Hypercube Algorithms Research Articles

High prevalence group testing in epidemiology with geometrically inspired algorithms

Demand for mass surveillance during peak times of the SARS-CoV-2 pandemic caused high workload for clinical laboratories. Efficient and cost conserving testing designs by means of group testing can substantially reduce resources during possible future emergency situations. The novel hypercube algorithm proposed by Mutesa et al. 2021 published in Nature provides methodological proof of concept and points out the applicability to epidemiological testing. In this work, the algorithm is explored and expanded for settings with high group prevalence. Numerical studies investigate the limits of the adapted hypercube methodology, allowing to optimize pooling designs for specific requirements (i.e. number of samples and group prevalence). Hyperparameter optimization is performed to maximize test-reduction. Standard deviation is examined to investigate resilience and precision. Moreover, empirical validation was performed by elaborately pooling SARS-CoV-2 virus samples according to numerically optimized pooling designs. Laboratory experiments with SARS-CoV-2 sample groups, ranging from 50 to 200 items, characterized by group prevalence up to 10%, are successfully processed and analysed. Test-reductions from 50 to 72.5% were achieved in the experimental setups when compared to individual testing. Higher theoretical test-reduction is possible, depending on the number of samples and the group prevalence, indicated by simulation results.

Strategies for predicting soil organic matter in the field using the Chinese Vis-NIR soil spectral library

The large-scale soil spectral library (SSL) can provide abundant information for predicting soil properties at a local scale, especially in places lacking data. However, since all the existing large-scale SSLs only contain dry spectra recorded under laboratory conditions, the challenge remains in using SSL for predicting soil information using in situ soil spectra. Previous studies have focused on the methods of transforming in situ spectra to dry spectra when using SSLs, while few studies have compared which strategies are optimal in predicting soil properties. To determine the optimal strategies for predicting soil organic matter (SOM) from an area not located in the archived Chinese Soil Spectral Library (CSSL), we investigated the prediction accuracy of memory-based learning (MBL) using spectra transformed by external parameter orthogonalization (EPO) on the CSSL (MBL_EPO) and on the CSSL spiked with subset samples selected by the conditioned Latin hypercube (cLH) algorithm (MBL_EPO_spiking) and using the data from CSSL spiked directedly by the subset in situ samples (MBL_wet_spiking). We also evaluated the prediction accuracy of the in situ and dry spectra using the selected subset and the partial least squares regression (PLSR) model directly. The results showed that the mean squared Euclidean distance (msd) calculated from spectra was an optimal indicator for selecting the representative samples for both the laboratory and in situ conditions. When only 20 samples with both in situ and dry spectra are available to predict SOM, MBL_EPO_spiking is suggested; otherwise, MBL_wet_spiking with the spiking of in situ spectra determined by the smallest msd is optimal. Our findings pave the way for efficient SOM prediction in situ by integrating a large-scale SSL.

Determination of the power smoothing effect in a photovoltaic-hydrokinetic system by experimental analysis and pattern search

Sizing optimization and power smoothing methodology in highly fluctuating grid-connected renewable systems represent important challenges nowadays, the total cost and energy quality depend on the effective balance between source and load with certain restrictions to increase reliability and enhance the use of renewable sources at all scales. This article presents a novel power smoothing effect and sizing optimization methodology for a grid-connected photovoltaic-hydrokinetic system by pattern search method comparatively with three optimization algorithms encompassing a multi-objective optimization. In this context, the method considers the reduction of power fluctuations using supercapacitors, where the objective function considers technical and economic indexes. The main contribution of the research is that the objective function decreases the fluctuation suppression ratio to sizing optimization of renewable system through three pattern search size optimization algorithms. Experiments have shown that pattern search method can find the global optimum by reducing the computational effort; the Latin hypercube algorithm reduces the photovoltaic capacity by approximately 5 % and 2.3 % with respect to the Nelder Mead and genetic algorithm respectively. The influence of the supercapacitor successfully reduces power fluctuations and causes an increase in self-consumption of 32.32 %, this result represents an annual cost savings of 42 % on energy purchased from the grid. Furthermore, this paper goes further by calculating the optimal location of a hydrokinetic turbine in a river using the HEC-RAS software to determine with high precision the HKT power fluctuations as input signal to novel proposed method.

Probabilistic Power Flow Method for Hybrid AC/DC Grids Considering Correlation among Uncertainty Variables

For a new power system using high-penetration renewable energy, the traditional deterministic power flow analysis method cannot accurately represent the stochastic characteristics of each state variable. The aggregation of renewable energy with different meteorological characteristics in the AC/DC interconnected grid significantly increases the difficulty of establishing a steady-state model. Therefore, this study proposes an improved Latin hypercube sampling algorithm using the van der Waerden scores and diffusion kernel density estimation to overcome the limitations of a priori assumption on probability distributions in uncertainty modeling and to retain the correlations among random variables in the sampling data. Interconnected grids are constructed with IEEE 9-bus and IEEE 14-bus and modified with IEEE 57-bus to describe common application cases of aggregated renewable energy. On this basis, the approximation errors of the proposed probabilistic power flow algorithm to the statistical characteristics of the power parameters are evaluated by setting the Nataf algorithm and the Latin hypercube algorithm using adaptive kernel density estimation as the control group. The results show that the improved Latin hypercube sampling algorithm can exhibit high computational accuracy and strong adaptability, both in severe operating scenarios with large amplitude of load fluctuations and with nonlinear power balance equations incorporating high dimensional random variables.

A Novel Hypercube-based Approach to Overlay Design Algorithms on Topic Distribution Networks

Data communication in peer-to-peer (P2P) network requires a fine-grained optimization for memory and processing to lower the total energy consumption. When the concept of Publish/subscribe (Pub/Sub) systems were used as a communication tool in a P2P network, the network required additional optimization algorithms to reduce the complexity. The major difficulty for such networks was creating an overlay design algorithm (ODA) to define the communication patterns. Although some ODAs may perform worse on a high-scale, some may have better average/maximum node degrees. Based on the experimentation and previous works, this study designed an algorithm called the Hypercube-ODA, which reduces the average/maximum node degree for a topic connected Pub/Sub network. The Hypercube-ODA algorithm creates the overlay network by creating random cubes within the network and arranging the nodes with the cubes they belong to. In this paper, the details of the proposed Hypercube algorithm were presented and its performance was compared with the existing ODAs. Results from the experiments indicate that the proposed method outperforms other ODA methods in terms of lower average node degree (lowering the average node degree by up to 60%).

Probability-based prediction of service life and energy-use due to silane-based treatment on chloride-exposed concrete structures

This study aims at performing probability-based prediction of chloride penetration into silane-treated concrete and the cumulative amount of energy-use due to silane-based treatment. There are five random variables to be considered. With various times of silane-based treatment, this study proposes three repair plans. Here, the Latin Hypercube algorithm is embedded in a Crank-Nicolson based finite difference framework. Using this self-developed framework, various studies are observed. It is found that the prolongation of the time, that the content of chloride ions at a specific depth approaches the critical chloride amount for the corrosion of reinforcing bars, occurs due to retarding the diffusion of chloride ions in concrete with silane-based treatment. If the application of silane-based treatment is earlier and more often, the extension of the service life is longer. Focusing on either the service life prolongation or the expected cumulative amount of energy-use, it is not simple to judge the best plan because the silane-based treatment plan with earlier and more often applications is both better in extending the service life and the worst due to causing higher expected cumulative amount of energy-use. To remedy this situation, this paper introduces the ratio of the expected cumulative amount of energy-use to the benefit of silane-based treatment. Using this proposed ratio, an optimum silane-based treatment plan can be judged.

Hull optimization of an underwater vehicle based on dynamic surrogate model

A parallel multidisciplinary optimization design is proposed for the lines design of an underwater vehicle. Resistance and energy consumption are concerned about in obtaining the optimized lines. Collaborative optimization strategy is adopted to construct the optimization framework. The simulated annealing algorithm in combination of the modified method of feasible directions are taken as the optimization algorithms. CFD based simulation is conducted to analyze the resistance of the underwater vehicle. To improve the optimization efficiency, a surrogate model is incorporated into the optimization framework to approximate the CFD calculation. Unlike the conventional static surrogate models used in the design of underwater vehicles, a dynamic surrogate based on trust region and lower confidence bound is proposed. An adaptive balance parameter is taken in the lower confidence bound approach to guarantee the tradeoff between the exploitation and exploration ability of the dynamic surrogate model. Optimal Latin hypercube algorithm is employed as the method of design of experiments. Radial basis function is employed to construct the surrogate model. SUBOFF model is used to demonstrate the proposed optimization scheme. Lines of the underwater vehicle are determined and optimization results of the resistance and energy consumption show the effectiveness of optimization scheme.

Sensitivity analysis and instability study of mechanical parameters of deterioration layer of RCC dam

Aiming at the characteristics of multi-layer structure of roller compacted concrete (RCC) dam and the complicated combination of layers, the three-dimensional calculation of the structure of RCC dam adopts the multi-layer elastoplastic constitutive model. The influence of various parameters of RCC dam deterioration on the strength of the dam body was analyzed in detail. The Latin hypercube algorithm was used to randomly sample each material parameter. The sensitivity of the degraded interlayer parameters was studied based on the Spearman rank correlation coefficient. To solve the simulation The non-stationary randomness of the degradation process, the introduction of the impact factor attenuation function to smooth the degradation, the establishment of a time-varying model of material parameter degradation, and on the basis of the strength reduction coefficient method, the degradation driving level of the RCC dam is calculated. Through the research results, it is possible to focus on monitoring the relevant aspects of the dam during operation, take relevant measures in real time, and feedback on the design and construction plan, which is of great significance for the research on the stability of the RCC dam against sliding.

Optimal deterministic distributed algorithms for maximal independent set in geometric graphs

Finding a maximal independent set (MIS) in a graph is one of the fundamental problems in distributed computing. Researchers in this community are trying to close the time complexity gap of computing a MIS in a graph, way back from Luby’s [STOC’85] O(logn) time (randomized) algorithm and Linial’s [SICOMP’92] Ω(log∗n) lower bound result (n is the number of nodes of the graph). Since then, an extensive research has been done on this problem, however, most of the results are randomized and we are lack of efficient deterministic solution. In fact, no polylogarithmic (in n) time deterministic algorithm is known so far in general graphs — an open question raised by Linial in 1993 [Comb. Probab. Comput.’93]. In this paper, we study the MIS problem in geometric graphs, a class of graphs which has both theoretical and practical importance. We present O(1)-time deterministic algorithms (hence, optimal) for computing MIS in unit interval graphs, unit square graphs and unit disk graphs. The same idea applies to develop optimal MIS algorithm for higher dimensional unit balls and unit hypercubes. We further extend the MIS algorithms to compute approximate maximum independent set (MaxIS) in the above geometric graph classes. The theoretical results are corroborated through extensive experiments to show the effectiveness and efficiency of our algorithms.Our algorithms are fully decentralized, scalable, and robust. In general, nodes exchange only O(logn) bits message through an edge per communication. Moreover, being local, our algorithms are also suitable in failure model, self-stabilizing and appropriate for dynamic environment.

Sampling Strategies for Soil Property Mapping Using Multispectral Sentinel-2 and Hyperspectral EnMAP Satellite Data

Designing a sampling strategy for soil property mapping from remote sensing imagery entails making decisions about sampling pattern and number of samples. A consistent number of ancillary data strongly related to the target variable allows applying a sampling strategy that optimally covers the feature space. This study aims at evaluating the capability of multispectral (Sentinel-2) and hyperspectral (EnMAP) satellite data to select the sampling locations in order to collect a calibration dataset for multivariate statistical modelling of the Soil Organic Carbon (SOC) content in the topsoil of croplands. We tested different sampling strategies based on the feature space, where the ancillary data are the spectral bands of the Sentinel-2 and of simulated EnMAP satellite data acquired in Demmin (north-east Germany). Some selection algorithms require setting the number of samples in advance (random, Kennard-Stones and conditioned Latin Hypercube algorithms) where others automatically provide the ideal number of samples (Puchwein, SELECT and Puchwein+SELECT algorithm). The SOC content and the spectra extracted at the sampling locations were used to build random forest (RF) models. We evaluated the accuracy of the RF estimation models on an independent dataset. The lowest Sentinel-2 normalized root mean square error (nRMSE) for the validation set was obtained using Puchwein (nRMSE: 8.7%), and Kennard-Stones (9.2%) algorithms. The most efficient sampling strategies, expressed as the ratio between accuracy and number of samples per hectare, were obtained using Puchwein with EnMAP and Puchwein+SELECT algorithm with Sentinel-2 data. Hence, Sentinel-2 and EnMAP data can be exploited to build a reliable calibration dataset for SOC mapping. For EnMAP, the different selection algorithms provided very similar results. On the other hand, using Puchwein and Kennard-Stones algorithms, Sentinel-2 provided a more accurate estimation than the EnMAP. The calibration datasets provided by EnMAP data provided lower SOC variability and lower prediction accuracy compared to Sentinel-2. This was probably due to EnMAP coarser spatial resolution (30 m) less adequate for linkage to the sampling performed at 10 m scale.

A new method for selecting sites for soil sampling, coupling global weighted principal component analysis and a cost-constrained conditioned Latin hypercube algorithm

Analysing spatial patterns of soil properties in a landscape requires a sampling strategy that adequately covers soil toposequences. In this context, we developed a hybrid methodology that couples global weighted principal component analysis (GWPCA) and cost-constrained conditioned Latin hypercube algorithm (cLHC). This methodology produce an optimized sampling stratification by analysing the local variability of the soil property, and the influence of environmental factors. The methodology captures the maximum local variances in the global auxiliary dataset with the GWPCA, and optimizes the selection of representative sampling locations for sampling with the cLHC. The methodology also suppresses the subsampling of auxiliary datasets from areas that are less representative of the soil property of interest. Consequently, the method stratifies the geographical space of interest in order to adequately represent the soil property. We present results on the tested method (R2 = 0.90 and RMSE = 0.18 m) from the Guinea savannah zone of Ghana.•It defines the local structure and accounts for localized spatial autocorrelation in explaining variability.•It suppresses the occurrence of model-selected sampling locations in areas that are less representative of the soil property of interest.

Soil hydraulic material properties and layered architecture from time-lapse GPR

Abstract. Quantitative knowledge of the subsurface material distribution and its effective soil hydraulic material properties is essential to predict soil water movement. Ground-penetrating radar (GPR) is a noninvasive and nondestructive geophysical measurement method that is suitable to monitor hydraulic processes. Previous studies showed that the GPR signal from a fluctuating groundwater table is sensitive to the soil water characteristic and the hydraulic conductivity function. In this work, we show that the GPR signal originating from both the subsurface architecture and the fluctuating groundwater table is suitable to estimate the position of layers within the subsurface architecture together with the associated effective soil hydraulic material properties with inversion methods. To that end, we parameterize the subsurface architecture, solve the Richards equation, convert the resulting water content to relative permittivity with the complex refractive index model (CRIM), and solve Maxwell's equations numerically. In order to analyze the GPR signal, we implemented a new heuristic algorithm that detects relevant signals in the radargram (events) and extracts the corresponding signal travel time and amplitude. This algorithm is applied to simulated as well as measured radargrams and the detected events are associated automatically. Using events instead of the full wave regularizes the inversion focussing on the relevant measurement signal. For optimization, we use a global–local approach with preconditioning. Starting from an ensemble of initial parameter sets drawn with a Latin hypercube algorithm, we sequentially couple a simulated annealing algorithm with a Levenberg–Marquardt algorithm. The method is applied to synthetic as well as measured data from the ASSESS test site. We show that the method yields reasonable estimates for the position of the layers as well as for the soil hydraulic material properties by comparing the results to references derived from ground truth data as well as from time domain reflectometry (TDR).

Random and Conditional (t, k)-Diagnosis of Hypercubes

Under the PMC model, we consider the (t, k)-diagnosability of a hypercube under the random fault model and conditional fault model separately. A system is called random (t, k)-diagnosable [or conditionally (t, k)-diagnosable] if at least k faulty vertices that can be identified in each iteration under the assumption that there is no any restriction on the fault distribution (or every vertex is adjacent to at least one fault-free vertex), provided that there are at most t faulty vertices, where \(t\ge k\). When the remaining number of faulty vertices is lower than k, all of them can be identified. In this paper, under the PMC and random fault models, we show that the sequential t-diagnosis algorithm for hypercubes proposed by [35] can be extended to the (t, k)-diagnosis algorithm for hypercubes, and we show that the n-dimensional hypercubes are randomly \(\left( \genfrac(){0.0pt}0{n}{\frac{n}{2}},n\right) \)-diagnosable if n is even, and randomly \(\left( 2\cdot \genfrac(){0.0pt}0{n-1}{\frac{n-1}{2}},n\right) \)-diagnosable if n is odd, where \(\genfrac(){0.0pt}0{p}{q}=\frac{p!}{q!(p-q)!}\). Moreover, we propose a conditional (t, k)-diagnosis algorithm for hypercubes by using properties of the conditional fault model and show that n-dimensional hypercubes are conditionally \(\left( \genfrac(){0.0pt}0{n}{\frac{n}{2}},2n-2\right) \)-diagnosable if n is even, and conditionally \(\left( 2\cdot \genfrac(){0.0pt}0{n-1}{\frac{n-1}{2}},2n-2\right) \)-diagnosable if n is odd. Furthermore, under the PMC model, our results improve the previous best low bounds on t under the random and conditional fault models.

Sampling based numerical seismic assessment of continuous span RC bridges

The seismic response of RC structures is dictated by a number of aleatory and epistemic uncertainties, related to material properties and seismic hazard. Typically, when following a probabilistic approach, the different variables are statistically defined using a random simulation tool that will take into account the different sources of uncertainty. This manuscript addresses the seismic reliability of RC bridge structures, particularly through the assessment of a random sampling scheme based on the Latin Hypercube algorithm for numerical simulation of material and ground motion properties. An optimal use of such sampling technique is thus sought so as to ensure higher accurateness in the calculation of the failure probability of a set of different existing bridges, through a relatively simple framework. The study demonstrates the robustness of the Latin Hypercube algorithm, which secured accuracy of the estimated probability of failure for both regular and irregular configurations. Recommendations to researchers and practitioners are made accordingly, particularly with respect to the minimal sampling size, which does not need to exceed one to two hundred realisations.

Matrix multiplication with a hypercube algorithm on multi-core processor cluster

The algorithm of multiplication of matrices of Dekel, Nassimi and Sahani or Hypercube is analysed, modified and implemented on multicore processor cluster, where the number of processors used is less than that required by the algorithm n3<sub>3</sub>. 2<sub>3</sub>, 4<sub>3</sub> and 8<sub>3</sub> processing units are used to multiply matrices of the order of 10x10, 10<sub>2</sub>x10<sub>2</sub> and 10<sub>3</sub>X10<sub>3</sub>. The results of the mathematical model of the modified algorithm and those obtained from the computational experiments show that it is possible to reach acceptable speedup and parallel efficiencies, based on the number of used processor units. It also shows that the influence of the external communication link among the nodes is reduced if a combination of the available communication channels among the cores in a multi-core cluster is used.

The g-extra conditional diagnosability and sequential t/k-diagnosability of hypercubes

The conditional diagnosis is a very important measure of the reliability and the fault-tolerance of networks. The ‘condition’ means that no faulty set contains all neighbours of any node. Under this assumption, for any system G, every component of has more than 1 node, where F is the faulty set of G. The g-extra conditional diagnosability is defined under the assumption that every component of has more than nodes. ‘A system with at most t faulty nodes is defined as sequentially t-diagnosable if at least one faulty node can be repaired, so that the testing can be continued using the repaired node to eventually diagnose all faulty nodes’ [E.P. Duarte Jr., R.P. Ziwich, and L.C.P. Albini, A survey of comparison-based system-level diagnosis, ACM Comput. Surv. 43(3) (2011), article 22]. To increase the degree of the sequential t-diagnosability of a system, sequential -diagnosis strategy is proposed in this paper. It is allowed that there are at most k misdiagnosed nodes. In this paper, we determine the g-extra conditional diagnosability of hypercubes and propose sequential -diagnosis algorithms for hypercubes with low time complexities under the Preparata, Metze, and Chien (PMC) model and the MM* model which is a special case of the Maeng and Malek (MM) model.

A Routing Algorithm Solving the Container Problem in a Hypercube with Bit Constraint

As reflected by the TOP500 list, hypercubes are popular interconnection networks for massively parallel systems, the main reason being the simplicity and ease of implementation of this network topology. In order to retain performance high and avoid bottleneck situation, routing algorithms are critical for these high-performance systems. Furthermore, disjoint path routing is a very desirable property of such communication algorithms. Effectively, selecting mutually node-disjoint paths guarantees that notorious parallel processing issues such as deadlocks, livelocks and starvations shall never occur. In this paper, we describe a routing algorithm for hypercubes that, given a bit constraint, selects internally node-disjoint paths between any pair of nodes satisfying the constraint, and such that the selected paths all satisfy the constraint. The introduction of such bit constraint enables the selection of multiple sets of disjoint paths between several node pairs each satisfying a distinct bit constraint, which is impossible with conventional routing algorithms. Selecting simultaneously disjoint paths between different node pairs induces increased communication performance and system dependability. The correctness and complexities of the described algorithm are formally proved, and analysis of the algorithm performance in practice is conducted by empirical evaluation.

Abstract 2357: Gene expression profiling robustly predicts the outcome of patients diagnosed with early stage lung adenocarcinoma

Abstract Background: Although the management of metastatic lung adenocarcinoma has been profoundly modified by the identification of actionnable molecular traits, decision making for early-stage lung adenocarcinoma still relies on the tumor stage (TNM) only. Most of these patients recur within 5 years after tumor resection and are given platinum based adjuvant chemotherapy. The recent availability of multiple gene expression data and the concomitant developments of novel algorithms - such as the Hypercube algorithm - could dramatically improve such predictive challenges. Methods: Datasets were selected from public repositories (Director's challenge (DC) n=274, Rousseaux n=86, Hou n=35, JBR.21 n=31) upon the following criteria: lung adenocarcinoma, stage IA-3A, no adjuvant therapy, R0 tumor resection. To generate robust predictors of 3-year overall survival, an iterative feature selection framework (Hypercube algorithm) was applied successively on DC and Hou. Only highly predictive rules in both datasets (AUC >.7) were then included in a subsequent logistic regression model and finally assessed for external validation. The predictive performance of the models was evaluated by ROC-AUC and by Kaplan Meier curves. Results: Our best model combining gene expression (10 genes) and clinical data (TNM) was robust across three lung adenocarcinoma datasets (AUC= .85, .80 and .59). Particularly, it was able to better predict the 3 year of survival compared to the TNM-based clinical model only (100% vs. 77%). Kaplan Meier estimates of overall survival in the validation datasets were dramatically discriminated according to the high- and low-risk groups predicted by our model (all P values =.01). Interestingly, this predictive classifier was specific to lung adenocarcinoma and has no predictive value in lung squamous datasets. Conclusion: We developed a robust and highly performing gene expression predictive signature of 3-year overall survival, specific to the lung adenocarcinoma setting. These promising results have the potential to change the decision making at bedside for early stage lung adenocarcinoma patients. Citation Format: Yann Gaston-Mathé, CHARLES FERTE, benoit gauthier, mathilde bateson, david planchard, benjamin besse, Jean-Pierre Armand, jean-charles soria. Gene expression profiling robustly predicts the outcome of patients diagnosed with early stage lung adenocarcinoma. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 2357. doi:10.1158/1538-7445.AM2014-2357

A distributed virtual hypercube algorithm for maintaining scalable and dynamic network overlays

SummaryNetwork overlays support the execution of distributed applications, hiding lower level protocols and the physical topology. This work presents DiVHA: a distributed virtual hypercube algorithm that allows the construction and maintenance of a self‐healing overlay network based on a virtual hypercube. DiVHA keeps logarithmic properties even when the number of nodes is not a power of two, presenting a scalable alternative to connect distributed resources. DiVHA assumes a dynamic fault situation, in which nodes fail and recover continuously, leaving and joining the system. The algorithm is formally specified, and the latency for detecting changes and the subsequent reconstruction of the topology is proved to be bounded. An actual overlay network based on DiVHA called HyperBone was implemented and deployed in the PlanetLab. HyperBone offers services such as monitoring and routing, allowing the execution Grid applications across the Internet. HyperBone also includes a procedure for detecting groups of stable nodes, which allowed the execution of parallel applications on a virtual hypercube built on top of PlanetLab. Copyright © 2014 John Wiley & Sons, Ltd.

Fault-tolerant routing based on approximate directed routable probabilities for hypercubes

Recently, parallel processing systems have been studied very actively, and many topologies have been proposed. A hypercube is one of the most popular topologies for interconnection networks. In this paper, we propose two new fault-tolerant routing algorithms for hypercubes based on approximate directed routable probabilities. Probabilities represent the ability of routing toward any node located at a specific distance and are calculated by considering from which direction the message has been received. Each node chooses one of its neighbor nodes to forward the message by comparing the approximate directed routable probabilities. We also conducted a computer experiment to verify the effectiveness of our algorithms.