Edge computing in the Internet of Things brings applications and content closer to the users by introducing an additional computational layer at the network infrastructure, between cloud and the resource-constrained data producing devices and user equipment. This way, the opportunistic nature of the operational environment is addressed by introducing computational power in location with low latency and high bandwidth. However, location-aware deployment of edge computing infrastructure requires careful placement scheme for edge servers. To provide the best possible Quality of Service for the user applications, their proximity needs to be optimized. Moreover, the deployment faces practical limitations in budget limitations, hardware requirements of servers and in online load balancing between servers. To address these challenges, we formulate the edge server placement as a capacitated location-allocation problem, while minimizing the distance between servers and access points of a real city-wide Wi-Fi network deployment. In our algorithm, we utilize both upper and lower server capacity constraints for load balancing. Furthermore, we enable sharing of workload between servers to facilitate deployments with low capacity servers. The performance of the algorithm is demonstrated in placement scenarios, exemplified by high capacity servers for edge computing and low capacity servers for Fog computing, with different parameters in a real-world data set. The data set consists of both dense deployment of access points in central areas, but also sparse deployment in suburban areas within the same network infrastructure. In comparison, we show that previous approaches do not sufficiently address such deployment. The presented algorithm is able to provide optimal placements that minimize the distances and provide balanced workload with sharing by following the capacity constraints.
We approximate the network topology using the geospatial distances, where proximity is measured using squared Euclidean distances. This produces k-means type clustering with spherical-like clusters with centralized cluster heads @cite_19 . This results in a star-like topology with spatially centralized edge servers in both dense and sparse areas, which contributes towards better proximity, i.e. QoS, particularly in the otherwise remote access points.
@cite_19
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A range assignment to the nodes in a wireless ad hoc network induces a topology in which there is an edge between two nodes if and only if both of them are within each other's transmission range. The critical transmission radius for k-connectivity is the smallest r such that if all nodes have the transmission radius r, the induced topology is k-connected. In this paper, we study the asymptotic critical transmission radius for k-connectivity in a wireless ad hoc network whose nodes are uniformly and independently distributed in a unit-area square or disk. We provide a precise asymptotic distribution of the critical transmission radius for k-connectivity. In addition, the critical neighbor number for k-connectivity is the smallest integer l such that if every node sets its transmission radius equal to the distance between itself and its l-th nearest neighbor, the induced (symmetric) topology is k-connected. Applying the critical transmission radius for k-connectivity, we can obtain an asymptotic almost sure upper bound on the critical neighbor number for k-connectivity. Recent papers have claimed that the result of K-means clustering for time series subsequences (STS clustering) is independent of the time series that created it. Our paper revisits this claim. In particular, we consider the following question: Given several time series sequences and a set of STS cluster centroids from one of them (generated by the K-means algorithm), is it possible to reliably determine which of the sequences produced these cluster centroids? While recent results suggest that the answer should be NO, we answer this question in the affirmative.We present cluster shape distance, an alternate distance measure for time series subsequence clusters, based on cluster shapes. Given a set of clusters, its shape is the sorted list of the pairwise Euclidean distances between their centroids. We then present two algorithms based on this distance measure, which match a set of STS cluster centroids with the time series that produced it. While the first algorithm creates DQG reuse this term more smaller "fingerprints" for the sequences, the second is more accurate. In our experiments with a dataset of 10 sequences, it produced a correct match 100 of the time.Furthermore, we offer an analysis that explains why our cluster shape distance provides a reliable way to match STS clusters to the original sequences, whereas cluster set distance fails to do so. Our work establishes for the first time a strong relation between the result of K-means STS clustering and the time series sequence that created it, despite earlier predictions that this is not possible. In recent years, the effect of the curse of high dimensionality has been studied in great detail on several problems such as clustering, nearest neighbor search, and indexing. In high dimensional space the data becomes sparse, and traditional indexing and algorithmic techniques fail from a efficiency and or effectiveness perspective. Recent research results show that in high dimensional space, the concept of proximity, distance or nearest neighbor may not even be qualitatively meaningful. In this paper, we view the dimensionality curse from the point of view of the distance metrics which are used to measure the similarity between objects. We specifically examine the behavior of the commonly used Lk norm and show that the problem of meaningfulness in high dimensionality is sensitive to the value of k. For example, this means that the Manhattan distance metric (L1 norm) is consistently more preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Using the intuition derived from our analysis, we introduce and examine a natural extension of the Lk norm to fractional distance metrics. We show that the fractional distance metric provides more meaningful results both from the theoretical and empirical perspective. The results show that fractional distance metrics can significantly improve the effectiveness of standard clustering algorithms such as the k-means algorithm. A range assignment to the nodes in a wireless ad hoc network induces a topology in which there is an edge between two nodes if and only if both of them are within each other's transmission range. The critical transmission radius for k-connectivity is the smallest r such that if all nodes have the transmission radius r,the induce topology is k-connected. The critical neighbor number for k-connectivity is the smallest integer l such that if every node sets its transmission radius equal to the distance between itself an its l-th nearest neighbor, the induce topology is k-connecte. In this paper, we study the asymptotic critical transmission radius for k-connectivity an asymptotic critical neighbor number for k-connectivity in a wireless ad hoc network whose nodes are uniformly an independently distribute in a unit-area square or disk. We provide a precise asymptotic distribution of the critical transmission radius for k-connectivity and an improve asymptotic almost sure upper bound on the critical neighbor number for k-connectivity.