Diameter Spanner, Eccentricity Spanner, and Approximating Extremal Graph Distances: Static, Dynamic, and Fault Tolerant

The diameter, vertex eccentricities, and the radius of a graph are some of the most fundamental graph parameters. Roditty and Williams [STOC 2013] gave an O(m√(n)) time algorithm for computing a 1.5 approximation of graph diameter. We present the first non-trivial algorithm for maintaining `< 2'- approximation of graph diameter in dynamic setting. Our algorithm maintain a (1.5+ϵ) approximation of graph diameter that takes amortized update time of O(ϵ^-1n^1.25) in partially dynamic setting. For graphs whose diameter remains bounded by some large constant, the total amortized time of our algorithm is O(ϵ^-2√(n)), which almost matches the best known bound for static 1.5-approximation of diameter. Backurs et al. [STOC 2018] gave an Õ(m√(n)) time algorithm for computing 2-approximation of eccentricities. They also showed that no O(n^2-o(1)) time algorithm can achieve an approximation factor better than 2 for graph eccentricities, unless SETH fails. We present the Õ(m) time algorithm for computing 2-approximation of vertex eccentricities in directed weighted graphs. We also present fault tolerant data-structures for maintaining 1.5-diameter and 2-eccentricities. We initiate the study of Extremal Distance Spanners. Given a graph G=(V,E), a subgraph H=(V,E0) is defined to be a t-diameter-spanner if the diameter of H is at most t times the diameter of G. We show that for any n-vertex directed graph G we can compute a sparse subgraph H which is a (1.5)-diameter-spanner of G and contains at most O(n^1.5) edges. We also show that this bound is tight for graphs whose diameter is bounded by n^1/4-ϵ. We present several other extremal-distance spanners with various size-stretch trade-offs. Finally, we extensively study these objects in the dynamic and fault-tolerant settings.

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1 Introduction

In this paper, we initiate the study of Extremal Distance Spanner. The notion of spanners (also known as distance spanner) was first introduced and studied in [7, 23, 37]. A spanner (also known as distance spanner) of a graph is a sparse subgraph that approximately preserves pair-wise distances of the underlying graph . Besides being theoretically interesting, they are known to have numerous applications in different areas of computer science such as distributed systems, communication networks and efficient routing schemes [21, 22, 38, 39, 42, 33, 4, 32], motion planning [25, 20], approximating shortest paths [18, 19, 26] and distance oracles [12, 43].

It is known that for any integer , there exists for an undirected graph with vertices a edges spanner with multiplicative stretch . The works of [40, 11] provided efficient constructions of such spanners. It is also widely believed that this size-stretch trade-off is tight. Assuming the widely believed Erdös girth conjecture [30], this size-stretch trade-off is tight. Other fascinating works have studied spanners for undirected graphs with additive stretch [10, 16, 2, 29, 44], spanners for different distance metrics [22, 39, 4], and so on. It was shown in [29] that for any and any , there exists a -spanner for -vertex unweighted graphs with edges, where .

Unfortunately, the landscape of distance spanners in the directed setting is far less understood. This is because we cannot have sparse spanners for general directed graphs. Even when underlying graph is strongly-connected, there exists graphs with edges such that excluding even a single edge from the graph results in a distance-spanner with stretch as high as diameter. In such a scenario, for directed graphs, a natural direction to study is construction of sparse subgraphs that approximately preserves the graph diameter.

This brings us to the following central question.

Given a directed graph and a “stretch factor” , can we construct a sparse subgraph of such that the distance between any two vertices in is bounded by times the maximum distance in ?

We define such graphs as -diameter spanners as they essentially preserve the diameter up to a multiplicative factor . We also consider the following related question of -eccentricity spanner.

Given a directed graph and a “stretch factor” , can we construct a sparse subgraph of such that the eccentricity of each vertex in is at most times the eccentricity of in ?

We study the unexplored terrain of extremal-distance spanners for directed graphs. First, it is not clear if there exists a -diameter spanner and -eccentricity spanner for directed graphs for small values of  (typically close to ). Next, suppose there exists a -diameter/eccentricity spanner for some parameter , how can we construct such spanners efficiently? Finally, we address the questions of maintaining these spanners in the dynamic setting and the fault-tolerant setting, where the underlying graph changes with time.

We believe that extremal-distance spanners are interesting mathematical objects in their own right. Nevertheless, such a sparsification of graphs indeed suffices for many of the original applications of the well-studied standard graph spanners, such as in communication networks, facility location problem, routing, etc. In particular, diameter spanners with a sparse set of edges are good candidates for backbone networks [32]. Our study of extremal-distance spanners has many additional implications that we present in the next subsection.

1.1 Our Contributions

In the following subsections, we present our results in detail.

1.1.1 Diameter Spanners

We show sparse diameter spanner constructions with various trade-offs between the size (number of edges) of the spanner and its stretch factor , and provide efficient algorithms to construct such spanners.

We provide efficient construction of -diameter spanners, and also show that our -stretch diameter spanner construction is essentially tight for graphs whose diameter is bounded by .

(a) There exists a Las Vegas algorithm that for any unweighted directed graph , computes a -diameter spanner of with at most edges. The computation time of is

with high probability. If

is edge-weighted, then satisfies the condition that , where is an upper bound on the weight of edges in .

(b) For every and every , there exists an unweighted directed graph with vertices and diameter , such that any subgraph of that satisfies is strictly less than contains at least edges.

For the scenario when , we provide a construction of -diameter spanners that are sparser than the -diameter spanners. We also show that our -stretch spanner construction is tight, for graphs whose diameter is bounded by .

(a) There exists a Las Vegas algorithm that for any directed graph having diameter , computes a -diameter spanner of that contains at most edges. The computation time of is with high probability111Though the computation time of is a function of , the algorithm does not need to apriori know the value ..

(b) For every and every , there exists a unweighted directed graph with vertices and diameter , such that any subgraph of for which is strictly less than contains at least edges.

We also show that for any directed graph we can either (i) compute a diameter spanner with arbitrarily low stretch, or (ii) compute a diameter spanner with arbitrarily low size.

For any arbitrarily small fractions , and any given directed graph , in  expected time, at least one of the following subgraphs can be computed.
(i) a -diameter spanner of containing at most edges.
(ii) a -diameter spanner of containing at most edges.

In Theorem 1.1.1 and Theorem 1.1.1, we show a lower bound on the number of edges in diameter spanners of stretch respectively and , for graphs with low diameter. A natural question to ask here is if for graphs with large diameter it is possible to obtain diameter spanners with low stretch (ideally ) that are also sparse (ideally having edges) in nature. The next theorem positively answers this question.

For any directed graph satisfying , we can compute a subgraph with edges satisfying .

1.1.2 Dynamic Maintenance of Diameter Spanners

We obtain the following dynamic algorithm for maintaining a -diameter spanner incrementally, as well as decrementally.

For any and -vertex directed graph, there exists an incremental (and decremental) algorithm that maintains a -diameter spanner that consists at most edges. The expected amortized update time of the algorithm is for the incremental setting and for the decremental setting, where denotes an upper bound on the diameter of the graph throughout the run of the algorithm.

For graphs whose diameter remains bounded by , we provide incremental (and decremental) algorithms for maintaining -diameter spanners. (In the dynamic setting as well these spanners are sparser than -stretch diameter spanners for graphs whose diameter is at most ).

For any and -vertex directed graph, there exists an incremental (and decremental) algorithm that maintains a -diameter spanner that consists at most edges, where denotes an upper bound on the graph diameter throughout the run of the algorithm. The expected amortized update time of the algorithm is for the incremental setting, and for the decremental setting.

An interesting immediate corollary of our dynamic maintenance of diameter spanners is an incremental (and decremental) algorithm that maintains a -approximation of the graph’s diameter, whose expected total update time is for the incremental setting, and for the decremental setting, where denotes an upper bound on the diameter of graph throughout the run of the algorithm. There is a very recent independent work by Ancona et al. [5] on dynamically maintaining the diameter value of a graph, using related techniques. In particular, they give a -approximation algorithm with total update time for the incremental setting, and a -approximation algorithm for incremental (resp., decremental) setting, whose total update time is with high probability. 222The authors of [5] informed us that they had a solution to the decremental problem since 2015 (private communication).

1.1.3 Eccentricity Spanners

Given a graph , we say that a subgraph of is a -eccentricity spanner of if the eccentricity of any vertex in is at most times its eccentricity in . Similarly, is said to be a -radius spanner, if the radius of is at most times the radius of graph .

We obtain a construction for -eccentricity spanner with edges that are computable in just time; also we show that there exists graphs whose -eccentricity spanner, for any , contains number of edges, where is the radius of the graph.

(a) There exists a Las Vegas algorithm that for any directed weighted graph computes in expected time a -eccentricity spanner (which is also a -radius spanner) containing at most edges.

(b) For every and every , there exists a unweighted directed graph with vertices and radius , such that any subgraph of that is a -eccentricity spanner of , for any , contains at least edges.

Implications: First near optimal time -approximation of vertex eccentricities.

For the problem of computing exact eccentricities in weighted directed graphs the only known solution is solving the all-pair-shortest-path problem and it takes time. Backurs et al. [8] showed that for any directed weighted graph there exists an algorithm for computing 2-approximation of eccentricities in time. They also showed that for any , there exists an time algorithm for computing a approximation of graph eccentricities.

We obtain (as a product of our -eccentricity spanner) the first time algorithm for computing -approximation of eccentricities.

For any directed weighted graph with vertices and edges, we can compute in expected time a -approximation of eccentricities of vertices in .

Our result is essentially tight. The approximation factor of cannot be improved since Backurs et al. showed in their paper that unless SETH fails no time algorithm can achieve an approximation factor better than for graph eccentricities [8]. Also the computation time of our algorithm is almost optimal as we need time to even scan the edges of the graph.

1.1.4 Dynamic Maintenance of Eccentricity Spanners

We obtain incremental and decremental algorithm for maintaining -eccentricity spanner.

For any , there exists an incremental (and decremental) algorithm that maintains for an -vertex directed graph a -eccentricity spanner (which is also a -radius spanner) containing at most edges. The expected amortized update time is for the incremental setting and for the decremental setting, where, denotes an upper bound on the diameter of the graph throughout the run of the algorithm.

Implication: Dynamic maintenance of -approximate eccentricities.

The above stated dynamic algorithm for -eccentricity spanner also imply a same time bound algorithm for maintaining a -approximation of vertex eccentricities.

For any , there exists an incremental (and decremental) algorithm that maintains for an -vertex directed graph a -approximation of graph eccentricities. The expected amortized update time is for the incremental setting and for the decremental setting, where, denotes an upper bound on the diameter of the graph throughout the run of the algorithm.

Additional dynamic algorithms with different approximation factors and runtime bounds for maintaining vertex eccentricities, were very recently given by Ancona et al. [5]; for more details, see [5].

1.2 More Related Work

The girth conjecture of Erdös [30] implies that there are undirected graphs on vertices, for which any -spanner will require edges. This conjecture has been proved for , and , and is widely believed to be true for any integer . Thus, assuming the girth conjecture, one can not expected for a better size-stretch trade-offs.

Althöfer et al. [3] were the first to show that any undirected weighted graph with vertices has a -spanner of size . The lower bound mentioned above implies that the size-bound of this spanner is essentially optimal. Althöfer et al. [3] gave an algorithm to compute such a spanner, and subsequently, a long line of works have studied the question of how fast can we compute such a spanner, until Baswana and Sen [11] gave a linear-time algorithm.

A -additive spanner of an undirected graph is a subgraph that preserves distances up to an additive constant . That is, for any pair of nodes in it holds that . This type of spanners were also extensively studied [10, 16, 2, 29]. The sparsest additive spanner known is a 6-additive spanner of size that was given by Baswana, Kavitha, Mehlhorn, and Pettie [10]. It was only recently that Abboud and Bodwin [1] proved that the additive spanner bound is tight, for any additive constant .

Since for directed graph distance spanners are impossible, the roundtrip distance metric was proposed. The roundtrip-distance between two vertices and is the distance from to plus the distance from to . Roditty, Thorup, and Zwick [39] presented the notion of roundtrip spanners for directed graphs. A roundtrip spanner of a directed graph is a sparse subgraph that approximately preserve the roundtrip distance between each pair of nodes and . They showed that any directed graph has roundtrip spanners, and gave efficient algorithms to construct such spanners.

The question of finding the sparsest spanner of a given graph was shown to be NP-Hard by Peleg and Schäffer [23], in the same work that graph spanner notion was introduced by Peleg and Schäffer [23].

Diameter spanners were mentioned by Elkin and Peleg [28, 27], but in the context of approximation algorithms for finding the sparsest diameter spanner (which is NP-Hard). To the best of our knowledge, there is no work that showed the existence of sparse diameter spanners with stretch less than , for directed graphs.

2 Preliminaries

Given a directed graph on vertices and edges, the following notations will be used throughout the paper.

  • :  the shortest path from vertex to vertex in graph .

  • :  the length of the shortest path from vertex to vertex in graph . We sometimes denote it by , when the context is clear.

  • .

  • :  the diameter of graph , that is, .

  • ():  an outgoing breadth-first-search (BFS) tree rooted at (supernode ).

  • ():  an incoming breadth-first-search (BFS) tree rooted at (supernode ).

  • :  the depth of tree .

  • :  the depth of tree .

  • :  the radius of graph , that is, .

  • :  the tree obtained from by truncating it at depth .

  • :  the tree obtained from by truncating it at depth .

  • :  the closest outgoing vertices of , where ties are broken arbitrarily.

  • :  the closest incoming vertices of , where ties are broken arbitrarily.

  • :  the depth of vertex in the rooted tree .

  • .

  • :  the power set of .

In all the above defined notations, when dealing with dynamic graphs, we use subscript to indicate the time-stamps.

Throughout the paper we assume the graph is strongly connected, as otherwise the diameter of is , and even an empty subgraph of preserves its diameter.

We first formally define the notion of the -diameter spanners that is used in the paper. [Diameter-spanner] Given a directed graph , a subgraph is said to be -diameter spanner of if .

Next we introduce the notion of -dominating-set-pair which are a generalization of traditional -dominating sets [34, 35].

[Dominating-set-pair] For a directed graph , and a set-pair satisfying , we say that is -dominating with size-bound , if , , and either (1) for each , , or (2) for each , .

Here, is said to be -out-dominating if it satisfies condition 1, and is said to be -in-dominating if it satisfies condition 2.

We point here that it was shown by Cairo, Grossi and Rizzi [15] that for any -vertex undirected graph with diameter , there is a -dominating-set of size . Using the -dominating-set they obtain a hierarchy of diameter and radius approximation algorithms. However, the construction extends to directed graph only when . In fact, we can prove that these bounds are unachievable for directed graphs whenever . For completeness, we revisit the construction for in Section 3.

We below state few results that will useful in our construction.

Let be an -vertex directed graph. Let be integers satisfying , and let be a uniformly random subset of of size . Then with a high probability, has non-empty intersection with and , for each . 333If the graph is undergoing either edge insertions, or edge deletions, then with high probability the relation holds for each of the instances of .

In order to dynamically maintain diameter-spanners, we will use the following result by Even and Shiloach [31] on maintaining single-source shortest-path-trees. Even and Shiloach gave the algorithm for maintaining shortest path tree in the decremental setting, and their algorithm can be easily adapted to work in the incremental setting as well.

[ES-tree [31]] There is a decremental (incremental) algorithm for maintaining the first levels of a single-source shortest-path-tree, in a directed or undirected graph, whose total running time, over all deletions (insertions), is , where is the initial (final) number of edges in the graph.

3 Static and Dynamic Maintenance of 1.5-Diameter Spanners

Our main idea for computing sparse diameter spanner comes from the recent line of works [41, 8, 17, 36, 15] on approximating diameter in directed or undirected graphs. Let be a uniformly random subset of of size . We take to be the vertex of the maximum depth in . Also, is set to . By Lemma 3, with high probability, the set contains a vertex of , if not, we can re-sample , and compute again. For convenience, throughout this paper, we refer to this constructed set-pair as a valid set-pair.

Using the idea of -dominating-set construction of Cairo et  al. [15] for undirected graphs, we show that a valid-set-pair is -dominating for any fractions satisfying , where is the diameter of the input graph. If depth of is at most , then is trivially -out-dominating. So let us consider the case that depth of is greater than . Observe that in such a case does not contain any vertex of , and so because intersects the set , whereas, is empty. Since is a strict subset of , we have that is at most if is unweighted, and at most is is edge-weighted with weights in range .

Let denote the subgraph of which is union of and , for . Graph contains at most edges since . Observe that computation of graph takes expected time (recall computation of is randomized). If is -out-dominating, then is a -diameter spanner. Indeed, for any , there is an satisfying , and , which implies . In the case when is not -out-dominating, then as shown is -in-dominating if is unweighted, and -in-dominating is is edge-weighted, so in this case, respectively, is bounded by or , for every . We thus have the following theorem.

For any directed unweighted graph with vertices and edges, we can compute in expected time a -diameter spanner of with at most edges.

Moreover, if is edge-weighted, then satisfies the condition that , where is an upper bound on the maximum edge-weight in .

The construction of -diameter spanners is quite trivial, however, their dynamic maintenance is challenging. In order to maintain a -diameter spanner dynamically, a naive approach would be to dynamically maintain the valid-sets. We face two obstacles: (i) the first being dynamic maintenance of a vertex having maximum depth in tree , and (ii) the second is dynamically maintaining the set . We see in the next subsection, how to tackle these issues.

3.1 Dynamic Maintenance of Dominating Sets

In this subsection, we provide efficient algorithms for maintaining a dominating-set-pair. We first observe that the static construction of dominating-set-pair can be even further generalized as given in the following lemma (see the Appendix for its proof).

For any integers satisfying , and any directed graph with vertices and edges, in expected time we can compute a set-pair of size bound which is -dominating for some vertex and any arbitrary fractions satisfying .

Our main approach for dynamically maintaining a dominating-set-pair is to use the idea of lazy updates. We formalize this through the following lemma. Let be a dynamic graph whose updates are insertions (or deletions) of edges, and be a (non-dynamic) subset of of size . Let be two time instances, and let , for some . Let and be such that and lie in the range . Then for any satisfying , set-pair is dominating at time if is non-empty, and
(i) if  , when restricted to edge deletions case.
(ii) if  , when restricted to edge insertions case.

Proof.

Let and respectively denote the values , and .

We first analyse the edge deletions case. If depth of at the time is bounded by , then is -out-dominating. So let us assume that is strictly greater than . Then . So . Note that is bounded below by , so, at time , the truncated tree must have empty intersection with . Since intersects , at time , must be contained in . The crucial point to observe is that in the decremental scenario, the set can only reduce in size with time. Thus . Since contains , we have . Thus, if is not -out-dominating, then is -in-dominating set.

We next analyse the edge insertions case. If , then is out-dominating at time since . If depth of in at time is greater than , then the truncated tree must have an empty intersection with set , however, the set has a non-empty intersection with , thus . So . Thus, is dominating at time . Observe that as edges are added to , the depth of vertices in and can only decrease with time, so the set-pair must also be dominating at time . ∎

We now present algorithms that for a given , and integers satisfying , incrementally (and decrementally) maintains for an -vertex graph a triplet such that at any time instance ,

  1. , and for some ,

  2. , , lies in range , where , and

  3. .

Incremental scenario. We first discuss the incremental scenario. The main obstacle in this setting is to dynamically maintain a vertex having large depth in . We initialize to a uniformly random subset of containing vertices, and store in the depth of tree . Next we compute a set far that consist of all those vertices whose distance from is at least , and set to be a uniformly random subset of far of size . We initialize to any arbitrary vertex in set , and set to . Throughout the algorithm whenever is empty, then we recompute , and . The probability of such an event is inverse polynomial in .

We use Theorem 2 to dynamically maintain and depth of individual vertices in . This takes time for any fixed . Whenever falls below the value , then we recompute and . For any fixed , this happens at most times, and so takes time in total. Whenever depth of a vertex lying in falls below the value , then we remove that vertex from . This step takes time in total. If falls below the value , then we replace by an arbitrary vertex in , and recompute .

If becomes empty and is still greater than , then we recompute , and . Observe that for any fixed this happens at most times. This is because if and is a partition of far such that the depth of all vertices in falls below earlier than the vertices in , then with high probability has a non-empty intersection with . This holds true as we assume adversarial model in which edge insertions are independent of choice of . Thus with high probability. each time is recomputed the size of set far decreases by at least half, assuming remains fixed. Since changes at most times, the set is recomputed at most times, and vertex can thus change times.

Finally, for vertex we maintain using ES-tree. Since, changes at most times, total time for maintaining is . Whenever falls by a factor of , then we re-set to . For a fixed , is updated at most times. So in total changes at most times, and the total time for maintaining set , throughout the edge insertions is . Thus, the total time taken by the algorithm is , where denotes the maximum diameter of throughout the sequence of edge updates. Also the expected number of times the triplet changes is .

Decremental Scenario. We now discuss the simpler scenario of edge deletions. As before, we initialize to be a uniformly random subset of containing vertices. Next we compute and set to be an arbitrary vertex having maximum depth in . Also is set to . We store in the depth of tree , and as in incremental setting use Theorem 2 to dynamically maintain the depth of . This takes time in total. Whenever exceeds the value , then we recompute and . For any fixed such an event happens at most times, and takes in total time. Also whenever is non-empty, then we recompute , and reinitialize the Even and Shiloach data-structure. The probability of such an event is inverse polynomial in . So the expected amortized update time for edge deletions is . Also if is the time when were last updated and is the current time then , . Thus the conditions (i) and (ii) hold. Also only increases with time, so condition (iii) trivially holds. So, the amortized update time of the procedure is in the decremental scenario, where denotes the maximum diameter of G throughout the sequence of edge updates; and the expected number of times when triplet changes .

The following theorem is immediate from the above discussion and Lemma 3.1.

For any , and any integers satisfying , there exists an algorithm that incrementally/decrementally maintains for an -vertex directed graph a set-pair of size-bound which is -dominating, for some , and any arbitrary fractions satisfying .

The expected amortized update time of the algorithm is in incremental setting and in decremental setting, where, denotes the maximum diameter of the graph throughout the sequence of edge updates. Also, the algorithm ensures that with high probability the triplet changes at most times in the incremental setting, and at most times in the decremental setting.

3.2 Dynamic Algorithms for -Diameter Spanners

We consider two model for maintaining the diameter spanners, namely, the explicit model and the implicit model. The explicit model maintains at each stage all the edges of a diameter spanner of the current graph. In the model of implicitly maintaining the diameter spanner, the goal is to have a data-structure that efficiently supports the following operations: (i) that adds to or remove from the graph the edge , and (ii) that checks if the diameter spanner contains edge .

We first consider the explicit maintenance of diameter spanners.

Let be an algorithm that uses Theorem 3.1 to incrementally (or decrementally) maintain at any time , a -dominating set-pair of size bound , where . We dynamically maintain a subgraph which is union of and , for . This takes in total