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    Dual Failure Resilient BFS Structure

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    We study {\em breadth-first search (BFS)} spanning trees, and address the problem of designing a sparse {\em fault-tolerant} BFS structure, or {\em FT-BFS } for short, resilient to the failure of up to two edges in the given undirected unweighted graph GG, i.e., a sparse subgraph HH of GG such that subsequent to the failure of up to two edges, the surviving part Hβ€²H' of HH still contains a BFS spanning tree for (the surviving part of) GG. FT-BFS structures, as well as the related notion of replacement paths, have been studied so far for the restricted case of a single failure. It has been noted widely that when concerning shortest-paths in a variety of contexts, there is a sharp qualitative difference between a single failure and two or more failures. Our main results are as follows. We present an algorithm that for every nn-vertex unweighted undirected graph GG and source node ss constructs a (two edge failure) FT-BFS structure rooted at ss with O(n5/3)O(n^{5/3}) edges. To provide a useful theory of shortest paths avoiding 2 edges failures, we take a principled approach to classifying the arrangement these paths. We believe that the structural analysis provided in this paper may decrease the barrier for understanding the general case of fβ‰₯2f\geq 2 faults and pave the way to the future design of ff-fault resilient structures for fβ‰₯2f \geq 2. We also provide a matching lower bound, which in fact holds for the general case of fβ‰₯1f \geq 1 and multiple sources SβŠ†VS \subseteq V. It shows that for every fβ‰₯1f\geq 1, and integer 1≀σ≀n1 \leq \sigma \leq n, there exist nn-vertex graphs with a source set SβŠ†VS \subseteq V of cardinality Οƒ\sigma for which any FT-BFS structure rooted at each s∈Ss \in S, resilient to up to ff-edge faults has Ξ©(Οƒ1/(f+1)β‹…n2βˆ’1/(f+1))\Omega(\sigma^{1/(f+1)} \cdot n^{2-1/(f+1)}) edges

    An optimal fixed-priority assignment algorithm for supporting fault-tolerant hard real-time systems

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    The main contribution of this paper is twofold. First, we present an appropriate schedulability analysis, based on response time analysis, for supporting fault-tolerant hard real-time systems. We consider systems that make use of error-recovery techniques to carry out fault tolerance. Second, we propose a new priority assignment algorithm which can be used, together with the schedulability analysis, to improve system fault resilience. These achievements come from the observation that traditional priority assignment policies may no longer be appropriate when faults are being considered. The proposed schedulability analysis takes into account the fact that the recoveries of tasks may be executed at higher priority levels. This characteristic is very important since, after an error, a task certainly has a shorter period of time to meet its deadline. The proposed priority assignment algorithm, which uses some properties of the analysis, is very efficient. We show that the method used to find out an appropriate priority assignment reduces the search space from O(n!) to O(n/sup 2/), where n is the number of task recovery procedures. Also, we show that the priority assignment algorithm is optimal in the sense that the fault resilience of task sets is maximized as for the proposed analysis. The effectiveness of the proposed approach is evaluated by simulation
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