We present two online algorithms for maintaining a topological order of a directed n-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles m arc additions in O(...
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We present two online algorithms for maintaining a topological order of a directed n-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles m arc additions in O(m(3/2)) time. For sparse graphs (m/n = O(1)), this bound improves the best previous bound by a logarithmic factor, and is tight to within a constant factor among algorithms satisfying a natural locality property. Our second algorithm handles an arbitrary sequence of arc additions in O(n(5/2)) time. For sufficiently dense graphs, this bound improves the best previous bound by a polynomial factor. Our bound may be far from tight: we show that the algorithm can take Omega(n(2)2 root(2lgn)) time by relating its performance to a generalization of the k-levels problem of combinatorial geometry. A completely different algorithm running in Theta (n(2) log n) time was given recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to the maintenance of strong components, without affecting the asymptotic time bounds.
The emerging federated cloud paradigm advocates sharing of resources among cloud providers, to exploit temporal availability of resources and diversity of operational costs for job serving. While extensive studies exi...
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ISBN:
(纸本)9781467359443
The emerging federated cloud paradigm advocates sharing of resources among cloud providers, to exploit temporal availability of resources and diversity of operational costs for job serving. While extensive studies exist on enabling interoperability across different cloud platforms, a fundamental question on cloud economics remains unanswered: When and how should a cloud trade VMs with others, such that its net profit is maximized over the long run? In order to answer this question by the federation, a number of important, correlated decisions, including job scheduling, server provisioning and resource pricing, need to be dynamically made, with long-term profit optimality being a goal. In this work, we design efficient algorithms for inter-cloud resource trading and scheduling in a federation of geo-distributed clouds. For VM trading among clouds, we apply a double auction-based mechanism that is strategyproof, individual rational, and ex-post budget balanced. Coupling with the auction mechanism is an efficient, dynamic resource trading and scheduling algorithm, which carefully decides the true valuations of VMs in the auction, optimally schedules stochastic job arrivals with different SLAs onto the VMs, and judiciously turns on and off servers based on the current electricity prices. Through rigorous analysis, we show that each individual cloud, by carrying out our dynamic algorithm, can achieve a time-averaged profit arbitrarily close to the offline optimum.
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