Robust Geometric Spanners††thanks: A preliminary version of this paper appears in the proceedings of the 29th ACM Symposium on Computational Geometry (SoCG 2013).
Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.
Key words. spanners, stretch-factor, spanning-ratio, tree-width, connectivity, expansion
AMS subject classifications. 68M10, 05C10, 65D18
The cost of building a network, such as a computer network or a network of roads, is closely related to the number of edges in the underlying graph that models this network. This gives rise to the requirement that this graph be sparse. However, sparseness typically has to be counter-balanced with other desirable graph (that is, network design) properties such as reliability and efficiency.
The classical notion of graph connectivity provides some guarantee of reliability. In particular, an -connected graph remains connected as long as fewer than vertices are removed. However these graphs are not sparse for large values of ; an -connected graph with vertices has at least edges.
For many applications, disconnecting a small number of nodes from the network is an inconvenience for the nodes that are disconnected, but has little effect on the rest of the network. In contrast, disconnecting a large part (say, a constant fraction) of the network from the rest is catastrophic. For example, it may be tolerable that the failure of one network component cuts off internet access for the residents of a small village. However, the failure of a single component that eliminates all communications between North America and Europe would be disastrous.
This global notion of connectivity is captured in graph theory by expanders and graphs of high treewidth, each of which can have a linear number of edges. These two properties of graphs have an enormous number of applications and have been the subject of intensive research for decades. See, for example, the book by Kloks  or the surveys by Bodlaender [11, 12] on treewidth and the survey by Hoory, Linial, and Wigderson  on expanders.
In this paper, we consider how to combine this global notion of connectivity with another desirable property of geometric graphs: low spanning ratio (a.k.a., low stretch factor or low dilation), the property of approximately preserving Euclidean distances between vertices. In particular, given a set of points in , we study the problem of constructing a graph on these points where the weights of the edges are given by the Euclidean distance between their endpoints. We wish to construct a graph such that
The graph is sparse: the graph has edges
The graph is a spanner: (weighted) shortest paths in the graph do not exceed the Euclidean distance between their endpoints by more than a constant factor; and
The graph has high global connectivity: removing a small number of vertices leaves a graph in which a set of vertices of size are all in the same component and all vertices in this set have spanning paths between them.
This is the first paper to consider combining low spanning ratio with high global connectivity. This is somewhat surprising, since many variations on sparse geometric spanners have been studied, including spanners of low degree [6, 19, 36], spanners of low weight [14, 24, 26], spanners of low diameter [8, 9], planar spanners [5, 21, 23, 29], spanners of low chromatic number , fault-tolerant spanners [2, 22, 31, 32], low-power spanners [4, 34, 37], kinetic spanners [1, 3], angle-constrained spanners , and combinations of these [7, 10, 15, 16, 17, 18]. The closest related work is that on fault-tolerant spanners [2, 22, 31, 32], but -fault-tolerance is analogous to the traditional definition of -connectivity in graph theory and suffers the same shortcoming: every -fault-tolerant spanner has edges.
In the next few subsections, we formally define robust spanners and discuss, at a more rigorous level, the relationship between robust-spanners, fault-tolerant spanners, and expanders. From this point onwards, all graphs we discuss have vertices that are points in ; refers to the number points/vertices; all distances between pairs of points are Euclidean distances; and any shortest path in a graph refers to the shortest (Euclidean) path that uses only edges of the graph.
Let be a set of points in . An undirected graph is a (geometric) -spanner of if, for every pair ,
where denotes the Euclidean distance between and and denotes the length of the Euclidean shortest path from to that uses only edges in . Here we use the convention that if there is no path, in , from to . We say simply that is a -spanner if it is a -spanner of (i.e., ). We point out that, although is always at least 1, it need not be an integer.
Geometric -spanners have been studied extensively and have applications in robotics, graph theory, data structures, wireless networks, and network design. A book  and handbook chapter  provide extensive discussions of geometric -spanners and their applications.
For a graph and a subset of ’s vertices, we denote by the subgraph of induced by . A graph is an -robust -spanner of if, for every subset , there exists a superset , , such that is a -spanner of .
An example is shown in Figure LABEL:fig:grid which suggests that the grid graph is an -robust 3-spanner. The set in this example is obtained by choosing “disjoint” squares that cover the vertices of and adding to any vertices contained in these squares. A short path between any two vertices in is obtained by starting with some shortest path in between these two vertices and then routing around any of the square holes encountered by this path. (A proof that the grid graph is indeed an -robust 3-spanner is sketched in Section LABEL:sec:summary.)
One can think of an -robust -spanner in terms of network reliability. If a network is an -robust -spanner, and nodes of the network fail, then the network remains a -spanner of of its nodes. Intuitively, most of the network survives the removal of nodes, provided that is small enough that .
A slightly stronger version of robustness, which is achieved by some of our constructions, requires that induces a spanner. Under this definition, the graph must be a -spanner of . For example, the grid graph in Figure LABEL:fig:grid satisfies this stronger definition since the vertices inside the squares are not used in the short paths between vertices outside the squares. In some applications, this stronger definition may be preferable since the nodes in , which no longer gain the full benefits of the network , are not required to help with the routing of messages between nodes of . (An open problem related to this stronger definition of robustness is discussed in Section LABEL:sec:summary.)
1.2 Robustness versus Fault-Tolerance
Robustness is related to, but different from, -fault tolerance. An -fault-tolerant -spanner, , has the property that is a -spanner of for any subset of size at most . In our terminology, an -fault tolerant spanner is -robust with
At a minimum, an -fault-tolerant spanner must remain connected after the removal of any vertices. This immediately implies that any -fault-tolerant spanner with vertices has at least edges, since every vertex must have degree at least . Several constructions of -fault-tolerant spanners with edges exist [22, 31, 32].
In contrast, surprisingly sparse -robust -spanners exist. For example, we show that for one-dimensional point sets, there exists -robust 1-spanners with edges; the removal of any set of vertices leaves a subgraph of size that is a -spanner. An -fault-tolerant spanner with also has this property, but all such graphs have edges.
We suggest that in many applications where an -fault-tolerant spanner is used, an -robust spanner may be a better choice. For example, one might build an -fault-tolerant spanner so that a network survives up to faults, perhaps because more than faults is viewed as unlikely. Using an -robust spanner instead means that, if faults do occur, then an additional nodes suffer, but the remaining nodes are unaffected. In one case, the network loses nodes while in the other case nodes are affected. For slow-growing functions this may be perfectly acceptable.
The use of an -robust spanner in place of an -fault-tolerant spanner has the additional advantage that the maximum number of faults need not be known in advance. In the unlikely event that faults occur, the network continues to remain usable. In particular, after faults, the usable network has size at least . In contrast, even with faults, an -fault-tolerant spanner may have no component of size larger than ; see Figure LABEL:fig:rft-problem for an example.
1.3 Robustness and Magnification
A function is called a magnification (or vertex-expansion) function [28, Page 390], for the graph if, for all ,
where denotes the set of vertices in that are adjacent to vertices in . Of particular interest are graphs that have a magnification function , for fixed , and all . Such graphs are called vertex expanders, and have a long history and an enormous number of applications .
If is -robust, then there exists a magnification function, , for that satisfies , for all and every ; see Figure LABEL:fig:magnification. This can be proven by contradiction: If must be less than for some , then there exists a set of size such that . Taking , where each is chosen arbitrarily from yields a set, , of size , such that , has no component of size greater than .
If we think of as a continuous increasing function (and hence invertible) then the above argument says that any -robust spanner with vertices has a magnification function such that . This implies, for example, that the smallest separator in an -robust spanner with vertices has size .
Unfortunately, achieving -robustness is considerably more difficult than just obtaining a magnification function of the preceding form; there exist vertex expanders with a linear number of edges , so they have magnification functions of the form , with fixed . However, these graphs can not be -robust since, in Theorem LABEL:thm:general-lower-bound-1d, we show that -robust spanners have a superlinear number of edges, for any function .
1.4 Overview of Results
In this paper, we prove upper and lower bounds on the size (number of edges) needed to achieve -robustness. These bounds are expressed as a dependence on the function . In particular, the number of edges depends on the function , which is the maximum number of times one can iterate the function on an initial input before exceeding . As a concrete example, if , then (with the initial input ).
Our most general lower-bound, Theorem LABEL:thm:general-lower-bound-1d, states that, for any constant, , there exists one-dimensional point sets of size for which any -robust -spanner has size . For one-dimensional point sets, we can almost match this lower-bound: Theorem LABEL:thm:general-1d states that any one-dimensional point set of size has an -robust 1-spanner of size . Furthermore, if is sufficiently fast-growing, this construction is -robust, and hence has optimal size. For point sets in dimension , our upper and lower bounds diverge by a factor of . Theorem LABEL:thm:dd shows that, for any set of points in and any fixed , there exists an -robust -spanner of size .
As a concrete example, we can consider a function . Removing any set of vertices from a vertex -robust -spanner leaves a set of at least vertices which continue to have -spanning paths between them. Our results show that, in one dimension, -robust spanners can be constructed that have edges and this is optimal. In two and higher dimensions, -robust spanners can be constructed that have edges.
The remainder of the paper is organized as follows: Section LABEL:sec:one-d gives results for 1-dimensional point sets, Section LABEL:sec:d-d gives results for -dimensional point sets, and Section LABEL:sec:summary summarizes and concludes with directions for further research.
2 One-Dimensional Point Sets
In this section, we consider constructions of robust -spanners for 1-dimensional point sets. Throughout this section is a set of real numbers with . We begin by giving a construction of an -robust 1-spanner having edges. This construction contains most of the ideas needed for the construction of -robust 1-spanners for more general .
2.1 An -robust spanner with edges
We now consider the following graph, which is closely related to the hypercube. The edge set, , of consists of
Notice that is a 1-spanner since it contains every edge of the form , for . Furthermore, has size since every vertex has degree at most . We now prove an upper-bound on the robustness of by using the probabilistic method.
Let be any set of real numbers. Then there exists an -robust -spanner of of size .
Let be any non-empty subset of and let . Select a random integer and consider the subgraph, , of consisting only of the edges of the form where . One can think of the edges of as a set monotone paths that all contain ; one of these paths contains every vertex in , another contains every second vertex, yet another contains every fourth vertex, and so on. (For readers with a background in data structures, looks a lot like a perfect skiplist in which appears at the top level; see Figure LABEL:fig:g-prime.)
For a vertex , let be the largest integer such that is a multiple of . Then we say that kills the vertices in ; see Figure LABEL:fig:killing. When this happens, the cost of is , which is the number of vertices killed by . Observe that, unless or , contains the edge that “jumps over” all the vertices killed by . Therefore, if we define to be the set of all vertices killed by vertices in , then (and hence also ) is a 1-spanner of ; it contains a path that visits all vertices of in order.
We say that a vertex is cheap if and expensive otherwise. We call our choice of a failure if
: some vertex of is expensive; or
: the total cost of all cheap vertices exceeds
We declare our choice of a success if neither nor holds. Observe that, in the case of a success, we obtain a set , , such that is a 1-spanner of . Therefore, all that remains is to show that the probability of success is greater than 0.
We first note that the probability any particular is expensive is at most . This is because is expensive if and only if . The probability of selecting with this property is only . Therefore, by the union bound,
To upper-bound the total expected cost of cheap vertices, we note that, if kills vertices, then . The probability that this happens is . Letting denote the set of cheap vertices in , the total expected cost of all cheap vertices is at most
Therefore, by Markov’s Inequality, . By the union bound
2.2 A General Construction
Let be a constant and let be any function that is convex, increasing over the interval , and such that . Let be the function iterated times on the initial value , i.e.,
We use the convention that for all . We define the iterated -inverse function
Notice that, for any , there exists such that
In particular, the sequence contains a value such that
Another important property is that, since is increasing, convex, and , the function is non-decreasing for : For every , and every , .
For a positive number , we define , as the smallest power of 2 greater than or equal to . From the function, , we define the graph to have the edge set:
The graph clearly has edges. The following theorem shows that this graph is a robust spanner:
Let , , , and be defined as above. Then the graph has edges and is
an -robust 1-spanner; and
an -robust 1-spanner if .
The proof is very similar to the proof of Theorem LABEL:thm:klogk-1d. Let be any non-empty subset of and let . Select a random integer from the set . We consider the subgraph of that contains only the edges where . We say that an edge has span .
For an integer , let be the smallest integer such that ; see Figure LABEL:fig:spanjump. Informally, if has any edge that jumps over , then it has an edge of span that jumps over . Then we say that kills where
As before, we define to be the set of all vertices killed by vertices in . It is easy to verify, since all edges have spans that are powers of 2, that the graph (and hence also ) contains a path that visits all the vertices of in order. Therefore, is a 1-spanner of .
What remains is to show that, with some positive probability, is sufficiently small to satisfy the appropriate condition, 1 or 2, of the theorem. Define as the number of vertices killed by . We say that is expensive if and cheap otherwise. If is expensive, then and . Therefore, the probability that is expensive is at most . Therefore, by the union bound, the probability that contains some expensive vertex is at most . All that remains is to bound the expected cost of all cheap vertices. Letting denote the set of cheap vertices in , we obtain
|(since is non-decreasing)|
Again, Markov’s Inequality implies that the probability that the total cost of all cheap vertices exceeds is at most . Therefore, the probability of finding a set of size at most is at least
which proves the existence of such a set .
To prove the second part of the theorem, we proceed exactly the same way, except that the sequence , , becomes geometric,111This is most easily seen by taking . Then it is straightforward to verify that , so that , so the sequence is exponentially increasing. Taking for a sufficiently small allows us to lower-bound any function this way. so it is dominated by its last term. This yields:
|(for some , since the sum is geometric)|
|(since is non-decreasing)|
as required. ∎
Applying Theorem LABEL:thm:general-1d with different functions yields the following results.
For any set of real numbers, and any constant , there exist -robust 1-spanners with
and edges; and
2.3 Lower Bounds
In this section, we give lower-bounds on the number of edges in -robust -spanners. These lower-bounds hold already for a specific 1-dimensional point set (the grid), therefore they apply to all dimensions .
2.3.1 A Lower Bound for Linear Robustness
We begin by focusing on the hardest case, .
Let and let be a constant. Then any -robust -spanner of has edges.
To simplify the following discussion, we will assume that is a -robust -spanner. Note that we have gone from -robust in the statement of the theorem to -robust in the proof. This does not cause a problem so long as we only consider values of greater than some constant hidden in the notation.
We claim that for every natural number divisible by 4 and every , has at least good edges, , such that and such that .
To see why the preceding claim is true, consider the set that contains as well as the left endpoint of each good edge (see Figure LABEL:fig:lower-bound). The set has size at most and, in , the only edges with and have length greater than . Now consider any , with . Since there is at least one element that is not in and at least one element that is not in . Now,
and, in , every path from to uses an edge of length greater than . Therefore,
This contradicts the assumption that is -robust -spanner, so we conclude that there are, indeed at least good edges.
Applying the above argument to , for implies that contains edges whose length is in the range . Applying this argument for proves that, for any constants , has edges. ∎
2.3.2 A General Lower Bound
Using the iterated functions from Section LABEL:sec:iterated, we obtain a whole class of lower-bounds.
Let , , and be defined as in Section LABEL:sec:iterated, let , and let be a constant. Then any -robust -spanner of has edges.
The proof is similar to the proof of Theorem LABEL:thm:simple-lower-bound-1d. We need only consider since, otherwise we can apply Theorem LABEL:thm:simple-lower-bound-1d. We group the edges of the graph into classes and show that each class contains vertices.
In particular, using the same argument one can show that, for any , any -robust -spanner of has edges whose lengths are in the range . Since is superlinear, there exists a constant such that, for any , . Thus, the number of edges in any -robust -spanner of is at least
Let and let and be constants. Then any -robust -spanner with
has edges; and
Note that the lower bounds in Parts 2 and 3 of this corollary match the corresponding upper-bounds while the lower-bound in Part 1 is off by a factor of .
The dependence of our lower bounds on the value of is not given in the statements of Theorems LABEL:thm:simple-lower-bound-1d and LABEL:thm:general-lower-bound-1d or in Corollary LABEL:cor:lower-bound. However, it is readily extracted from their proofs. In Theorem LABEL:thm:simple-lower-bound-1d, each value of shows the existence of edges and there are values of , so the lower-bound is .
In Theorem LABEL:thm:general-lower-bound-1d, each value of shows the existence of edges, but now the number of values of is where is the minimum value such that . (Informally, is where the slope of exceeds .) Thus, in Theorem LABEL:thm:general-lower-bound-1d, the lower-bound is . It is fairly straightforward to apply this bound to the choices of used in Corollary LABEL:cor:lower-bound or to other choices of . For example, applying it to Case 3 of Corollary LABEL:cor:lower-bound we get and the result that any -robust -spanner has edges.
3 Higher Dimensions
In this section, we give a family of constructions for point sets , . These constructions make use of dumbbell tree spanners [33, Chapter 11]. In particular, they make use of binary dumbbell trees, first used by Arya et al.  in the construction of low-diameter spanners. A full description of the construction (and proof of existence) of binary dumbbell trees can be found in the notes by Smid .
A (binary) dumbbell tree spanner of is defined by a set of binary trees , each having leaves. Each node, , in each of these trees is associated with one element, . For each , and each , contains exactly one leaf, , such that and at most one internal node, , such that . For any two points , there exists some tree, , with two leaves, and , such that , and the path, in defines a path whose Euclidean length is at most , where is a parameter in the construction of the dumbbell tree. Thus, the graph obtained by taking
is a -spanner of .
The size (number of edges) of a dumbbell tree spanner is clearly . For a fixed dimension, , as a function of and as approaches 1, the number of trees, , is . In particular, for , .
In the following, we will often treat the nodes of each tree, , in a dumbbell tree decomposition as if the nodes are elements of . This will happen, for example, when we make statements like “the path in from the leaf containing to the leaf containing has length at most .” We do this to avoid the cumbersome phraseology required to distinguish between a node and the node associated with . Hopefully the reader can tolerate this informality.
Let , , and be defined as in Section LABEL:sec:iterated and let and be constants. Let be any set of points in . Then, for any constant , there exists an -robust -spanner of with edges.
Fix a value and recall that, in any binary tree, , with nodes, there exists a vertex whose removal disconnects into at most 3 components each of size at most . Repeatedly applying this fact to any component of size greater than yields a set of vertices whose removal disconnects into components each of size at most [33, Lemma 12.1.5]; see Figure LABEL:fig:dumbbell-chop.
Perform the above decomposition for each of the trees defining a dumbbell tree -spanner, , of with . This yields a set, , of vertices whose removal disconnects every dumbbell tree into components each of size at most . Using any of the -fault-tolerant spanner constructions cited in the introduction, we can construct a -fault-tolerant -spanner for having edges. Let denote the graph whose edge set contains all edges of the dumbbell spanner and all edges of a -fault-tolerant spanner on .
Suppose that we are now given a set , . Any vertex appears at most twice in each tree . For each , we say that kills all the vertices in any component of that contains . Furthermore, if is an element of , then kills all the vertices in the (at most 3) components of whose that have a vertex adjacent to . The total number of vertices killed by is therefore ; see Figure LABEL:fig:dumbbell-kill.
Let be the set of all vertices killed by all vertices in . The size of is . Consider some pair of vertices . There exists a tree such that the path, in , from the leaf containing to the leaf containing has length at most . If and are in the same component of then this path is also a path in .
If and are in different components of then consider the path from the leaf containing to the leaf containing in . Let denote the first node on this path that is in and let denote the last node on this path that is in . The graph contains a path, from the leaf containing , to , to , and then finally to , where the path from to uses the -fault tolerant spanner; see Figure LABEL:fig:dumbbell-kill. Therefore,
Since this is true for every pair , this means that is a -spanner of .
We have just shown how to construct a graph that has edges and is robust provided that . To obtain a graph that is -robust for any value of , we take the graph containing the edges of each for . The graph has edges. For any set , we can apply the above argument on the subgraph with , to show that is -robust. ∎
Applying Theorem LABEL:thm:dd with different functions yields the following results.
For any constants , , , and any set of points in , there exist -robust -spanners with
and edges; and
Note that, like our lower bounds, Theorem LABEL:thm:dd and Corollary LABEL:cor:dd do not express the relationship between the number of edges and the spanning ratio, . As before, this relationship is not hard to work out. The number, , of spanning trees in the dumbbell tree spanner is , where . Each -fault-tolerant spanner has edges  and we construct one of these for different values of . Thus, the total number of edges in our constructions is .
3.1 Linear-Size (Kind of) Robust Spanners
The lower bound in Theorem LABEL:thm:general-lower-bound-1d shows that linear-size -robust -spanners do not exist for any function . In this section, we show that there are linear sized graphs that satisfy a weaker definition of robustness.
We say that a graph is -hardy if, for every subset , there exists a superset , , such that is a -spanner of . Note that this definition is almost identical to that of robustness except that the size of may also depend on . In particular, any -robust -spanner is also an -hardy -spanner with .
If -hardy -spanners with edges exist for all , then -hardy -spanners with edges exist for all .
Perform the same dumbbell tree decomposition used in the proof of Theorem LABEL:thm:dd to obtain a set of