-robust spanners in one dimension
A geometric -spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most times the Euclidean distance between the points. Informally, a spanner is -robust if deleting vertices only harms other vertices. We show that on any one-dimensional set of points, for any , there exists an -robust -spanner with edges. Previously it was only known that -robust spanners with edges exists and that there are point sets on which any -robust spanner has edges.
Geometric networks are graphs whose vertices are points in the -dimensional Euclidean space and whose edges are weighted by the Euclidean distance between their endpoints. A desirable property of geometric networks is to have relatively short paths between any pair of points. We say that a geometric network is a -spanner of for some if
where denotes the length of the shortest path between and in and is the Euclidean distance between and . If is not specified, then we mean . Spanners have been studied extensively and there are many algorithms to construct spanners with various properties. Narasimhan and Smid  give a comprehensive overview of spanners.
An interesting property of spanners is their resistance against failures. Assume that some points in the network fail, that is, these points with all their edges are removed from the graph. The first natural question is whether the remaining part of the network is still a -spanner or not, perhaps for a larger value of . It is obvious that any point can be isolated by deleting its neighbors, so the remaining graph cannot in all cases remain a spanner.
Bose et al.  introduced the following notion of robustness. Let be a (geometric) -spanner for some and let be an arbitrary function. Assume that a set of points fail. Then we say that is an -robust -spanner if for any , there exists a set with , such that the subgraph induced by is a -spanner of . This means that it is allowed to ignore further points in the sense that we do not require the spanner property for them, but they are not deleted from . An alternative and stronger definition could be that the subgraph induced by is a -spanner. That is, we delete not only the failed vertices from but all the vertices that are in .
Bose et al.  proved various bounds on the size (number of edges) of -robust spanners in one and higher dimensions for general functions . In particular, they prove that an -robust spanner on vertices may needs edges, even for one-dimensional point sets. They show no upper bounds for the size of -robust spanners below the trivial . As open problem they pose obtaining tighter bounds for the size of -robust spanners, even in the simple setting where the input is the set . In this paper, we address this problem for general one-dimensional point sets.
The definition of -robustness is not the only way to obtain failure resistant networks. An alternative concept is -fault tolerance [2, 3]. A graph is a -fault tolerant -spanner if for any set of points , where , the graph after the removal of the points of is a -spanner. Fault tolerance is a suitable definition if the number of failures is bounded by some constant. In many cases, however, the number of failures is not known in advance and might be large. Clearly, the degree of each vertex must be at least to achieve -fault tolerance, therefore the size of the graph is large if is big. The definition of -robustness allows us to avoid this, at the cost of possibly having to ignore a small number of additional vertices.
In Section 2 we present a simple construction for which we show that it is an -robust -spanner on vertices with edges for any one-dimensional point set. In Section 3 we further improve the upper bound by generalizing the construction given in Section 2. We prove that for any one-dimensional point set there exists an -robust -spanner of size .
2 A simple construction
Let be an arbitrary point set with for and for . For the sake of simplicity, assume, that , where is a positive integer. We construct a graph as follows. Let for . We call the cluster. There are clusters and each of them has exactly points. Note that adjacent clusters are half-overlapping, that is . We define half-clusters as the sets obtained by splitting each cluster in the middle. Therefore, the number of half-clusters we have is exactly and each of them contains exactly points. Let and denote the left and right half of , respectively. Also note that . The structure of the clusters is illustrated in Figure 1.
We add two types of edges to the graph. First, we form cliques in each of the clusters. Since the size of the clusters is , we add at most edges. Second, for any pair of half-clusters we add an arbitrary complete matching between the two half-clusters. The number of edges we add is again , since each matching consists of exactly edges and there are pairs of half-clusters.
For any -dimensional point set (with ), the graph constructed above is an -robust -spanner with edges.
Let’s assume for now that ; later we will show how to adapt the construction to arbitrary . Clearly, the size of is . Firstly, it remains to construct the set for any set of failed points such that . Secondly, we have to prove that holds for any two points , where is the subgraph induced by , i.e., is a -spanner for on .
Consider the case that a set of points fail. To start, set . Then for each half-cluster (), if , we update by adding the clusters and to , see Figure 2. Formally,
It is clear that .
Fix and assume that . There are three separate cases. First, if they are in the same cluster, then and the claim holds. Second, if they are not in the same, but overlapping clusters, then in the intersection of the clusters there is a vertex , which shares an edge with both and . The third case is when and , with . Then, we know that , otherwise . Similarly, we know that , otherwise holds. Therefore, by the pigeonhole principle there is a vertex and a vertex for which . It is clear that and , since within a cluster all nodes are connected (Figure 3). Therefore, the length of this path between and is exactly , because by construction.
Now we show the extension to arbitrary . Let’s assume that . Split into two parts such that , where and . Build the same graph on the set and then extend the same structure to . Since , at most full clusters can be added, which are half-overlapping and have size . If necessary, we add one more cluster that has a smaller right half that containing the points which were not added to any of the clusters yet. The size of the graphs is still and construction of and the proof of the existence of paths without any detour remains the same.
3 Iterated construction
Let be the same as before, and assume that , where is an additional parameter, which determines the number of layers in the construction of the spanner . In each layer there are clusters with the same size, namely the clusters in layer have size for . Let denote the cluster in layer , then the clusters of the layer are
for any and . In each layer the clusters are half-overlapping, so that . Again, we define half-clusters by splitting each cluster into two parts of equal size. Let and denote the left and right half of , respectively. Note that a cluster in layer contains half-clusters from layer . The case corresponds to the simple construction in Section 2.
We define the edgeset of in the following way. For any cluster in the first (lowest) layer, form a clique on its vertices. This adds edges. In layer (), for any cluster , form a complete matching between any pair of half-clusters from layer that are both contained in , see Figure 4. The number of edges that are added is in each layer. Finally we add a complete matching between any pair of half-clusters in the top level. This adds another edges. Thus, has edges.
For any and any -dimensional point set (with ), the graph constructed above is an -robust -spanner with edges for .
We follow the structure of the proof of Theorem 1, that is, first we show that the graph constructed above is an -robust -spanner if for some , then we show the generalization to arbitrary . Let us start with the construction of . To start, set . Then apply the following rule repeatedly for each layer from bottom to top. Assume that is the current layer. First, set . Then, for each half-cluster in layer , if , we update by adding the clusters and . Finally, let . Clearly, . Therefore, the size of is at most .
Fix and assume . Let be the smallest cluster that contains both and . We use induction on the size of the smallest cluster that contains and to prove that a path of length between and in the subgraph induced by exists. Let and . There are two cases that we distinguish. The first case is when holds, that is, the clusters and do not intersect. Then, there is a vertex and such that . By induction there is a path of length between and and a path of length between and . Then using the edge we get a path of length between and , see Figure 5.
The second case is when holds, that is, the clusters and intersect. As it is illustrated in Figure 6, there are clusters and one layer below that contain the vertices in the left half and in the right half, respectively. Since , therefore and there exists a cluster such that and holds. Therefore, by the pigeonhole principle, we can choose the points and such that . Similarly, we can choose and such that . Again, by induction there are paths with length of the Euclidean distance between and and . Note that or can occur, but these cases are simpler. The induction terminates, since at each step the level is decreased at least by one and at the bottom layer all points are connected within one cluster.
Now to show the generalization to arbitrary . Assume that . Repeat the same construction on the first points as before. Then extend each layer continuously with half-overlapping clusters as long as they fit. If there are some points left at the end, add one more cluster that has a smaller sized right half. Form matchings between the proper half-clusters as before and form cliques within each cluster in the lowest layer. Regarding the failures, the construction of remains the same. This extension preserves the desired properties and the proof to find a path of length between and works the same.
Acknowledgements. The work in this paper is supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003.
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