67 research outputs found

    Efficiently Realizing Interval Sequences

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    We consider the problem of realizable interval-sequences. An interval sequence comprises of nn integer intervals [ai,bi][a_i,b_i] such that 0aibin10\leq a_i \leq b_i \leq n-1, and is said to be graphic/realizable if there exists a graph with degree sequence, say, D=(d1,,dn)D=(d_1,\ldots,d_n) satisfying the condition aidibia_i \leq d_i \leq b_i, for each i[1,n]i \in [1,n]. There is a characterisation (also implying an O(n)O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdos-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n2)o(n^2) time. In this paper, we provide an O(nlogn)O(n \log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length pp non-graphic sequence to a graphic one.Comment: 19 pages, 1 figur

    Algorithms for self-healing networks

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    Many modern networks are reconfigurable, in the sense that the topology of the network can be changed by the nodes in the network. For example, peer-to-peer, wireless and ad-hoc networks are reconfigurable. More generally, many social networks, such as a company\u27s organizational chart; infrastructure networks, such as an airline\u27s transportation network; and biological networks, such as the human brain, are also reconfigurable. Modern reconfigurable networks have a complexity unprecedented in the history of engineering, resembling more a dynamic and evolving living animal rather than a structure of steel designed from a blueprint. Unfortunately, our mathematical and algorithmic tools have not yet developed enough to handle this complexity and fully exploit the flexibility of these networks. We believe that it is no longer possible to build networks that are scalable and never have node failures. Instead, these networks should be able to admit small, and, maybe, periodic failures and still recover like skin heals from a cut. This process, where the network can recover itself by maintaining key invariants in response to attack by a powerful adversary is what we call self-healing. Here, we present several fast and provably good distributed algorithms for self-healing in reconfigurable dynamic networks. Each of these algorithms have different properties, a different set of gaurantees and limitations. We also discuss future directions and theoretical questions we would like to answer

    Minimum Cost Design of Cellular Networks in Rural Areas with UAVs, Optical Rings, Solar Panels and Batteries

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    Bringing the cellular connectivity in rural zones is a big challenge, due to the large installation costs that are incurred when a legacy cellular network based on fixed Base Stations (BSs) is deployed. To tackle this aspect, we consider an alternative architecture composed of UAV-based BSs to provide cellular coverage, ground sites to connect the UAVs with the rest of the network, Solar Panels (SPs) and batteries to recharge the UAVs and to power the ground sites, and a ring of optical fiber links to connect the installed sites. We then target the minimization of the installation costs for the considered UAV-based cellular architecture, by taking into account the constraints of UAVs coverage, SPs energy consumption, levels of the batteries and the deployment of the optical ring. After providing the problem formulation, we derive an innovative methodology to ensure that a single ring of installed optical fibers is deployed. Moreover, we propose a new algorithm, called DIARIZE, to practically tackle the problem. Our results, obtained over a set of representative rural scenarios, show that DIARIZE performs very close to the optimal solution, and in general outperforms a reference design based on fixed BSs
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