Sparse Fault-Tolerant BFS Trees

This paper addresses the problem of designing a sparse {\em fault-tolerant} BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph $T$ of the given network $G$ such that subsequent to the failure of a single edge or vertex, the surviving part $T'$ of $T$ still contains a BFS spanning tree for (the surviving part of) $G$. Our main results are as follows. We present an algorithm that for every $n$-vertex graph $G$ and source node $s$ constructs a (single edge failure) FT-BFS tree rooted at $s$ with O(n \cdot \min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS tree rooted at $s$. This result is complemented by a matching lower bound, showing that there exist $n$-vertex graphs with a source node $s$ for which any edge (or vertex) FT-BFS tree rooted at $s$ has $\Omega(n^{3/2})$ edges. We then consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees} for short, aiming to provide (following a failure) a BFS tree rooted at each source $s\in S$ for some subset of sources $S\subseteq V$. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every $n$-vertex graph and source set $S \subseteq V$ of size $\sigma$ constructs a (single failure) FT-MBFS tree $T^*(S)$ from each source $s_i \in S$, with $O(\sqrt{\sigma} \cdot n^{3/2})$ edges, and on the other hand there exist $n$-vertex graphs with source sets $S \subseteq V$ of cardinality $\sigma$, on which any FT-MBFS tree from $S$ has $\Omega(\sqrt{\sigma}\cdot n^{3/2})$ edges. Finally, we propose an $O(\log n)$ approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no $\Omega(\log n)$ approximation algorithm for these problems under standard complexity assumptions

Resource Efficient Maintenance of Wireless Network Topologies 1

Abstract: Multiple hop routing in mobile ad hoc networks can minimize energy consumption and increase data throughput. Yet, the problem of radio interferences remain. However if the routes are restricted to a basic network based on local neighborhoods, these interferences can be reduced such that standard routing algorithms can be applied. We compare different network topologies for these basic networks, i.e. the Yao-graph (aka. Θ-graph) and some also known related models, which will be called the SymmYgraph (aka. YS-graph), the SparsY-graph (aka. YY-graph) and the BoundY-graph. Further, we present a promising network topology called the HL-graph (based on Hierarchical Layers). We compare these topologies regarding degree, spanner-properties, and communication features. We investigate how these network topologies bound the number of (uni- and bidirectional) interferences and whether these basic networks provide energy-optimal or congestion-minimal routing. Then, we compare the ability of these topologies to handle dynamic changes of the network when radio stations appear and disappear. For this we measure the number of involved radio stations and present distributed algorithms for repairing the network structure. Key Words: ad hoc networks, topology control, distributed algorithms Category: F.2, G.2.3, I.3.

I/O-efficient well-separated pair decomposition and applications

We present an external-memory algorithm to compute a well-separated pair decomposition (WSPD) of a given point set S in d in O(sort(N)) I/Os, where N is the number of points in S and sort(N) denotes the I/O-complexity of sorting N items. (Throughout this paper we assume that the dimension d is fixed.) As applications of the WSPD, we show how to compute a linear-size t-spanner for S within the same I/O-bound and how to solve the K-nearest-neighbour and K-closest-pair problems in O(sort(KN)) and O(sort(N+K)) I/Os, respectively

Sparse fault-tolerant spanners for doubling metrics with bounded hop-diameter or degree

We study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k-vertex-fault-tolerant t-spanner ((k, t)-VFTS or simply k-VFTS), if for any subset S â\u8a\u86 X with |S | â\u89¤ k, it holds that dH\S(x, y) â\u89¤ t · d(x, y), for any pair of x, y â\u88\u88 X \ S. For any doubling metric, we give a basic construction of k-VFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hop-diameter, which is studied in the context of fault-tolerance for the first time even for Euclidean spanners. We provide a construction of k-VFTS with bounded hop-diameter: for m â\u89¥ 2n, we can reduce the hop-diameter of the above k-VFTS to O(α(m, n)) by adding O(km) edges, where α is a functional inverse of the Ackermannâ\u80\u99s function. Finally, we construct a fault-tolerant single-sink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic k-VFTS. As a result, we get a k-VFTS with O(k 2 n) edges and maximum degree O(k²)

Efficient Construction of Low Weight Bounded Degree Planar Spanner

Given a set V of n points in a two-dimensional plane, we give an O(n log n)-time centralized algorithm that constructs a planar t-spanner for V, for t <= +1} C del , such that the degree of each node is bounded from above by 19 + and the total edge length is proportional to the weight of the minimum spanning tree of V , where 0 < # < #/2 is an adjustable parameter..