Time Complexity Research Articles

Adapting stable matchings to forced and forbidden pairs

We introduce the problem of adapting a stable matching to forced and forbidden pairs. Given a stable matching M1, a set Q of forced pairs, and a set P of forbidden pairs, we want to find a stable matching that includes all pairs from Q, no pair from P, and is as close as possible to M1. We study this problem in four classic stable matching settings: Stable Roommates (with Ties) and Stable Marriage (with Ties). Our main contribution is a polynomial-time algorithm, based on the theory of rotations, for adapting Stable Roommates matchings to forced pairs. In contrast, we show that the same problem for forbidden pairs is NP-hard. However, our polynomial-time algorithm for forced pairs can be extended to a fixed-parameter tractable algorithm with respect to the number of forbidden pairs. Moreover, we study the setting where preferences contain ties: Some of our algorithmic results can be extended while other problems become intractable.

A Linear Time Shrinking-SL(t)-ViT Approach for Brain Tumor Identification and Categorization

Automated brain tumor detection and classification systems have gained popularity in recent years because traditional diagnosis procedures are time-consuming and costly in nature. Deep learning(DL) methods, specifically pre-trained convolutional neural networks (CNNs), have shown promising results in accurately and rapidly classifying brain tumors. However, the lack of diverse magnetic resonance imaging (MRI) datasets has hindered the ability of DL algorithms to generalize effectively. To address this issue, this paper proposes a DL model using generative adversarial networks (GANs) in conjunction with data augmentation strategies and a structural similarity loss function employed for generating annotated images. A novel DL model inspired from Vision Transformer, Shrinking Linear Time Vision Transformer (SL(t)-ViT) network model is proposed for disease data classification. The model underwent extensive evaluation across multiple datasets, employing standard performance metrics to assess its efficacy in brain tumor identification. The proposed model achieved remarkable testing accuracies of 0.995, 0.996, 0.9954, 0.998, and 0.997 for binary classification tasks across multiple datasets, and 0.986, 0.982, 0.985, and 0.993 for multi-class classification tasks. These results underscore the superior performance of our proposed model, showcasing its capability to outperform state-of-the-art techniques. Specifically, it demonstrated a substantial margin of improvement, ranging from 1-2% for binary classification and 9-10% for multi-class classification, solidifying its position as a leading approach in brain tumor identification. Results demonstrate the efficacy of the proposed model, outperforming other models. Overall, this study highlights the potential of SL(t)-ViT and GANs in improving the accuracy, resource consumption, and efficiency of brain tumor diagnosis.

Parameterized complexity of coverage in multi-interface IoT networks: Pathwidth

The Internet of Things (IoT) has emerged as one of the growing fields in digital technology over the past decade. A primary goal of IoT is to connect physical objects to the Internet to provide various services. Due to the vast number and diversity of these objects, referred to as devices, IoT must tackle both traditional and novel theoretical and practical network problems. Among these, multi-interface problems are well-known and have been extensively studied.This research focuses on one of the newest multi-interface models that fits well within the IoT context. It is known as the Coverage in the budget-constrained multi-interface problem, where the budget represents the total amount of energy in the network, and coverage refers to the model’s goal of activating all required communications among IoT devices. Since most IoT devices are battery-powered, energy consumption must be considered to extend the network’s lifespan. This means selecting the most energy-efficient interface configuration that allows all desired connections to function. To achieve this, both global energy consumption and the local number of active interfaces are limited. Moreover, this model also incentivize devices to turn on the available interfaces to create a more performant network. Finally, this model also takes into account the performance of the networks assigning a profit to devices that activate interfaces and realize connections.This problem can be represented using an undirected graph where each vertex represents a device, and each edge represents a desired connection. Every device is equipped with a set of available interfaces that can be used to facilitate transmission among the devices. The final goal is to activate a subset of the available interfaces that maximize the total profit, while not violating the constraints.This problem has been recognized as NP-hard, which is why we decided to investigate the decision version from the perspective of fixed-parameter tractability (FPT) theory. FPT is an advanced area of complexity theory that aims to identify the core complexity of a combinatorial problem by incorporating parameters into the time complexity domain.We provide two fixed-parameter tractability results, each describing an FPT algorithm. One algorithm is based on the well-known pathwidth parameter, the number of available interfaces, and the maximum available energy. The other algorithm considers pathwidth, the number of available interfaces, and an upper bound on the optimal profit. Finally, we show that these two algorithms can be applied to the maximization version of the problem.

Quantum Attacks on MIBS Block Cipher Based on Bernstein–Vazirani Algorithm

Because of the substantial progress in quantum computing technology, the safety of traditional cryptologic schemes is facing serious challenges. In this study, we explore the quantum safety of the lightweight cipher MIBS and propose quantum key-recovery attacks on the MIBS cipher by utilizing Grover’s algorithm and Bernstein–Vazirani algorithm. We first construct linear-structure functions based on the 5-round MIBS cipher according to the characteristics of the linear transformations, and then we obtain a quantum distinguisher of the 5-round MIBS cipher by applying Bernstein–Vazirani algorithm to the constructed functions. Finally, utilizing this distinguisher and Grover’s algorithm, we realize a 7-round key-recovery attack on the MIBS cipher, and then we expand the attack to more rounds of MIBS based on a similar idea. The quantum attack on the 7-round MIBS requires 156 qubits and has a time complexity of 210.5. An 8-round attack requires 179 qubits and has a time complexity of 222. Compared with existing quantum attacks, our attacks have better time complexity when attacking the same number of rounds.

Determinants of Exchange Rate Jumps in Türkiye

This study examines the exchange rate jumps in Türkiye between 2013 and 2021 and the factors influencing these jumps. The Turkish Lira experienced a consistent depreciation against other currencies during this period. A closer examination of the depreciation timeline revealed that the Turkish Lira's depreciation was occasionally abrupt and exceedingly pronounced. The primary objective of this research is to identify these episodes of spikes, which can be characterized as jumps in exchange rates amid regular increases. To achieve this, the Pruned Exact Linear Time (PELT) algorithm was employed to detect sudden shifts in the exchange rate. Taking these points as dependent variables, a rare event logistic regression model was utilized to determine the probability of an exchange rate jump. The findings indicate that increased dollarization raises the likelihood of an exchange rate jump, while higher deposit rates and central bank reserves reduce the probability of a jump.

Optimizing Seismic Design of Multi-Tower Buildings Using Sky Bridge Isolation and BIM: A Case Study

This research aims to extend prior knowledge of sky bridge isolations in a design case study that complies with building codes, focusing on the design of a multi-tower building linked with a sky bridge and its isolation system. Building Information Modeling (BIM) is used during the design process. Linear time history analysis is performed to capture seismic responses without statistical distortion of response combinations. Link elements are used to simulate the isolations, and the ground motions are excited in bidirectional directions. The experimental results demonstrate that using an isolation system at the sky bridge connection improves torsional behavior, as evidenced by a 12% reduction in torsional mass contribution in the fundamental mode shape of the buildings. Other notable improvements include better lateral force distributions and optimization of reinforcement volume by 36.91% at maximum. Additionally, convenient post-design procedures, such as automated design visualizations and quantity surveys of reinforcements are reported through using BIM.

A fast local search based memetic algorithm for the parallel row ordering problem

The parallel row ordering problem (PROP) is concerned with arranging two groups of facilities along two parallel lines with the goal of minimizing the sum of the flow cost-weighted distances between the pairs of facilities. As the main result of this paper, we show that the insertion neighborhood for the PROP can be explored in optimal time Θ(n2) by providing an O(n2)-time procedure for performing this task, where n is the number of facilities. As a case study, we incorporate this procedure in a memetic algorithm (MA) for solving the PROP. We report on numerical experiments that we conducted with MA on PROP instances with up to 500 facilities. The experimental results demonstrate that the MA is superior to the adaptive iterated local search algorithm and the parallel hyper heuristic method, which are state-of-the-art for the PROP. Remarkably, our algorithm improved best known solutions for six largest instances in the literature. We conjecture that the time complexity of exploring the interchange neighborhood for the PROP is Θ(n2), exactly as in the case of insertion operation.

Reconfiguring Shortest Paths in Graphs

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P≠NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\ exttt {P}}\\,}}\ e {{\\,\\mathrm{\ exttt {NP}}\\,}}$$\\end{document}), even for relaxed variants of the problem (assuming P≠PSPACE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\ exttt {P}}\\,}}\ e {{\\,\\mathrm{\ exttt {PSPACE}}\\,}}$$\\end{document}). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 2$$\\end{document}) contiguous vertices on a shortest path can be changed at a time.

A Deterministic Method to Construct a Common Supercell Between Two Similar Crystalline Surfaces.

Here, a deterministic algorithm is proposed, that is capable of constructing a common supercell between two similar crystalline surfaces without scanning all possible cases. Using the complex plane, the 2D lattice is defined as the 2D complex vector. Then, the relationship between two surfaces becomes the eigenvector-eigenvalue relation where an operator corresponds to a transformation matrix. It is shown that this transformation matrix can be directly determined from the lattice parameters and rotation angle of the two given crystalline surfaces with O(log Nmax) time complexity, where Nmax is the maximum index of repetition matrix elements. This process is much faster than the conventional brute force approach ( ). By implementing the method in Python code, experimental 2D heterostructures and their moiré patterns and additionally find new moiré patterns that have not yet been reported are successfully generated. According to the density functional theory (DFT) calculations, some of the new moiré patterns are expected to be as stable as experimentally-observed moiré patterns. Taken together, it is believed that the method can be widely applied as a useful tool for designing new heterostructures with interestingproperties.

Unearthing the Disturbing Ghosts of Postcolonial LandsThat Linger After the Sermon

What lingers over a wounded community after preaching? This paper argues that the disturbing ghosts of postcolonial lands linger in the pulpit and haunt the wounded hearers after the sermon. Preachers should not only acknowledge but also preach about these ghosts whose presence are found in the Bible and in Christ. Mirroring the features of the ghosts, such preaching, which I term haunto-postcolonial preaching, disrupts the conventional notions of linear time and segregated space and connects the ghosts to the hearers and the Bible. Its goal is to create a haunting space in the pulpit, unearthing unresolved postcolonial trauma to bring rupture to the existing oppressive structures in the church and society. By allowing the ghosts to haunt the congregation about a lost or a reality yet-to-be-attained, preachers invite their hearers to reimagine and pursue a better and just society beyond the sermon where the unappeased souls can be consoled.

Generalization of the Synthetic Aperture Radar Azimuth Multi-Aperture Processing Scheme—MAPS

This paper analyzes the advantages and the drawbacks of using the Synthetic Aperture Radar (SAR) azimuth multichannel technique known as Multi-Aperture Processing Scheme (MAPS), in a set of relevant application cases that are far from the canonical ones. In the scientific literature on this topic, equally distributed azimuth channels with the quasi-monostatic deployment are assumed. With this research, we aim at extending the models from the current literature to (i) a generic bistatic acquisition geometry, (ii) a set of cases where the number of receiving tiles is not the same for each channel, or (iii) the tiles are shared between adjacent channels thus creating an overlapping configuration. The paper introduces the mathematical models for the listed non-conventional MAPS cases. Dealing with the bistatic MAPS, we first solve the problem by interpreting multichannel acquisition as a bank of Linear Time Invariant (LTI) filters. Then, a more physical approach, based on discrimination of the direction of arrivals (DoAs) is pursued. The effectiveness of the two methods and the advantages of the second approach on the first are proved by using a simplified 1D end-to-end simulation. Even limiting to the monostatic configuration, the azimuth antenna tiles have always been supposed equally partitioned among the RX channels. Overcoming this limit has two advantages: (i) more MAPS possible solutions in case few azimuth tiles are available, as in the ROSE-L mission; (ii) the number of channels can be designed independently of the number of tiles, also allowing asymmetric solutions, useful for a phase array antenna with an odd number of tiles such as in the SAOCOM-1 mission. Conversely, sharing one or more receiving tiles in different receiving channels makes the input noise partially correlated. The drawback is an increase in the noise level. A trade-off is determined for the different solutions obtained using simulations with real mission parameters. The theoretical performance and the end-to-end simulations are compared.

Real-Time Optimal Scheduling of a Water Diversion System Using an Improved Wolf-Pack Algorithm and Scheme Library

A water diversion system (WDS) with cascade pumping stations (CPSs) plays an important role in the application of water resources. However, high energy consumption is reported due to unreasonable scheduling schemes and long decision times. Herein, this paper presents a new method to achieve optimal scheduling schemes effectively, including the head allocation of CPSs, the number of running pumps, and pump blade angles. A double-layer mathematical model for a WDS was established with the goal of achieving minimal energy consumption, considering the constraints of flow rate, water level, and the characteristics of pump units. The inner-layer model was used to obtain scheduling schemes of single-stage pumping stations, as well as the water levels and flow rates of water channels, while the outer-layer model was used to optimize inter-stage head allocation. An improved wolf-pack algorithm (IWPA) was proposed to solve the model, using a Halton sequence to obtain the uniform initial population distribution and introducing simulated annealing (SA) to improve the global searchability. Moreover, an idea for a pre-established scheme library was suggested for inner-layer models to obtain the solutions in real time with less calculation workload. Taking an actual project as a case, in contrast with the actual schemes, the optimal scheduling method could result in energy savings of 14.37–20.39%, a CO2 emission reduction of 13–32 tons per day, and water savings of 0.14–18.34%. Moreover, the time complexity decreased to square order, and the CPU time of the optimal method was about 1% that of the traditional method. This study provides an efficient method for the high-value utilization of energy and water resources for a WDS.

Frustrated random walks: A fast method to compute node distances on hypergraphs.

A hypergraph is a generalization of a graph that arises naturally when attribute-sharing among entities is considered. Compared to graphs, hypergraphs have the distinct advantage that they contain explicit communities and are more convenient to manipulate. An open problem in hypergraph research is how to accurately and efficiently calculate node distances on hypergraphs. Estimating node distances enables us to find a node's nearest neighbors, which has important applications in such areas as recommender system, targeted advertising, etc. In this paper, we propose using expected hitting times of random walks to compute hypergraph node distances. We note that simple random walks cannot accurately compute node distances on highly complex real-world hypergraphs, which motivates us to introduce frustrated random walks (FRW) for this task. We further benchmark our method against DeepWalk, and show that while the latter can achieve comparable results, FRW has a distinct computational advantage in cases where the number of targets is fairly small. For such cases, we show that FRW runs in significantly shorter time than DeepWalk. Finally, we analyze the time complexity of our method, and show that for large and sparse hypergraphs, the complexity is approximately linear, rendering it superior to the DeepWalk alternative.

3DRecNet: A 3D Reconstruction Network with Dual Attention and Human-Inspired Memory

Humans inherently perceive 3D scenes using prior knowledge and visual perception, but 3D reconstruction in computer graphics is challenging due to complex object geometries, noisy backgrounds, and occlusions, leading to high time and space complexity. To addresses these challenges, this study introduces 3DRecNet, a compact 3D reconstruction architecture optimized for both efficiency and accuracy through five key modules. The first module, the Human-Inspired Memory Network (HIMNet), is designed for initial point cloud estimation, assisting in identifying and localizing objects in occluded and complex regions while preserving critical spatial information. Next, separate image and 3D encoders perform feature extraction from input images and initial point clouds. These features are combined using a dual attention-based feature fusion module, which emphasizes features from the image branch over those from the 3D encoding branch. This approach ensures independence from proposals at inference time and filters out irrelevant information, leading to more accurate and detailed reconstructions. Finally, a Decoder Branch transforms the fused features into a 3D representation. The integration of attention-based fusion with the memory network in 3DRecNet significantly enhances the overall reconstruction process. Experimental results on the benchmark datasets, such as ShapeNet, ObjectNet3D, and Pix3D, demonstrate that 3DRecNet outperforms existing methods.

A linear algorithm for radio k-coloring of powers of paths having small diameters

The radio k-chromatic number rck(G) of a graph G is the minimum integer λ such that there exists a function ϕ:V(G)→{0,1,⋯,λ} satisfying |ϕ(u)−ϕ(v)|≥k+1−d(u,v), where d(u,v) denotes the distance between u and v. A considerable amount of attention has been given to find the exact values or providing polynomial time algorithms to determine rck(G) for several basic graph families such as paths, cycles, trees, and powers of paths, usually for some specific values of k. In this article, we find the exact values of rck(G) where G is a power of a path with diameter strictly less than k. Our proof readily provides a linear time algorithm for assigning a radio k-coloring of G. Furthermore, our proof technique is a potential tool for solving the same problem for other classes of graphs having “small” diameters.

WaveFusionNet: Infrared and visible image fusion based on multi-scale feature encoder–decoder and discrete wavelet decomposition

To merge complementary information from multimodal images, such as thermal saliency from infrared images and texture details from visible images, traditional multi-scale transform-based methods have been extensively studied, with deep learning-based methods gaining significant popularity in recent years. However, there has been limited research on optimally combining the advantages of these two categories in fusion. In this paper, we propose a novel infrared and visible image fusion (IVIF) framework, WaveFusionNet, which integrates precise frequency feature decomposition from the discrete wavelet transform (DWT) with the comprehensive feature extraction from the multi-scale encoder. Firstly, we train an encoder–decoder network for multi-scale feature extraction and image reconstruction. DWT is used for down-sampling with minimal information loss by decomposing extracted features into low and high-frequency sub-bands. Next, a dual-band feature fusion (DBFF) module is trained to merge these sub-bands by integrating a spatial feature transform-based sub-network for low-frequency fusion and a maximum absolute value selection strategy for fusing high-frequencies. Finally, all fused sub-bands are fed into the pre-trained decoder to reconstruct the final image. Experimental results on three benchmark datasets (TNO, Roadscene, and MSRS) demonstrate that the proposed fusion method outperforms recent IVIF methods in both quantitative assessment and visual perception while maintaining competitive time complexity.

Finite-horizon energy allocation scheme in energy harvesting-based linear wireless sensor network

Linear wireless sensor networks (LWSNs) are a specialized topology of wireless sensor networks (WSNs) widely used for environmental monitoring. Traditional WSNs rely on batteries for energy supply, limiting their performance due to battery capacity constraints, while renewable energy harvesting technology is an effective approach to alleviating the battery capacity bottleneck. However, the stochastic nature of renewable energy makes designing an efficient energy management scheme for network performance improvement a compelling research problem. In this paper, we investigate the problem of maximizing throughput over a finite-horizon time period for an energy harvesting-based linear wireless sensor network (EH-LWSN). The solution to the original problem is very complex, and this complexity mainly arises from two factors. First, the optimal energy allocation scheme has temporal coupling, i.e., the current optimal strategy relies on the energy harvested in the future. Second, the optimal energy allocation scheme has spatial coupling, i.e., the current optimal strategy of any node relies on the available energy of other nodes in the network. To address these challenges, we propose an iterative energy allocation algorithm for EH-LWSN. Firstly, we theoretically prove the optimality of the algorithm and analyze the time complexity of the algorithm. Next, we design the corresponding distributed version and consider the case of estimating the energy harvest. Finally, through experiments using a real-world renewable energy dataset, the results show that the proposed algorithm outperforms the other two heuristics energy allocation schemes in terms of network throughput.

Grid-Based DBSCAN Clustering Accelerator for LiDAR’s Point Cloud

Autonomous robots operate on batteries, rendering power efficiency essential. The substantial computational demands of object detection present a significant burden to the low-power cores employed in these robots. Therefore, we propose a grid-based density-based spatial clustering of applications with a noise (DBSCAN) clustering accelerator for light detection and ranging (LiDAR)’s point cloud to accelerate computational speed and alleviate the operational burden on low-power cores. The proposed method for DBSCAN clustering leverages the characteristics of LiDAR. LiDAR has fixed positions where light is emitted, and the number of points measured per frame is also fixed. These characteristics make it possible to impose grid-based DBSCAN on clustering a LiDAR’s point cloud, mapping the positions and indices where light is emitted to a 2D grid. The designed accelerator with the proposed method lowers the time complexity from O(n2) to O(n). The designed accelerator was implemented on a field programmable gate array (FPGA) and verified by comparing clustering results, speeds, and power consumption across various devices. The implemented accelerator speeded up clustering speeds by 9.54 and 51.57 times compared to the i7-12700 and Raspberry Pi 4, respectively, and recorded a 99% reduction in power consumption compared to the Raspberry Pi 4. Comparisons of clustering results also confirmed that the proposed algorithm performed clustering with high visual similarity. Therefore, the proposed accelerator with a low-power core successfully accelerated speed, reduced power consumption, and effectively conducted clustering.

The circular balancing problem

We propose a balancing problem with a minmax objective in a circular setting. This balancing problem involves the arrangement of an even number of items with different weights on a circle while minimizing the maximum total weight of items arranged on any half circle. Due to its generic structure, it may have applications in fair resource allocation schemes. We show the NP-hardness of the problem and develop polynomial-time algorithms when the number of distinct weights is a fixed constant. We propose for the general case a tight 7/6-approximation algorithm and show that it performs better than two existing algorithms designed for an equivalent problem in the literature. The worst-case performance ratio is derived through a linear combination of valid inequalities that are obtained from the problem definition, the properties of the proposed algorithm, and the optimal circular permutation structure. Furthermore, we formulate a more general problem of minimizing the maximum total weight of items on equally divided circular sectors and present its computational complexity and a tight approximation algorithm.

Study on Ground Motion Amplification in Upper Arch Bridge Due to “W”-Type Deep Canyon Using Boundary-Integral and Peak Frequency Shift Methods

The study of the dynamic response characteristics of “W”-type deep canyon terrain to double-span concrete arch bridges under earthquake action holds great practical significance. In this research, a bridge in Sichuan Province is taken as the object of study. The boundary-integral equation method and peak frequency shift method are combined to apply an embedded linear time–history analysis algorithm to the finite element spatial dynamic calculation model of the entire bridge, resulting in an improved model. By comparing these two methods with model test results, the seismic response characteristics of the middle part of a “W” concrete arch bridge under different foundation depths and seismic intensities are examined. The boundary integral equation method was utilized to calculate ground motion response at any point on site, revealing a significant amplifying effect of increased seismic wave intensity on acceleration response at the top of the arch bridge. When input seismic wave intensity increased from 0.1 g to 0.3 g, maximum acceleration at buried depths of 3 m and 8 m in the middle of the arch bridge foundation increased by 102.63% and 79.16%, respectively, indicating that shallow buried depth structures are more sensitive to seismic wave intensity. Furthermore, using peak frequency shift rules for analyzing seismic wave propagation characteristics in “W”-type deep canyon topography confirms the sensitivity of shallow buried depth structures to seismic wave intensity and reveals the mechanism through which topography influences seismic wave propagation. This study provides a helpful method for understanding the propagation law and energy distribution characteristics of seismic waves in complex terrain. It was observed that the displacement at the top of the arch bridge increased significantly with an increase in seismic intensity. When subjected to 0.1 g, 0.2 g, and 0.3 g EI-Centro seismic waves, the maximum displacement at the top of the arch bridge model with a foundation buried depth of 3 m was 8 mm, 32 mm, and 142 mm, respectively. For arch bridge models with an 8-m foundation buried depth, these displacements were measured at 6.2 mm, 21 mm, and 68 mm, respectively. The results from model tests verified that increasing the depth of foundation burial effectively reduces the displacement at the top of the structure. Furthermore, by combining a boundary-integral equation method and peak-frequency shift method, this study accurately predicted significant influences on W-shaped double deep canyon topography from seismic response, and successfully captured stress concentration and seismic wave amplification/focusing effects on arch foot structures. The calculated results from both methods align well with model test data which confirm their effectiveness and complementarity when analyzing seismic responses under complex terrain conditions for bridge structures.