Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

Fault-Tolerant Spanners: Better and Simpler

A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults $r$, for all stretch bounds. For stretch $k \geq 3$ we design a simple transformation that converts every $k$-spanner construction with at most $f(n)$ edges into an $r$-fault-tolerant $k$-spanner construction with at most $O(r^3 \log n) \cdot f(2n/r)$ edges. Applying this to standard greedy spanner constructions gives $r$-fault tolerant $k$-spanners with $\tilde O(r^{2} n^{1+\frac{2}{k+1}})$ edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on $n$ but exponentially on $r$ (approximately like $k^r$). For the case $k=2$ and unit-length edges, an $O(r \log n)$-approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to $O(\log n)$, which is, notably, independent of the number of faults $r$. We further strengthen this bound in terms of the maximum degree by using the \Lovasz Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.Comment: 17 page

A Spanner for the Day After

We show how to construct $(1+\varepsilon)$-spanner over a set $P$ of $n$ points in $\mathbb{R}^d$ that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters $\vartheta,\varepsilon \in (0,1)$, the computed spanner $G$ has $O\bigl(\varepsilon^{-c} \vartheta^{-6} n \log n (\log\log n)^6 \bigr)$ edges, where $c= O(d)$. Furthermore, for any $k$, and any deleted set $B \subseteq P$ of $k$ points, the residual graph $G \setminus B$ is $(1+\varepsilon)$-spanner for all the points of $P$ except for $(1+\vartheta)k$ of them. No previous constructions, beyond the trivial clique with $O(n^2)$ edges, were known such that only a tiny additional fraction (i.e., $\vartheta$) lose their distance preserving connectivity. Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black box fashion