We consider the problem of localizing wireless devices in an ad-hoc network embedded in a d-dimensional Euclidean space. Obtaining a good estimation of where wireless devices are located is crucial in wireless network applications including environment monitoring, geographic routing and topology control. When the positions of the devices are unknown and only local distance information is given, we need to infer the positions from these local distance measurements. This problem is particularly challenging when we only have access to measurements that have limited accuracy and are incomplete. We consider the extreme case of this limitation on the available information, namely only the connectivity information is available, i.e., we only know whether a pair of nodes is within a fixed detection range of each other or not, and no information is known about how far apart they are. Further, to account for detection failures, we assume that even if a pair of devices is within the detection range, it fails to detect the presence of one another with some probability and this probability of failure depends on how far apart those devices are. Given this limited information, we investigate the performance of a centralized positioning algorithm MDS-MAP introduced by , and a distributed positioning algorithm, introduced by , called HOP-TERRAIN. In particular, for a network consisting of n devices positioned randomly, we provide a bound on the resulting error for both algorithms. We show that the error is bounded, decreasing at a rate that is proportional to R Rc, where Rc is the critical detection range when the resulting random network starts to be connected, and R is the detection range of each device.
Our first result, specifically the analysis of MDS-MAP , has a similar flavour as in @cite_17 . We provide a theoretical guarantee that backs up experimental results. We use shortest paths as our primary guess for the missing entries in the distance matrix and apply MDS to find the topology of the network. In contrast to @cite_17 , we require weaker assumptions for our results. More specifically, we assume that only neighbouring sensors have information about each other and that only connectivity information is known. Furthermore, the knowledge of detection probabilities plays no role in our analysis or the algorithm. And last, in our analysis we assume that the average degree grows logarithmically-not linearly- with the number of sensors, which results in needing many less revealed entries in the distance matrix. In particular, the last condition is quite realistic: If the average degree grows any slower then the network is not even connected (more on this issue in ). As the shortest paths algorithm works for both rage-free and range-aware cases, our analysis includes both and provides the first error bounds on the performance of MDS-MAP .
@cite_17
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Sensor localization from only connectivity information is a highly challenging problem. To this end, our result for the first time establishes an analytic bound on the performance of the popular MDS-MAP algorithm based on multidimensional scaling. For a network consisting of n sensors positioned randomly on a unit square and a given radio range r = o(1), we show that resulting error is bounded, decreasing at a rate that is inversely proportional to r, when only connectivity information is given. The same bound holds for the range-based model, when we have an approximate measurements for the distances, and the same algorithm can be applied without any modification. It is often useful to know the geographic positions of nodes in a communications network, but adding GPS receivers or other sophisticated sensors to every node can be expensive. MDS-MAP is a recent localization method based on multidimensional scaling (MDS). It uses connectivity information - who is within communications range of whom - to derive the locations of the nodes in the network, and can take advantage of additional data, such as estimated distances between neighbors or known positions for certain anchor nodes, if they are available. However, MDS-MAP is an inherently centralized algorithm and is therefore of limited utility in many applications. In this paper, we present a new variant of the MDS-MAP method, which we call MDS-MAP(P) standing for MDS-MAP using patches of relative maps, that can be executed in a distributed fashion. Using extensive simulations, we show that the new algorithm not only preserves the good performance of the original method on relatively uniform layouts, but also performs much better than the original on irregularly-shaped networks. The main idea is to build a local map at each node of the immediate vicinity and then merge these maps together to form a global map. This approach works much better for topologies in which the shortest path distance between two nodes does not correspond well to their Euclidean distance. We also discuss an optional refinement step that improves solution quality even further at the expense of additional computation. Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 Despite many efforts over the past decade, the ability to generate topological maps of the Internet at the router-level accurately and in a timely fashion remains elusive. Mapping campaigns commonly involve traceroute-like probing that are usually non-adaptive and incomplete, thus revealing only a portion of the underlying topology. In this paper we demonstrate that standard probing methods yield datasets that implicitly contain information about much more than just the directly observed links and routers. Each probe, in addition to the underlying domain knowledge, returns information that places constraints on the underlying topology, and by integrating a large number of such constraints it is possible to accurately infer the existence of unseen components of the Internet. We describe DomainImpute, a novel data analysis methodology designed to accurately infer the unseen hop-count distances between observed routers. We use both synthetic and a large empirical dataset to validate the proposed methods. On our empirical real world dataset, we show that our methods can estimate over 55 of the unseen distances between observed routers to within a one-hop error. We give, for the first time, a precise mathematical analysis of the connectivity and security properties of sensor networks that make use of the random predistribution of keys. We also show how to set the parameters---pool and key ring size---in such a way that the network is not only connected with high probability via secure links but also provably resilient, in the following sense: We formally show that any adversary that captures sensors at random with the aim of compromising a constant fraction of the secure links must capture at least a constant fraction of the nodes of the network. In the context of wireless sensor networks where random predistribution of keys is employed, we are the first to provide a mathematically precise proof, with a clear indication of parameter choice, that two crucial properties---connectivity via secure links and resilience against malicious attacks---can be obtained simultaneously. We also show in a mathematically rigorous way that the network enjoys another strong security property. The adversary cannot partition the network into two linear size components, compromising all the links between them, unless it captures linearly many nodes. This implies that the network is also fault tolerant with respect to node failures. Our theoretical results are complemented by extensive simulations that reinforce our main conclusions. Abstract Visual tracking algorithms based on structured output support vector machine (SOSVM) have demonstrated excellent performance. However, sampling methods and optimization strategies of SOSVM undesirably increase the computational overloads, which hinder real-time application of these algorithms. Moreover, due to the lack of high-dimensional features and dense training samples, SOSVM-based algorithms are unstable to deal with various challenging scenarios, such as occlusions and scale variations. Recently, visual tracking algorithms based on discriminative correlation filters (DCF), especially the combination of DCF and features from deep convolutional neural networks (CNN), have been successfully applied to visual tracking, and attains surprisingly good performance on recent benchmarks. The success is mainly attributed to two aspects: the circular correlation properties of DCF and the powerful representation capabilities of CNN features. Nevertheless, compared with SOSVM, DCF-based algorithms are restricted to simple ridge regression which has a weaker discriminative ability. In this paper, a novel circular and structural operator tracker (CSOT) is proposed for high performance visual tracking, it not only possesses the powerful discriminative capability of SOSVM but also efficiently inherits the superior computational efficiency of DCF. Based on the proposed circular and structural operators, a set of primal confidence score maps can be obtained by circular correlating feature maps with their corresponding structural correlation filters. Furthermore, an implicit interpolation is applied to convert the multi-resolution feature maps to the continuous domain and make all primal confidence score maps have the same spatial resolution. Then, we exploit an efficient ensemble post-processor based on relative entropy, which can coalesce primal confidence score maps and create an optimal confidence score map for more accurate localization. The target is localized on the peak of the optimal confidence score map. Besides, we introduce a collaborative optimization strategy to update circular and structural operators by iteratively training structural correlation filters, which significantly reduces computational complexity and improves robustness. Experimental results demonstrate that our approach achieves state-of-the-art performance in mean AUC scores of 71.5 and 69.4 on the OTB2013 and OTB2015 benchmarks respectively, and obtains a third-best expected average overlap (EAO) score of 29.8 on the VOT2017 benchmark. Previous theoretical work on deep learning and neural network optimization tend to focus on avoiding saddle points and local minima. However, the practical observation is that, at least in the case of the most successful Deep Convolutional Neural Networks (DCNNs), practitioners can always increase the network size to fit the training data (an extreme example would be [1]). The most successful DCNNs such as VGG and ResNets are best used with a degree of "overparametrization". In this work, we characterize with a mix of theory and experiments, the landscape of the empirical risk of overparametrized DCNNs. We first prove in the regression framework the existence of a large number of degenerate global minimizers with zero empirical error (modulo inconsistent equations). The argument that relies on the use of Bezout theorem is rigorous when the RELUs are replaced by a polynomial nonlinearity (which empirically works as well). As described in our Theory III [2] paper, the same minimizers are degenerate and thus very likely to be found by SGD that will furthermore select with higher probability the most robust zero-minimizer. We further experimentally explored and visualized the landscape of empirical risk of a DCNN on CIFAR-10 during the entire training process and especially the global minima. Finally, based on our theoretical and experimental results, we propose an intuitive model of the landscape of DCNN's empirical loss surface, which might not be as complicated as people commonly believe. We discuss methodology for multidimensional scaling (MDS) and its implementation in two software systems, GGvis and XGvis. MDS is a visualization technique for proximity data, that is, data in the form of N × N dissimilarity matrices. MDS constructs maps (“configurations,” “embeddings”) in IRk by interpreting the dissimilarities as distances. Two frequent sources of dissimilarities are high-dimensional data and graphs. When the dissimilarities are distances between high-dimensional objects, MDS acts as a (often nonlinear) dimension-reduction technique. When the dissimilarities are shortest-path distances in a graph, MDS acts as a graph layout technique. MDS has found recent attention in machine learning motivated by image databases (“Isomap”). MDS is also of interest in view of the popularity of “kernelizing” approaches inspired by Support Vector Machines (SVMs; “kernel PCA”).This article discusses the following general topics: (1) the stability and multiplicity of MDS solutions; (2) the analysis of struc... Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure-tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes (Dimakis et. al., 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property. Explicit constructions of such codes in the literature are only available for the cases where there are at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math where @math The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. This paper establishes information-theoretic limits for estimating a finite-field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse-a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n × n-sensing matrices contain, on average, Ω(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the aforementioned results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive minimum distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Given two locations s and t in a road network, a distance query returns the minimum network distance from s to t, while a shortest path query computes the actual route that achieves the minimum distance. These two types of queries find important applications in practice, and a plethora of solutions have been proposed in past few decades. The existing solutions, however, are optimized for either practical or asymptotic performance, but not both. In particular, the techniques with enhanced practical efficiency are mostly heuristic-based, and they offer unattractive worst-case guarantees in terms of space and time. On the other hand, the methods that are worst-case efficient often entail prohibitive preprocessing or space overheads, which render them inapplicable for the large road networks (with millions of nodes) commonly used in modern map applications. This paper presents Arterial Hierarchy (AH), an index structure that narrows the gap between theory and practice in answering shortest path and distance queries on road networks. On the theoretical side, we show that, under a realistic assumption, AH answers any distance query in O(log α) time, where α = dmax dmin, and dmax (resp. dmin) is the largest (resp. smallest) L∞ distance between any two nodes in the road network. In addition, any shortest path query can be answered in O(k + log α) time, where k is the number of nodes on the shortest path. On the practical side, we experimentally evaluate AH on a large set of real road networks with up to twenty million nodes, and we demonstrate that (i) AH outperforms the state of the art in terms of query time, and (ii) its space and pre-computation overheads are moderate. Abstract We propose a novel approach to solving the approximate k-nearest neighbor search problem in metric spaces. The search structure is based on a navigable small world graph with vertices corresponding to the stored elements, edges to links between them, and a variation of greedy algorithm for searching. The navigable small world is created simply by keeping old Delaunay graph approximation links produced at the start of construction. The approach is very universal, defined in terms of arbitrary metric spaces and at the same time it is very simple. The algorithm handles insertions in the same way as queries: by finding approximate neighbors for the inserted element and connecting it to them. Both search and insertion can be done in parallel requiring only local information from the structure. The structure can be made distributed. The accuracy of the probabilistic k-nearest neighbor queries can be adjusted without rebuilding the structure. The performed simulation for data in the Euclidean spaces shows that the structure built using the proposed algorithm has small world navigation properties with log 2 ( n ) insertion and search complexity at fixed accuracy, and performs well at high dimensionality. Simulation on a CoPHiR dataset revealed its high efficiency in case of large datasets (more than an order of magnitude less metric computations at fixed recall) compared to permutation indexes. Only 0.03 of the 10 million 208-dimensional vector dataset is needed to be evaluated to achieve 0.999 recall (virtually exact search). For recall 0.93 processing speed 2800 queries s can be achieved on a dual Intel X5675 Xenon server node with Java implementation. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes ( , 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property (also known as minimum storage regenerating codes, or MSR codes). Explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math , where @math . The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math . Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. In sparse approximation theory, the fundamental problem is to reconstruct a signal A∈ℝn from linear measurements 〈Aψi〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1 e, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + e and improves the reconstruction time from poly(n) to poly(k logn) Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement Wireless ad hoc and sensor networks (WSNs) often require a connected dominating set (CDS) as the underlying virtual backbone for efficient routing. Nodes in a CDS have extra computation and communication load for their role as dominator, subjecting them to an early exhaustion of their battery. A simple mechanism to address this problem is to switch from one CDS to another fresh CDS, rotating the active CDS through a disjoint set of CDSs. This gives rise to the connected domatic partition (CDP) problem, which essentially involves partitioning the nodes V(G) of a graph G into node disjoint CDSs. We have developed a distributed algorithm for constructing the CDP using our maximal independent set (MlS)-based proximity heuristics, which depends only on connectivity information and does not rely on geographic or geometric information. We show that the size of a CDP that is identified by our algorithm is at least [delta+1 beta(c+1)] - f, where delta is the minimum node degree of G, beta les 2, c les 11 is a constant for a unit disk graph (UDG), and the expected value of f is epsidelta|V|, where epsi Lt 1 is a positive constant, and delta ges 48. Results of varied testing of our algorithm are positive even for a network of a large number of sensor nodes. Our scheme also performs better than other related techniques such as the ID-based scheme. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math . We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of @math nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level @math . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. In practical military or first responder deployment scenarios, information flows need to adhere to specified policies regardless of the physical connectivity of nodes. Nodes in such networks are associated with various levels in a command-and-control hierarchy, and therefore typically form a logical hierarchical tree network that is used to route both command and data traffic. Associated with this logical hierarchical network is a communication network that represents the connectivity of these nodes in the deployed scenario. Such composite networks introduce constraints that can result in information flows having to traverse much longer paths in the underlying communication network. In this paper, we look at the problem of adding edges to a logical hierarchical network (or any other social network) so as to minimize the number of hops required to route data traffic in the underlying communication network from a node to other specified nodes. The edges added are a subset of all possible edges in the complementary logical hierarchical graph and have to satisfy specified hierarchical constraints. First, we consider the general problem of minimizing the eccentricity of a source node 's' (where eccentricity of 's' is the maximum of the shortest paths from 's' to all other nodes) in a metric graph on adding upto 'B' unequal cost metric edges from the set of all edges in the complementary graph. We develop an efficient constant factor approximation algorithm for this case that outperforms existing constant factor algorithms for eccentricity minimization. Here the added edge metric cost as well as the graph edge metric cost correspond to the number of hops in the shortest path required to route traffic in the actual deployed topology (i.e., underlying communication network). Next, we consider the case where the set of possible added edges is a specified subset of the edges in the complementary graph and the set of destinations is a subset of the graph nodes. For this case, we develop heuristic algorithms based on the previous eccentricity minimizing algorithm that show good performance. We validate our algorithms using two realistic military deployment scenarios. We find that adding even a low number of hierarchically constrained edges (of the order of 10) can cause a significant decrease (around 50 ) in the eccentricity of a node in the logical hierarchical network and thus can reduce the number of hops required for data traffic traversal.