Abstract
In this work, we consider the problem of placing replicas in a data center or storage area network, represented as a digraph, so as to lexico-minimize a previously proposed reliability measure which minimizes the impact of all failure events in the model in decreasing order of severity. Prior work focuses on the special case in which the digraph is an arborescence. In this work, we consider the broader class of multitrees: digraphs in which the subgraph induced by vertices reachable from a fixed node forms a tree. We parameterize multitrees by their number of “roots” (nodes with in-degree zero), and rule out membership in the class of fixed-parameter tractable problems (FPT) by showing that finding optimal replica placements in multitrees with 3 roots is NP-hard. On the positive side, we show that the problem of finding optimal replica placements in the class of untangled multitrees is FPT, as parameterized by the replication factor \(\rho \) and the number of roots k. Our approach combines dynamic programming (DP) with a novel tree decomposition to find an optimal placement of \(\rho \) replicas on the leaves of a multitree with n nodes and k roots in \(O(n^2\rho ^{2k+3})\) time.
N. Mittal—This work was supported in part by NSF grants CNS-1115733 and CNS-1619197.
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Notes
- 1.
Recall that in a full binary tree every node has 0 or 2 children.
- 2.
Using a subset as opposed to a multiset rules out the possibility of placing multiple replicas on the same server, which would defeat the purpose of replication.
- 3.
That is, if M is an untangled multitree, then for every \(U \subseteq V\), the vertex-induced subgraph \(M[U] = (U, (U \times U) \cap E)\) is also an untangled multitree.
- 4.
Where admissibility follows by child-descendant completeness of \(\varGamma _u\).
- 5.
An algorithm for undirected hypergraphs with the same running time exists. In any case, undirected hypergraphs can be handled via [1] by adding an extra hyperedge going in the reverse direction.
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Mills, K.A., Chandrasekaran, R., Mittal, N. (2017). Lexico-Minimum Replica Placement in Multitrees. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10628. Springer, Cham. https://doi.org/10.1007/978-3-319-71147-8_9
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