of MFCC decreases as hop length increases, and the optimal number of coefficients is observed to be approximately 30. The performance of MFCC varies with frame length across the datasets: for the COVID-19 datasets (Cambridge COVID-19 Sound database and Coswara dataset), performance declines with longer frame lengths, while for the SVD dataset, performance improves with increasing frame length (from 50 ms to 500 ms). Furthermore, we investigate the optimized combination of these parameters and observe substantial enhancements in accuracy. Compared to the worst combination, the SVM model achieves an accuracy of 81.1%, 80.6%, and 71.7%, with improvements of 19.6%, 16.10%, and 14.90% for the Cambridge COVID-19 Sound database, the Coswara dataset, and the SVD dataset respectively. To validate the generalization of these findings, we employ the Long Short-Term Memory (LSTM) model as a validation model. Remarkably, the LSTM model also demonstrates improved accuracy of 14.12%, 10.10%, and 6.68% across the datasets when utilizing the optimal combination of parameters. The optimal parameters are validated using an external voice pathology dataset (TACTICAS dataset). The results demonstrate the generalization capabilities of the optimized parameters across various pathologies, machine-learning models, and languages.

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity.

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Rietveld refined structural, optical and temperature dependent impedance spectroscopy of NiO–ZnO heterostructure composite: Synthesized through solid state method for high-frequency devices

Multifunctional NiO–ZnO heterostructure composite powder is economically synthesized through a solid-state route having a crystallite size of 23 (±2) nm. Chemical composition is identified through FTIR, the peak associated with Zn–O is detected at 1049 cm−1. However, the peaks detected at 870 cm−1 and 590 cm−1 show Ni–O stretching vibrations. UV-DRS technique is employed to estimate the energy band gap (Eg) around 3.18 eV. The impedance studies confirmed the presence of a non-Debye-type relaxation mechanism, electrical parameters confirm the contribution of electroactive regions i.e. grains and interfaces. An equivalent circuit model (R-CPE) is employed to measure various electrical parameters. Adiabatic small polaron hopping model is applied to estimate activation energies for Ni+2 (0.30 eV), Zn+2 (0.35 eV), and bulk (0.32 eV) respectively. The hopping length at the interface is estimated at around 0.8 Å by employing Mott's variable range hopping model. Jonscher's power law (JPL) is used to fit ac conductivity data, and conclude activation energy around 0.3 eV. The conduction mechanism through the whole frequency range (1–107 Hz) is governed by correlated barrier hopping. Various physical parameters i.e. binding energy (Wm), hopping length, and barrier height compared throughout the whole temperature range (313–393 K). The rate of increase in real permittivity at low frequency is related to the disorder of the cation at sublattices. These results deduced potential for NiO–ZnO heterostructure composite for high dielectric permittivity with low tangent loss for high energy storage applications.

Insights into crystal structure, temperature-dependent magnetization and dielectric relaxation mechanism in Bi substituted samarium iron garnet

The single phase Sm3−xBixFe5O12 with x=0.0,0.2,0.4 and 0.6 samples are synthesized by solid-state reaction method. We have systematically studied the structural, morphological, temperature dependent magnetic and electric properties. All the samples crystallize in simple cubic crystal structure and belongs to Ia3‾d space group. The lattice constant and bond angle between Fe(a)−O−Fe(d) network is enhanced with Bi substitution. As a result, the ferrimagnetic transition temperature is also enhanced from 565K to 569K. We have examined the applicability of Bloch's and Cojocaru's laws to our temperature-dependent magnetization data and find that Bloch's T3/2 law is not suitable due to geometrical asymmetry, long wavelength excitation, and local atomic disorders. However, Cojocaru's law which accounts for the geometrical aspects, provides a better fit to our magnetization data. Further, the impedance plots at room temperature exhibit no relaxation which is attributed to the limited mobility of charge carriers. On the other hand, the high temperature impedance data represents dielectric relaxation at a particular frequency. This frequency is used to determine the relaxation time which is fitted to Arrhenius law and activation energy is evaluated. The activation energy lies in between 0.24−0.32eV which corresponds to the singly ionized oxygen vacancies. The conduction mechanism is analyzed with Variable range hopping model and the average hopping length and hopping energies are also determined for these garnet samples.

Weak grip strength among persons with multiple sclerosis having minimal disability is not related to agility or integrity of the corticospinal tract

IntroductionMobility impairment is common in multiple sclerosis (MS); however, agility has received less attention. Agility requires strength and neuromuscular coordination to elicit controlled propulsive rapid whole-body movement. Grip strength is a common method to assess whole body force production, but also reflects neuromuscular integrity and global brain health. Impaired agility may be linked to loss of neuromuscular integrity (reflected by grip strength or corticospinal excitability). ObjectivesWe aimed to determine whether grip strength would be associated with agility and transcranial magnetic stimulation (TMS)-based indices of corticospinal excitability and inhibition in persons with MS having low disability. We hypothesized that low grip strength would predict impaired agility and reflect low corticospinal excitability. MethodsWe recruited 34 persons with relapsing MS (27 females; median [range] age 45.5 [21.0-65.0] years) and mild disability (median [range] Expanded Disability Status Scale 2.0 [0-3.0]), as well as a convenience sample of age- and sex-matched apparently healthy controls. Agility was tested by measuring hop length during bipedal hopping on an instrumented walkway. Grip strength was measured using a calibrated dynamometer. Corticospinal excitability and inhibition were examined using TMS-based motor evoked potential (MEP) and corticospinal silent period (CSP) recruitment curves, respectively. ResultsMS participants had significantly lower grip strength than controls independent of sex. Females with and without MS had weaker grip strength than males. There were no statistically significant sex or group differences in agility. After controlling for sex, weaker grip strength was associated with shorter hop length in controls only (r = 0.645, p < .05). Grip strength did not significantly predict agility in persons with MS, nor was grip strength predicted by corticospinal excitability or inhibition. ConclusionsIn persons with MS having low disability, grip strength (normalized to body mass) was reduced despite having intact agility and walking performance. Grip strength was not associated with corticospinal excitability or inhibition, suggesting peripheral neuromuscular function, low physical activity or fitness, or other psychosocial factors may be related to weakness. Low grip strength is a putative indicator of early neuromuscular aging in persons with MS having mild disability and normal mobility

Mid‐Air Collisions Control the Wavelength of Aeolian Sand Ripples

AbstractThe wavelength of aeolian sand ripples increases with wind velocity, but the cause for this increase remained unclear until now. Using numerical simulations, we find that the relationship between the wind strength and the initial wavelength disappears without mid‐air collisions, which means that mid‐air collisions crucially contribute to the initial wavelength of ripples. As wind strength increases, the average hop length of non‐colliding particles decreases. Affected by the bed surface topography and the non‐uniform mid‐air colliding probability along the wavy surface, non‐colliding particles cause a surface modulated erosion/deposition flux. The first order contribution of the flux fluctuation destabilizes the original bed surface and has a wind‐dependent phase shift with respect to the surface profile. This phase shift leads to an initial wavelength that increases with wind velocity. Based on these findings we derive the scaling law for the initial wavelength of aeolian sand ripples, which agrees well with experimental results. A weak non‐linear relationship between ripple wavelength and wind velocity has been found, especially for large wind strengths.

How Concerted Are Ionic Hops in Inorganic Solid-State Electrolytes?

Despite being fundamental to the understanding of solid-state electrolytes (SSEs), little is known on the degree of coordination between mobile ions in diffusive events, thus hindering a detailed comprehension and possible rational design of SSEs. Here, we introduce an unsupervised k-means clustering approach that is able to identify ion-hopping events and correlations between many mobile ions and apply it to a comprehensive ab initio MD database comprising several families of inorganic SSEs and millions of ionic configurations. It is found that despite two-body interactions between mobile ions being the largest, higher-order n-ion (2 < n) correlations are most frequent. Specifically, we prove a general exponential decaying law for the probability density function governing the number of concerted mobile ions. For the particular case of Li-based SSEs, it is shown that the average number of correlated mobile ions amounts to 10 ± 5 and that this result is practically independent of the temperature. Interestingly, our data-driven analysis reveals that fast-ionic diffusion strongly and positively correlates with ample hopping lengths and long hopping spans but not with high hopping frequencies and short interstitial residence times. Finally, it is shown that neglection of many-ion correlations generally leads to a modest overestimation of the hopping frequency that roughly is proportional to the average number of correlated mobile ions.

Dissolution‑recrystallization growth and structural characteristics of facile decomposition-processed NiFe2O4 octahedrons

In this paper, NiFe2O4 powders were prepared by a thermal decomposition process with various temperatures. The structural characteristic and intrinsic defect were characterized by X-ray diffraction (XRD), Fourier transform infrared (FT-IR) spectroscopy, ultraviolet–visible (UV–vis) spectroscopy, and photoluminescence (PL). The morphology was investigated by scanning electron microscope (SEM) and transmission electron microscope (TEM). XRD pattern reveals the pure NiFe2O4 phase as the calcination temperature of 900 and 1,000 °C. The calcination of 900 °C depicted the micro-octahedral structures with a smooth surface. The calculated hopping length and the FT-IR spectra reveal the cation shifting from octahedral to tetrahedral structure. The decreasing energy bandgap of NiFe2O4 was investigated as the calcination temperature increased. The PL transition of 3A2g(3F) → 3T1g(3F) refers to the band-to-band emission. The structural characteristics, morphology, and defects were investigated and discussed in terms of phase, bonding vibration, crystal growth mechanism, emerging energy level and electron transition.

Comparison of some kinematic variables of the triple jump between the two world championships (IAAF) (2017) and (2018)

The study aimed to compare some of the kinematic variables and achievement between the competitions (2017) and (2018) in triple jump to find the percentages of the teams, which may indicate the reasons for the different level of the two championships. Statistical difference, although we may not find these differences in all variables, especially achievement, but at such global levels the differences of the few numerical parts make a difference in global achievements, numbers and medals. The researcher adopted the descriptive approach using the method of comparisons to extract the results of the study, and the research community consisted of (27) jumpers (12) competitors in the (2017) championship and (15) competitors in the (2018) championship, and the statistical bag (SPSS) was used to extract: (Mean ), (Std.Divition), (Independent-Sample.T.Test). This study found that the Hop angle and Length of speed Jump is the most influential interval in the best achievement. The researcher recommends following up on such international reports and comparisons because they may give solutions and orientation in training for the most influential factors at different levels of achievement.

Influence of magnetism on vertical hopping transport in CrSBr

We investigate the c-direction conduction in CrSBr in the linear regime, which is not accessible in other van der Waals (vdW) magnetic semiconductors, because of the unmeasurably low current. The resistivity, which is 108−1011 times larger than in the a and b directions, exhibits magnetic state dependent thermally activated and variable range hopping transport. In the spin-flip phase at 2 T, the activation energy is 20 meV lower than in the antiferromagnetic state due to a downshift of the conduction band edge, in agreement with calculations. In the variable range hopping regime, the average hopping length decreases from twice the interlayer distance to the interlayer distance at 2 T because in the antiferromagnetic state the large exchange energy impedes electrons hopping between adjacent layers. Our work demonstrates that the linear transport regime provides new information about electronic processes in vdW magnetic semiconductors and shows how magnetism influences these processes both in real and reciprocal space. Published by the American Physical Society 2024

Clubfoot patients show more anterior-posterior displacement during one-leg-standing and less ankle power and plantarflexor moment during one-leg-hopping than typically developing children

BackgroundClubfoot patients show good-to-excellent foot correction after the Ponseti method. Nevertheless, underlying functional problems that limit motor abilities such as one-leg-standing and one-leg-hopping still persist. These restrictions have been proposed to arise due to problems with maintaining balance and the limited force-generating capacity of clubfoot patients. More insight is needed to understand the underlying limiting factors to improve overall motor ability in clubfoot patients. Research questionThe aim of this study was to determine the differences between clubfoot patients and typically developing children (TDC) in force and balance parameters during walking, one-leg-standing and hopping. MethodsThree-dimensional motion analysis was performed in 19 TDC and 16 idiopathic Ponseti-treated clubfoot patients between 5–9 years old. Kinematic and kinetic parameters were calculated during walking and one-leg-hopping. To describe the balance parameters, center of pressure (CoP) data was assessed during walking, one-leg-hopping and one-leg-standing. Mean group values were calculated and compared using nonparametric statistical tests. A general linear model with repeated measures was used to determine which activity showed the largest group differences. ResultsClubfoot patients showed lower peak plantarflexor moment and peak ankle power absorption and generation during one-leg-hopping compared to TDC. Furthermore, clubfoot patients showed a lower hop length and velocity than TDC. The difference in peak plantarflexor moment and ankle power between the study groups was larger during one-leg-hopping than during walking. Finally, clubfoot patients showed a higher anterior-posterior CoP range during one-leg-standing. SignificanceDeviations in force parameters seemed to limit one-leg-hopping in clubfoot patients, and impaired anterior-posterior static balance was thought to be the underlying cause of problems with one-leg-standing. Furthermore, one-leg-hopping was more sensitive to distinguish between clubfoot patients and TDC than walking. Individualized physiotherapy targeting static balance and force parameters, with extra emphasis on including eccentric contractions, might improve the overall motor abilities of clubfoot patients.

Mechanical properties of polypyrrole/SnO2 nanocomposites and its LPG sensing application

Polypyrrole/SnO2 nanocomposites were created using in-situ polymerization techniques. The nanocomposites were described using scanning electron microscopy (SEM) and transmission electron microscopy (TEM), and their thermal properties were studied using a Differential Scanning Calorimeter (DSC). The DC conductivity of the samples was measured as a function of temperature from 30 ℃ to 1900 ℃, and it was observed that increasing the concentration of tin oxide particles improves conductivity due to polaron hopping and composite chain length extension. The tensile strength of PPy nanocomposites doped in PVA thin film up to 6 wt% indicates 64.2 MPa, which may be related to the homogenous distribution of PPy nanocomposite in PVA. The study reveals that because 50 wt% of the nanocomposites have the highest conductivity and sensitivity, these nanocomposites may be useful in future applications.

Enhanced electrical conductivity and charge conduction mechanisms in Nano-cubical Sunset Yellow dye incorporated with titanium dioxide nanoparticles

This manuscript focuses on the investigation of the effect of titanium dioxide nanoparticles doping on the electrical conductivity of Nano-cubical Sunset Yellow Dye. The successful preparation of the composite was confirmed through XRD analysis, and EDX analysis verified the composite's composition. DC conductivity measurements revealed two conduction mechanisms with activation energies of 0.683 eV and 0.466 eV for the dye without NPs at low and high-temperature regions, respectively, and 0.532 eV and 0.351 eV for the same regions in the presence of NPs. AC conductivity analysis showed single polaron hopping as the dominant charge conduction mechanism for both devices. The device with the dye-NP composite exhibited lower activation energy and improved conductivity compared to the dye-only device. The composite also displayed a higher density of localized states, and by incorporation of NPs reduced the hopping length, contributing to the enhanced conductivity of the composite device.

Modified graph-based algorithm to analyze security threats in IoT.

In recent years, the growing and widespread usage of Internet of Things (IoT) systems has led to the emergence of customized structures dependent on these systems. Industrial IoT (IIoT) is a subset of IoT in terms of applications and usage areas. IIoT presents many participants in various domains, such as healthcare, transportation, agriculture, and manufacturing. Besides the daily life benefits, IIoT technology provides major contributions via the Industrial Control System (ICS) and intelligent systems. The convergence of IoT and IIoT systems brings some integration and interoperability problems. In IIoT systems, devices interact with each other using information technologies (IT) and network space. However, these common usages and interoperability led to some security risks. To avoid security risks and vulnerabilities, different systems and protocols have been designed and published. Various public databases and programs identify and provide some of the security threats to make it easier for system administrators' missions. However, effective and long-term security detection mechanisms are needed. In the literature, there are numerous approaches to detecting security threats in IoT-based systems. This article presents two major contributions: First, a graph-based threat detection approach for IoT-based network systems is proposed. Threat path detection is one of the most critical steps in the security of IoT-based systems. To represent vulnerabilities, a directed acyclic graph (DAG) structure is constructed using threat weights. General threats are identified using Common Vulnerabilities and Exposures (CVE). The proposed threat pathfinding algorithm uses the depth first search (DFS) idea and discovers threat paths from the root to all leaf nodes. Therefore, all possible threat paths are detected in the threat graph. Second, threat path-reducing algorithms are proposed considering the total threat weight, hop length, and hot spot thresholds. In terms of available threat pathfinding and hot spot detecting procedures, the proposed reducing algorithms provide better running times. Therefore, all possible threat paths are founded and reduced by the constructed IoT-based DAG structure. Finally, simulation results are compared, and remarkable complexity performances are obtained.

High-temperature variable range hopping conduction and dielectric relaxation in CoFe2O4 ceramic

CoFe2O4 ceramic was fabricated using a sol-gel auto-combustion process. The cubic spinel structure with F d −3 m space group symmetry was confirmed through the Rietveld refinement of the XRD pattern of CoFe2O4 sintered at 800 °C for 4 h. The cobalt vacancies at the tetrahedral site were verified by electron density distribution calculated through the maximum entropy method. The impedance spectroscopy study revealed that the grain boundary was dominant compared to the grain in the electrical properties of CoFe2O4. The total dc resistivity was fitted by the variable range hopping (VRH) model, demonstrating that the charge carriers for conduction mechanism confirm to the VRH model. The density of localized states (N(EF)) near the Fermi level was found to be 1.07 × 1019 eV−1-cm−3. The hopping energy (EH) and hopping length (R) of the charge carriers were estimated to be 0.29 eV and 2.28 nm, respectively, at 473 K.

Exploring Study of Magnetic and Electrical Properties of Tl3+ Doped Co0.5Ni0.5Fe2O4 Spinel Ferrites

Co0.5Ni0.5Fe2-xTlxO4 (x=0, 0.05, 0.1, 0.15, and 0.2) spinel ferrites were prepared via the sol-gel technique. The XRD analysis revealed a single-phase spinel structure. The lattice constant ‘a’ increased from 8.223-8.269Å with doping of Tl+3 due to larger ionic radii of Tl+3 than Fe+3 ions. The mass susceptibility at 300K decreased from 7.46 x 10-3-4.15 x 10-3 cm3/g due tothe weakening of AB interactions followed by a decrease in Curie temperature from 453K to 408K. The electrical permittivity follows the Maxwell-Wagner model and Koop’s theory. The dc resistivity of ferrites at 300K increased from 1.82x109-9.23x109 ?-cm with increasing Tl+3 contents due to increased hopping lengths. The activation energy obtained from Arrhenius plots increased from 0.126 to 0.131eV, increasing Tl+3contents

Structure and electrical conductivity of nanocomposite SiOxNy(Si) and SiAlzOxNy(Si) films

In this study, the peculiarities of the structure and electrical conductivity of nanocomposite SiOxNy(Si) and SiAlzOxNy(Si) films in the temperature range of 95 – 340 K are determined. The composite films were obtained by ion-plasma sputtering of Si and co-sputtering of Si and Al targets in oxygen and nitrogen containing atmosphere (Ar+O2 +N2) followed by high temperature annealing. During the annealing of the deposited Si rich oxynitride films, Si nanoinclusions were formed in an amorphous dielectric matrix due to phase separation. Al atoms are found to be in the bound state (no Al metal nanoparticles present). The films have low absorbance and considerable transparency in the ultraviolet – visible – near infrared spectral region. Reduction of some part of Si-O bonds by Al during annealing the Al-containing samples leads to the separation of higher Si concentration as compared to the Al-free samples. The electrical conductivity of the films at low electric fields is found to be by variable-range electron hopping through the traps near the Fermi level. The characteristics of the traps and transport process such as the density of electron traps near the Fermi level, the activation energy of hopping and the hopping length are determined. It is found that the electrical conductivity of the SiAlzOxNy(Si) films annealed in argon is higher as compared to the films annealed in nitrogen and the Al-free films due to higher concentration of electron traps. The region of the conductivity decrease with the increase of electric field is revealed on the current-voltage characteristics of both the SiAlzOxNy(Si) films annealed both in nitrogen and argon and the SiOxNy(Si) films annealed in nitrogen. This effect is explained based on the peculiarities of electric field polarization in the films. The important role is played by the interface traps at the Si nanoparticle / amorphous matrix interfaces that capture electrons. At high electric fields, the film conductivity corresponds to the space charge limited current mechanism with the exponential distribution of the traps in the energy gap.

Unravelling the linear and biquadratic magnetoelectric coupling in Ba0.95Sn0.05Ti0.95 Ga0.05O3 – CoFe1.8Ga0.2O4 particulate multiferroic composites

Since multiferroic composites have substantially stronger magnetoelectric coupling between the order parameters than the single-phase multiferroics, they are more interesting prospects for next-generation multi-state memories and spintronic devices. In light of this, the current work outlines the fabrication of ferroelectric perovskite BaTiO3 and ferrimagnetic spinel CoFe2O4-based particulate multiferroic composites with general formulla (1-x) Ba0.95Sn0.05Ti0.95 Ga0.05O3 – x CoFe1.8Ga0.2O4 (x = 0.04, 0.08 & 0.12). At room temperature, the gallium and tin doping enhanced the dielectric properties of BaTiO3 (ε ´ =9200), and particulate composite (ε ´ =6000) with 12% of CoFe1.8Ga0.2O4 phase. In the same composite, the reduced leakage current density of 3.4 μA/cm2 and improved magnetoelectric properties, with linear and quardatic coupling coefficients of 3.41 mVcm−1Oe−1 and 3.4 × 10−4 mVcm−1Oe−2 were achieved. In composites with a larger magnetic phase, the decrease in cell volume and increased tetragonality produces the compressive strain, that in turn improves the ME coupling. Diffused ferroelectric transitions were ascertained by using the Curie- Weiss Law. Enhanced dielectric properties are linked to decrease in hopping and bond lengths, caused by the incorporation of non-magnetic Ga3+ ions in the CoFe2O4 lattice. A reduction in leakage current density was observed on Sn doping. The calculated square-ness ratios reveal an ordered multi-domain structure. Improved dielectric response along with linear and biquadratic coupling makes these Sn and Ga doped BaTiO3-CoFe2O4 particulate composites as potential candidates for magnetoelectric devices.

Crossover from the Mott Variable Range Hopping Conduction Regime to Nearest Neighbor Site Hopping Regime in ZnSxSe1-x Thin Films

In this work, we study the behavior of resistivity as a function of temperature and for several samples of ZnSxSe1-x thin films with x = 0, 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0 respectively. In fact, we re-analyze in our investigation experimental measurements obtained by M. Popa et al. [1] in the range of temperature from 300 K to 500 K. We showed that the resistivity follows a nearest neighbor site hopping conduction mechanism with 𝝆 = 𝝆𝟎𝐞𝐱𝐩 ( 𝑬𝒂 𝒌𝑩𝑻 ) for very high temperatures and Mott variable range hopping conduction with 𝝆 = 𝝆𝑪𝐞𝐱𝐩 (𝑻𝟎 𝑻 )𝟏/𝟒 for relatively low temperatures. The crossover between the two regimes can be explained by the competition between the localization length scale 𝝃𝒍𝒐𝒄and the hopping length scale 𝑹𝒉𝒐𝒑.

Hop-Constrained Oblivious Routing

We prove the existence of an oblivious routing scheme that is -competitive in terms of , thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize , defined as follows: The dilation is the maximum path hop length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have within a factor of the best possible value. More precisely, for any integer hop constraint , this oblivious routing scheme selects paths of length at most and is -competitive in terms of congestion in comparison to the best possible congestion achievable via paths of length at most hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002; Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008], which are -competitive in terms of congestion but are not competitive in terms of dilation.