1 Introduction
Geometric graphs are such that their vertices are points in the -dimensional Euclidean space and edges are straight line segments. The quality or efficiency of a geometric graph is often measured in terms of the ratio of shortest path distances and geometric distances between its vertices. Let be a geometric graph, where is a set of points and is the set of edges. The shortest path distance between two points in the graph is denoted by (or just ). The graph is a -spanner for some constant , if holds for all pairs of points , where stands for the Euclidean distance of and . The spanning ratio, stretch factor, or dilation of a graph is the minimum number for which is a -spanner. A path between and is a -path if its length is at most .
The main interest in spanners is to show that they posses further desirable properties beyond short connections, such as failure resistance, small weight, small diameter or bounded degree to mention some without completeness. We focus our attention to construct spanners that can survive massive failures of vertices. The most studied notion is fault tolerance [LNS98, LNS02, Luk99], which provides a properly functioning residual graph if there are no more failures than a predefined parameter . It is clear, that a -fault tolerant spanner must have edges to avoid small degree nodes, which can be isolated by deleting their neighbors. Therefore, fault tolerant spanners must have quadratic size to be able to survive a failure of a constant fraction of vertices. Another notion is robustness [BDMS13], which gives more flexibility by allowing the loss of some additional nodes by not guaranteeing -paths for them. For a function a -spanner is -robust, if for any set of failed points there is an extended set with size at most such that the residual graph has a -path for any pair of points . The function controls the robustness of the graph - the slower the function grows the more robust the graph is. The benefit of robustness is that a near linear number of edges are enough to achieve it, even for the case when is linear, there are constructions with nearly edges. For , a spanner that is -robust with is a -reliable spanner [BHO19]. This is the strongest form of robustness, since the dilation can increase for only a tiny additional fraction of points beyond . The fraction is relative to the number of failed vertices and controlled by the parameter .
Recently, the authors [BHO19] showed a construction of reliable -spanners of size in one dimension, and of reliable -spanners of size in higher dimensions (the constant in the depends both on the dimension, , and the reliability parameter). An alternative construction, with slightly worse bounds, was given by Bose et al. [BCDM18].
Limitations of previous constructions.
The construction of Buchin et al. [BHO19] (and also the construction of Bose et al. [BCDM18]) relies on using expanders to get a monotone spanner for points on the line, and then extending it to higher dimensions. The spanner (in one dimension) has edges. Unfortunately, even in one dimension, such a reliable spanner requires edges, as shown by Bose et al. [BDMS13]. Furthermore, the constants involved in these constructions [BHO19, BCDM18] are quite bad, because of the usage of expanders. See Table 1.1 for a summary of the sizes of different constructions (together with the new results).
dim | # edges | constants | |
Reliable spanners | |||
Buchin et al. [BHO19] | |||
Bose et al. [BCDM18] | ? | ||
Reliable spanners in expectation | |||
New results | |||
The problem.
As such, the question is whether one can come up with simple and practical constructions of spanners that have linear or near linear size, while still possessing some reliability guarantee – either in expectation or with good probability.
Some definitions.
Given a graph , an attack is a set of vertices that are being removed. The damaged set , is the set of all the vertices which are no longer connected to the rest the graph, or are badly connected to the rest of the graph – that is, these vertices no longer have the desired spanning property. The loss caused by , is the quantity (where we take the minimal damaged set). The loss rate of is . A graph is -reliable if for any attack , the loss rate is at most .
Randomness and obliviousness.
As mentioned above, reliable spanners must have size . A natural way to get a smaller spanner, is to consider randomized constructions, and require that the reliability holds in expectation (or with good probability). Randomized constructions are (usually) still sensitive to adversarial attacks, if the adversary is allowed to pick the attack set after the construction is completed (and it is allowed to inspect it). A natural way to deal with this issue is to restrict the attacks to be oblivious – that is, the attack set is chosen before the graph is constructed (or without any knowledge of the ).
In such an oblivious model, the loss rate is a random variable (for a fixed attack
). It is thus natural to construct the graph randomly, in such a way that , or alternatively, that the probability is small.-spanner.
Surprisingly, the one-dimensional problem is the key for building reliable spanners. Here, the graph is constructed over the set of vertices . An attack is a subset . Given an attack , the requirement is that for all , such that , there is a monotonically increasing path from to in – here, the length of the path between and is exactly . Since there is no distortion in the length of the path, such graphs are -spanners.
Reliability vs. distortion.
Building reliable graphs is relatively easy by using expanders. Expanders, however, have hop diameter. In the oblivious model, even simpler constructions are possible (essentially a random star). As these constructions require (or even logarithmic number of) hops, their distortion is at least . As such, the complexity in the construction arises out of the need to keep the distortion small (i.e., ). Even in the one-dimensional case, keeping the distortion under control does not seem obvious, even in the oblivious model. This is inherently the main challenge in this work.
Our results.
We give a randomized construction of a -spanner in one dimension, that is -reliable in expectation, and has size . Formally, the construction has the property that . This construction can also be modified so that holds with some desired probability. This is the main technical contribution of this work.
Next, following in the footsteps of the construction of reliable spanners, we use the one-dimensional construction to get -spanners that are -reliable either in expectation or with good probability. The new constructions have size roughly .
Main idea.
We borrow the notion of shadow from our previous work. A point is in the -shadow if there is a neighborhood of , such that an -fraction of it belongs to the attack set. One can think about the maximum such that is in the -shadow of as the depth of (here, the depth is in the range ). A point with depth close to one, are intuitively surrounded by failed points, and have little hope of remaining well connected. Fortunately, only a few points have depth truly close to one . The flip side is that the attack has little impact on shallow points (i.e., points with depth close to ). Similar to people, shallow points are surrounded by shallow points. As such, only a small fraction of the shallow points needs to be strongly connected to other points in the graph, as paths from (shallow) points around them can then travel via these hub points.
To this end, similar in spirit to skip-lists, we define a random gradation of the points , where – this is done via a random tournament tree. In each level, each point of is connected to all its neighbors within a certain distance (which increases as increases). Intuitively, because of the improved connectivity, the probability that a point is well-connected (after the attack) increases if they belong to higher level of the gradation. Thus, the probability of a shallow point to remain well connected is, intuitively, good. Specifically, we can quantify the probability of a vertex to lose its connectivity as a function of its depth. Combining this with bounds on the number of points of certain depths, results in bounds on the expected size of the damaged set.
Comparison to previous work.
While we borrow some components of the previous work, the basic scheme in the one-dimensional case, is new, and significantly different – the previous construction used expanders in a hierarchical way. The new construction requires different analysis and ideas. The extension to higher dimension is relatively straightforward and follows the ideas in the previous work, although some modifications and care are necessary.
Paper organization.
2 Preliminaries
Let be a -spanner for some . An attack on is a set of vertices that fail, and no longer can be used. An attack is oblivious, if the set is picked without any knowledge of .
Definition 2.1 (Reliable spanner).
Let be a -spanner for some constructed by a (possibly) randomized algorithm. Given an oblivious attack , its damaged set is the smallest set, such that for any pair of vertices , we have
that is, -paths are preserved for all pairs of points not contained in . The quantity is the loss of under the attack . The loss rate of is . For , the graph is -reliable if holds for any attack . Further, we say that the graph is -reliable in expectation if holds for any oblivious attack . For , we say that the graph is -reliable with probability if holds for any oblivious attack .
Notice, that the set is not unique, since one can (possibly) choose the point to include in for a pair that does not have a -path in . However, this does not cause a problem in defining the loss rate.
Definition 2.2.
Let denote the interval . Similarly, for and , let denote the interval .
We use the shadow notion as it was introduced by Buchin et al. [BHO19].
Definition 2.3.
Consider an arbitrary set and a parameter . A number is in the left -shadow of , if and only if there exists an integer , such that Similarly, is in the right -shadow of , if and only if there exists an integer , such that and The left and right -shadow of is denoted by and , respectively. The combined shadow is denoted by
Lemma 2.4 ([Bho19]).
For any set , and , we have that .
Lemma 2.5 ([Bho19]).
Fix a set , and let be a parameter. We have that .
Definition 2.6.
Given a graph over , a monotone path between , such that , is a sequence of vertices , such that , for .
A monotone path between and has length . Throughout the paper we use and to denote the base and natural base logarithm of , respectively. For any set , let denote the complement of . For two integer numbers , let .
3 Reliable spanners in one dimension
We show how to build a graph on that still has monotone paths almost for all vertices that are not directly attacked. First, in Section 3.2, we show that our construction is -reliable in expectation. Then, in Section 3.3, we show how to modify the construction to obtain a -spanner that is -reliable with probability .
3.1 Construction
The input consists of a parameter and the point set . The backbone of the construction is a random elimination tournament, see Figure 3.1 as an example. We assume that is a power of as otherwise one can construct the graph for the next power of two, and then throw away the unneeded vertices.
The tournament is a full binary tree, with the leafs storing the values from to , say from left to right. The value of a node is computed randomly and recursively. For a node, once the values of the nodes were computed for both children, it randomly copies the value of one of its children, with equal probability to choose either child. Let be the values stored in the th bottom level of the tree. As such, , and is a singleton. Each set can be interpreted as an ordered set (from left to right, or equivalently, by value).
Let
(3.1) |
where is a sufficiently large constant. Let be the smallest integer for which holds (i.e., ). For , and for all connect with the
(3.2) |
successors (and predecessors) of in . Let be the set of all edges in level . The graph on is defined as the union of all edges over all levels – that is, . Note, that top level of the graph is a clique.
Remark 3.7.
Before dwelling on the correctness of the construction, note that the obliviousness of the attack is critical. Indeed, it is quite easy to design an attack if the structure of is known. To this end, let be the set of values of closest to – namely, we are taking out the middle-part of the graph, that belongs to the th level. Consider the attack . It is easy to verify that this attack breaks into at least two disconnected graphs, each of size at least .
3.2 Analysis
Lemma 3.8.
The graph has edges.
Proof:
The number of edges contributed by a point in is at most at level , and . Thus, we have
Fix an attack . The high-level idea is to show that if a point is far enough from the faulty set, then, with high probability, there exist monotone paths reaching far from in both directions. For two points , we show that if both and have far reaching monotone paths, then the path going to the right from , and the path going to the left from must cross each other, which in turn implies, that there is a monotone path between and . Therefore, it is enough to bound the number of points that does not have far reaching monotone paths.
Definition 3.9 (Stairway).
Let be an arbitrary point. The path is a right (resp., left) stairway of to level , if
(resp., ),
if , then , for ,
, for . Furthermore, a stairway is safe if none of its points are in the attack set . A right (resp., left) stairway is usable, if (resp., ) forms a clique in . Let denote the set of points that have safe and usable stairways to both directions.
Let , for . Let be the -shadow of , for . Observe that , and there is an index such that , if
. A point is classified according to when it get “buried” in the shadow. A point
, for , is a th round point, if . Intuitively, a th round point is more likely to have a safe stairway the larger the value of is.Lemma 3.10.
For any , if then does not contain any point of .
Proof:
If contains any point of , then but that would readily imply that .
Definition 3.11.
A point is bad if it belongs to , or it does not have a right or left stairway that is safe and usable. Formally, a point is bad, if and only if .
Lemma 3.12.
For any two points that are not bad, there is a monotone path connecting and in the residual graph .
Proof:
Suppose we have . Let be a safe usable right stairway starting from and be a safe usable left stairway from . These stairways exist, since . Let and consider the stairways and . Notice that both are safe and at least one of them is usable.
Let be the first index such that , if there is any. We distinguish two cases based on whether holds or not. In case , the path is a monotone path from to , since implies . On the other hand, if we have , the path is a monotone path between and , since implies .
Finally, if holds for all , then the path is a monotone path between and . We have , since at least one of the stairways is usable. This concludes the proof that there is a monotone path from to .
Lemma 3.13.
For a fixed set , we have that .
Proof:
Let , and observe that knowing that certain points of are not in , increases the probability of another point to be in . That is, . As such, we have
Lemma 3.14.
Assume that and let be a th round point for some . The probability that is bad is at most .
Proof:
For any integer , let and let see Figure 3.2. Recall that , so is the next power of . Let be the largest integer such that . For , the points of form a clique in , since
Indeed, any two vertices of with distance smaller than are connected by an edge in . As such, it is enough to prove that there is a right safe stairway from , that climbs on the levels to level . Since forms a clique, it follows that such a stairway would be usable.
Let be the event that is empty, for . Since , we have that In the other hand, we have As such, if then . This happens if which happens if , given that . Notice that holds for all , if .
So assume that . Let be all points of , which are the possible candidates to be contained in . By Eq. (3.1), there are at least
such points. Observe, that by the structure of the construction, a point is more likely to be contained in conditioned on the event there are some other points which are not contained in . Therefore, by Lemma 3.13, we have for The sequence has a fast decay in , since
if holds. Thus, we have
for , using the conditions , and the fact that .
Let be the leftmost point in , for . Since , for all , it follows that . Furthermore, since is a clique in , and , it follows that , if , for all . We conclude that is a safe and usable right stairway in .
The bound now follows by applying the same argument symmetrically for the left stairway. Indeed, using the union bound, we obtain .
Lemma 3.15.
Let and be an oblivious attack. Then, for the expected number of bad points, we have .
Proof:
We may assume that all the points of are bad. Fortunately, by Lemma 2.5, we have , since and for . By Lemma 2.4, we have
For , we have, by Lemma 3.14, that
Since, , we have, by linearity of expectation, that
since .
Theorem 3.16.
Let and be fixed. The graph , constructed in Section 3.1, has edges, and it is a -reliable -spanner of in expectation. Formally, for any oblivious attack , we have .
Proof:
By Lemma 3.8 the size of the construction is Let be an oblivious attack and consider the bad set . By Lemma 3.12, for any two points outside the bad set, there is a monotone path connecting them. Further, by Lemma 3.15, we have for any oblivious attack. Therefore, we obtain .
3.3 Probabilistic bound
One can replace the guarantee, in Theorem 3.16, on the bound of the loss rate (which holds in expectation), by an upper bound that holds with probability at least , for some prespecified . A straightforward application of Markov’s inequality implies that taking the union of independent copies () of the construction of Theorem 3.16 with parameter , results in a graph with the desired property. Indeed, we have
Here we show how one can do better to avoid the multiplicative factor .
Construction.
The input consists of two parameters and the set . Let be the graph constructed in Section 3.1 with parameters
where is a sufficiently large constant. First, we need a variant of Lemma 3.14 to bound the probability of a th round point being bad, using the new value of .
Lemma 3.17.
Assume that , and let be a th round point for some . The probability that is bad is at most .
Proof:
The proof is the same as the proof of Lemma 3.14. The only difference is due to the new value of , which results in , using the same notation. Therefore, we have
for . See Lemma 3.14 for a complete proof.
Lemma 3.18.
Let , be fixed and be an oblivious attack. Then, with probability at least , the number of bad points is at most . That is, we have .
Proof:
The idea is to give bounds on the number of bad th round points for all . Let be the event that happens, for . Recall, by the choice of , we have . Notice, that at least one of the events must happen, for , in order to have , since
Using Markov’s inequality and Lemma 3.17 we get
Therefore, we obtain
which is equivalent to .
Theorem 3.19.
Let , and be fixed. The graph , constructed above, is a -reliable -spanner of , with probability at least . Formally, we have for any oblivious attack . Furthermore, the graph has edges.
Proof:
The bound on the size follows directly from Lemma 3.8. Let be an oblivious attack and consider the bad set . By Lemma 3.12, for any two points outside the bad set, there is a monotone path connecting them. Further, by Lemma 3.18, we have for any oblivious attack.
4 Reliable spanners in higher dimensions
Now we turn to the higher-dimensional setting, and show that one can construct spanners with near linear size that are reliable in expectation or with some fixed probability (which can be provided as part of the input). We use the same technique as Buchin et al. [BHO19], that is, we use our one-dimensional construction as a black box in combination with a result of Chan et al. [CHPJ18]. Let the dimension be fixed. In the following we assume