Joint dimensioning of server and network infrastructure for resilient optical grids/clouds

We address the dimensioning of infrastructure, comprising both network and server resources, for large-scale decentralized distributed systems such as grids or clouds. We design the resulting grid/cloud to be resilient against network link or server failures. To this end, we exploit relocation: Under failure conditions, a grid job or cloud virtual machine may be served at an alternate destination (i.e., different from the one under failure-free conditions). We thus consider grid/cloud requests to have a known origin, but assume a degree of freedom as to where they end up being served, which is the case for grid applications of the bag-of-tasks (BoT) type or hosted virtual machines in the cloud case. We present a generic methodology based on integer linear programming (ILP) that: 1) chooses a given number of sites in a given network topology where to install server infrastructure; and 2) determines the amount of both network and server capacity to cater for both the failure-free scenario and failures of links or nodes. For the latter, we consider either failure-independent (FID) or failure-dependent (FD) recovery. Case studies on European-scale networks show that relocation allows considerable reduction of the total amount of network and server resources, especially in sparse topologies and for higher numbers of server sites. Adopting a failure-dependent backup routing strategy does lead to lower resource dimensions, but only when we adopt relocation (especially for a high number of server sites): Without exploiting relocation, potential savings of FD versus FID are not meaningful

Triangles and Girth in Disk Graphs and Transmission Graphs

Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest

Efficient algorithms for finding disjoint paths in grids

The reconfiguration problem on VLSI/WSI processor arrays in the presence of faulty processors can be stated as the following integral multi-source routing problem: Given a set of N nodes (faulty processors or sources) in an m×n rectangular grid where m, n≤N, the problem to be solved is to connect the N nodes to distinct nodes at the grid boundary using a set of `disjoint' paths. This problem can be referred to as an escape problem which can be solved trivially in O(mnN) time. By exploiting all the properties of the network, planarity and regularity of a grid, integral flow, and unit capacity source/sink/flow, we can optimally compress the size of the grid from O(mn) to O(√mnN) and solve the problem in O(d√mnN), where d is the maximum number of disjoint paths found, for both the edge-disjoint and vertex-disjoint cases. In the worst case, d, m, n are O(N) and the result is O(N2.5). Note that this routing problem can also be solved with the same time complexity even if the disjoint paths have to be ended at another set of N nodes (sinks) in the grid instead of the grid boundary.published_or_final_versio

Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary

We study the classical Node-Disjoint Paths (NDP) problem: given an undirected n-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set {P} of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy O(sqrt{n})-approximation algorithm. Until recently, the best negative result was an Omega(log^{1/2-epsilon}n)-hardness of approximation, for any fixed epsilon, under standard complexity assumptions. A special case of the problem, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for this special case achieves an O~(n^{1/4})-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor 2^{Omega(sqrt{log n})}, even if the underlying graph is a subgraph of a grid, and all source vertices lie on the grid boundary. In a very recent follow-up work, the authors further show that NDP in grid graphs is hard to approximate to within factor Omega(2^{log^{1-epsilon}n}) for any constant epsilon under standard complexity assumptions, and to within factor n^{Omega(1/(log log n)^2)} under randomized ETH. In this paper we study the NDP problem in grid graphs, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized 2^{O(sqrt{log n}* log log n)}-approximation algorithm for this problem. Our result in a sense complements the 2^{Omega(sqrt{log n})}-hardness of approximation for sub-graphs of grids with sources lying on the grid boundary, and should be contrasted with the above-mentioned almost polynomial hardness of approximation of NDP in grid graphs (where the sources and the destinations may lie anywhere in the grid). Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an Omega(sqrt n) integrality gap, even in grid graphs, with all source and destination vertices lying on the grid boundary. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods