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In this paper, we introduce a centralized algorithm which constructs the generalized Cech complex. The generalized Cech complex represents the topology of a wireless network whose cells are different in size. This complex is useful to address a wide variety of problems in wireless networks such as: boundary holes detection, disaster recovery or ene...
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... construction of k-simplices where k ≥ 2 is more complex. The rest of this section is devoted to the details of the construction of k-simplices where k ≥ 2. From the defintion of thě Cech complex, each k-simplex represents a group of (k + 1) cells which has a non-empty intersection. The number of combinations of (k + 1) cells in all N cells is huge. We should first find out a candidate group that has an opportunity to be a k-simplex. Let us assume that u = {v 0 , v 1 , . . . , v k } is a k-simplex. Then we can deduce that each pair (v i , v j ), where 0 ≤ i = j ≤ k, are neighbors. This suggest thatû thatˆthatû = {ˆv{ˆv 0 Look at the example from Figure 2, the cell i is the smallest cell and it is inside the others. So the four cells i, j, m, and n compose a 3-simplex (i, j, m, n). If the first case is not satisfied, we consider the second case that the smallest cell v * is not inside others and there exists a intersection point x ij ∈ X that is inside cell v t for all 0 ≤ t ≤ k and t = i, j. We then conclude thatûthatˆthatû = {ˆv{ˆv 0 , ˆ v 1 , . . . , ˆ v k } is a k-simplex. In Figure 3, an intersection point of cell i and cell j is inside other cells m and n. This point is marked as a red point in Figure 3. So, we conclude that (i, j, m, n) is a 3-simplex. If both first case and second case are not satisfied, we conclude thatûthatˆthatû = {ˆv{ˆv 0 , ˆ v 1 , . . . , ˆ v k } is not a k-simplex. The smallest cell v * is not inside others and no intersection point x ij ∈ X is inside cell v t for any 0 ≤ t ≤ k and t = i, j. So there must exist i * , j * and m * such that x i * and x j * are not inside the cell m * . So (i * , j * , m * ) isn't a 2-simplex then it can not be part of a k-simplex with k ≥ 2. In Figure 4, we can not find a pair of cells that one of their intersection points is inside all the other cells. In addition, both intersection points of cell i and cell j are not inside cell m. So the three cells i, j and m do not compose a 2-simplex. Then, four cells i, j, m and n do not compose a 3-simplex. The algorithm to verify a candidate if it's a k-simplex, where k ≥ 2, is given ...
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... Rescaling the radii by the same factor, we obtain a filtered generalizedČech complex, where the associated simplicial complexes evolve as the scale parameter varies. In particular, in [10], algorithms are provided to calculate the generalizedČech complex in R 2 , and [11] presents an algorithm to determine theČech scale for a collection of disks in the plane. ...
In this article, we study the intersection of a finite collection of disks in Euclidean space by examining spheres of various dimensions and their poles (extreme values with respect to canonical projections) contained within the intersection’s boundary. We derive explicit formulae for computing these extreme values and present two applications. The first application involves computing the smallest common rescaling factor for the radii of the disk system, which brings the system to a single point of intersection. This calculation allows us to compute the generalized Čech filtration, a crucial tool for the topological data analysis of weighted point clouds. The second application focuses on determining the minimal Axis-Aligned Bounding Box (AABB) for the intersection of a finite collection of disks in Euclidean space, addressing a significant problem in computational geometry. We consider that this work aims to contribute to the fields of topological data analysis and computational geometry by providing new tools for analyzing complex geometric structures.
... Rescaling the radii by the same factor, we obtain a filtered generalizedČech complex, where the associated simplicial complexes evolve as the scale parameter varies. In particular, in [2], algorithms are provided to calculate the generalizedČech complex in R 2 , and [3] presents an algorithm to determine theČech scale for a collection of disks in the plane. ...
... Finally, for i = q, we can use the last expression to substitute it in (2) and obtain the desired result. ...
In this work we study the intersection properties of a finite disk system in the euclidean space. We accomplish this by utilizing subsets of spheres with varying dimensions and analyze specific points within them, referred to as poles. Additionally, we introduce two applications: estimating the common scale factor for the radii that makes the re-scaled disks intersects in a single point, this is the \v{C}ech scale, and constructing the minimal Axis-Aligned Bounding Box (AABB) that encloses the intersection of all disks in the system.
... Čech complex and Rips complex [45], [46], [47] has been adopted by authors to find out coverage holes. A comparative analysis of Čech complex and Rips complex [44] has been done in respect of coverage holes by varying ratio of communication radii (Rc) to sensing radii (Rs) of a sensor node. ...
The paper aims to put in the forefront a survey on the existing holes problem in the Wireless Underground Sensor Networks (WUSNs). Anomalies in wireless sensor networks can weaken the network by impeding their functionalities. These existing anomalies commonly known as holes may cause irreversible devastating problems and may prove fatal to humans. There can be a different kind of holes in the network such as coverage hole, routing hole, jamming hole, and sink/ worm hole, etc. This paper intends to discuss in entirely various holes problems in WUSNs, reasons behind their occurrence and different techniques to mitigate the issues. An efficient hole detection technique shall increase network lifetime, use lesser resources and give accurate results with zero false positive. To put a light on the present scenario, methods/ techniques used to cope with different types of holes have been discussed in detail, presenting their analysis, constraints, and advantages. Besides, it would also touch upon the comparative analysis in terms of soundness and limitations of these techniques. The study is concluded with future research directions.
... Assume that one has access to the set E t (X n ) of edges of X n of length smaller than 2t. Then, authors in [28] propose an algorithm of complexity C D n i=1 N (X i ) D to compute Cech(t, X n ). When t is of order (ln n/n) 1/d , N (X i ) is on average of order ln n and we obtain an average complexity of order C D n(ln n) D . ...
... In high dimension, the complexity can be reduced if one has access to the dimension d by computing Conv d (t, X n ) instead (see Remark 3.8). Indeed, according to [28], the set of simplices of Cech(t, X n ) of dimension smaller than d can be computed with average time complexity of order C d Dn(ln n) d . We also have to consider the computation of the edges E t (X n ). ...
... If there is a possibility to change from to in such a way that is approximate to a continuous one, we have to replace (1) the sign of the sum with the integral sign. In Step 2 there are several possibilities to construct simplicial complexes, for example, Cech complexes [18], Delaunay complexes [19] and alpha complexes [20]. Each of them has own advantages and disadvantages. ...
In that, the digital image segmentation algorithm is pre-processing for many systems of a machine vision, object detection, etc. and there is no universal one for any type of digital image, it is necessary to obtain a new approach to it. Topological data analysis is issued not so long ago but its instruments have a universal character. We obtained the algorithm which is based on the persistent homologies as the effective mode of topological data analysis and its implementation in C# (.NET 4.5). Its testing on different types of images (data of aerial photography, compositions, etc.) was held and the results were compared with the K-means algorithm.
... The most famous one, the Cech complex, is built by checking the intersection of every collection of simplices, making it extremely cumbersome computationally. Indeed, building a Cech complex from N points up to it highest dimension n can cost O(N 2 +N 2n ), making the direct calculation absolutely unfeasible in most cases [60]. The standard alternative is the Rips-Vietoris complex, which can be defined as an abstract simplicial complex defined on the basis of a metric space M with distance d by constructing a simplex for each set of points with diameter at most d, or equivalently, as the clique complex associated to a metrical graph. ...
Technological advances are providing researchers with larger and larger amounts of data, creating a rich landscape for data scientists to explore and search for meaningful patterns. Novel biomedical imaging techniques over the next 5 to 10 years, for instance, will develop their resolution to the point at which single subject scans might provide terabytes of data. While traditional data analytics cannot handle these large volumes of data, various Big Data statistical methods should be investigated and developed. This is clearly a great challenge for all stages of data management, but most importantly for the analysis and interpretation. In this chapter, we discusse a very recent paradigm and a set of techniques rooted in algebraic topology, which are able to scale and provide useful information within the context of abundant but often low-quality data sets. These techniques, collectively referred to as Topological Data Analysis, can extract meaningful information without the need to rely on specific a priori models or hypotheses. The potential computational and interpretative pitfalls as well as the potential benefits of using a robust-by-design high-order relational data description are also highlighted. Among these, we focus on persistent homology, its combination with machine learning, as well as topological simplification, together with their applications in biomedical signal processing.
... Therefore, this algorithm only works with a collection of same sized cells. Although an algorithm to compute theČech complex for a collection of differently sized cells is proposed in [6], this algorithm is still centralized. ...
... The higher dimensional simplex is, the higher overlap times is. 6 ] means the four corresponding cells: c 0 , c 1 , c 2 and c 6 , together, have a common intersection. In contrast, a chain of 1-simplices indicates a coverage hole inside corresponding cells of the chain. ...
... When the collection of neighbors is available, each cell computes its simplices by verifying the intersection among it and its neighbors. For the details about the intersection verification, see [6]. As each pair of neighbors forms a 1simplex, the collection of 1-simplices of each cell is easily found. ...
In this paper, we introduce a distributed algorithm to compute th\v{e} Cech complex. This algorithm is aimed at solving coverage problems in self organized wireless networks. Two applications based on the distributed computation of th\v{e} Cech complex are proposed. The first application detects coverage holes while the later one optimizes coverage of wireless networks.
... Les stations de base peuvent être déployées partout et être de toute nature. La topologie du réseau peut être capturée par un objet mathématique appelé complexe de Čech [43]. Ce dernier rend compte exactement de la topologie d'un réseau [23]. ...
... The process of the Čech complex computation is shown by corresponding illustrated pictures. This work has been published in [43]. ...
... The minimal dimensional Čech complex that satisfies requirements of these applications can be constructed in only polynomial time. This work has been published in [43]. ...
Simplicial homology is a useful tool to access important information about the topology of wireless networks such as : coverage and connectivity. In this thesis, we model the wireless network as a random deployment of cells. Firstly, we introduce an algorithm to construct the Cech complex, which describes exactly the topology of the network. Then, the Cech complex is used in further applications. The first application is to save transmission power for wireless networks. This application not only maximizes the coverage of the network but also minimizes its transmission power. At the same time, the coverage and the transmission power are optimized. The second application is to balance the traffic load in wireless networks. This application controls the transmission power of each cell in the network, always under the coverage constraint. With the controlled transmission power, the users are redirected to connect to the lower traffic load cells. Consequentially, the balanced traffic load is obtained for the network.
... The construction of the Čech complex is based on the verification of intersection between cells. For the detail of this construction, see the document [13]. The Betti numbers can be computed by equation (1). ...
Energy saving is one of the most investigated problems in wireless networks. In this paper, we introduce two homology based algorithms: a simulated annealing one and a downhill one. These algorithms optimize the energy consumption at network level while maintaining the maximal coverage. By using simplicial homology, the complex geometrical calculation of the coverage is reduced to simple matrix computation. The simulated annealing algorithm gives a solution that approaches the global optimal one. The downhill algorithm gives a local optimal solution. The simulated annealing algorithm and downhill algorithm converge to the solution with polynomial and exponential rate, respectively. Our simulations show that this local optimal solution also approaches the global optimal one. Our algorithms can save at most 65% of system's maximal consumption power in polynomial time. The probability density function of the optimized radii of cells is also analyzed and discussed.
