Collective Additive Tree Spanners of Bounded Tree-Breadth Graphs with Generalizations and Consequences

Collective Additive Tree Spanners of Bounded Tree-Breadth Graphs with Generalizations and Consequences

Feodor F. Dragan Algorithmic Research Laboratory, Department of Computer Science,
Kent State University, Kent, OH 44242, USA
{dragan, mabuata}@cs.kent.edu
   Muad Abu-Ata Algorithmic Research Laboratory, Department of Computer Science,
Kent State University, Kent, OH 44242, USA
{dragan, mabuata}@cs.kent.edu
Abstract

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour’s tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph admits a system of collective additive tree -spanners (resp., multiplicative tree -spanners) if there is a system of at most spanning trees of such that for any two vertices of a spanning tree exists such that (resp., ). When one gets the notion of additive tree -spanner (resp., multiplicative tree -spanner). It is known that if a graph has a multiplicative tree -spanner, then admits a Robertson-Seymour’s tree-decomposition with bags of radius at most in . We use this to demonstrate that there is a polynomial time algorithm that, given an -vertex graph admitting a multiplicative tree -spanner, constructs a system of at most collective additive tree -spanners of . That is, with a slight increase in the number of trees and in the stretch, one can “turn” a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph admits a multiplicative -spanner with tree-width , then admits a Robertson-Seymour’s tree-decomposition each bag of which can be covered with at most disks of of radius at most each. This is used to demonstrate that, for every fixed , there is a polynomial time algorithm that, given an -vertex graph admitting a multiplicative -spanner with tree-width , constructs a system of at most collective additive tree -spanners of .

Keywords: graph algorithms; approximation algorithms; tree spanner problem; collective tree spanners; spanners of bounded tree-width; Robertson-Seymour’s tree-decomposition; balanced separators.

1 Introduction

One of the basic questions in the design of routing schemes for communication networks is to construct a spanning network (a so-called spanner) which has two (often conflicting) properties: it should have simple structure and nicely approximate distances in the network. This problem fits in a larger framework of combinatorial and algorithmic problems that are concerned with distances in a finite metric space induced by a graph. An arbitrary metric space (in particular a finite metric defined by a graph) might not have enough structure to exploit algorithmically. A powerful technique that has been successfully used recently in this context is to embed the given metric space in a simpler metric space such that the distances are approximately preserved in the embedding. New and improved algorithms have resulted from this idea for several important problems (see, e.g., [4, 7, 18, 34, 44, 51]).

There are several ways to measure the quality of this approximation, two of them leading to the notion of a spanner. For , a spanning subgraph of is called a (multiplicative) -spanner of if for all [19, 55, 56]. If and , for all , then is called an additive -spanner of [50, 59, 60]. The parameter is called the stretch (or stretch factor) of , while the parameter is called the surplus of . In what follows, we will often omit the word “multiplicative” when we refer to multiplicative spanners.

Tree metrics are a very natural class of simple metric spaces since many algorithmic problems become tractable on them. A (multiplicative) tree -spanner of a graph is a spanning tree with a stretch [17], and an additive tree -spanner of is a spanning tree with a surplus [59]. If we approximate the graph by a tree spanner, we can solve the problem on the tree and the solution interpret on the original graph. The tree -spanner problem asks, given a graph and a positive number , whether admits a tree -spanner. Note that the problem of finding a tree -spanner of minimizing is known in literature also as the Minimum Max-Stretch spanning Tree problem (see, e.g., [39] and literature cited therein).

Unfortunately, not many graph families admit good tree spanners. This motivates the study of sparse spanners, i.e., spanners with a small amount of edges. There are many applications of spanners in various areas; especially, in distributed systems and communication networks. In [57], close relationships were established between the quality of spanners (in terms of stretch factor and the number of spanner edges), and the time and communication complexities of any synchronizer for the network based on this spanner. Another example is the usage of tree -spanners in the analysis of arrow distributed queuing protocols [46, 54]. Sparse spanners are very useful in message routing in communication networks; in order to maintain succinct routing tables, efficient routing schemes can use only the edges of a sparse spanner [58]. The Sparsest -Spanner problem asks, for a given graph and a number , to find a -spanner of with the smallest number of edges. We refer to the survey paper of Peleg [53] for an overview on spanners.

Inspired by ideas from works of Alon et al. [1], Bartal [4, 5], Fakcharoenphol et al. [40], and to extend those ideas to designing compact and efficient routing and distance labeling schemes in networks, in [32], a new notion of collective tree spanners111Independently, Gupta et al. in [44] introduced a similar concept which is called tree covers there. were introduced. This notion slightly weaker than the one of a tree spanner and slightly stronger than the notion of a sparse spanner. We say that a graph admits a system of collective additive tree -spanners if there is a system of at most spanning trees of such that for any two vertices of a spanning tree exists such that (a multiplicative variant of this notion can be defined analogously). Clearly, if admits a system of collective additive tree -spanners, then admits an additive -spanner with at most edges (take the union of all those trees), and if then admits an additive tree -spanner.

Recently, in [28], spanners of bounded tree-width were introduced, motivated by the fact that many algorithmic problems are tractable on graphs of bounded tree-width, and a spanner of with small tree-width can be used to obtain an approximate solution to a problem on . In particular, efficient and compact distance and routing labeling schemes are available for bounded tree-width graphs (see, e.g., [31, 44] and papers cited therein), and they can be used to compute approximate distances and route along paths that are close to shortest in . The -Tree-width -spanner problem asks, for a given graph , an integers and a positive number , whether admits a -spanner of tree-width at most . Every connected graph with vertices and at most edges is of tree-width at most and hence this problem is a generalization of the Tree -Spanner and the Sparsest -Spanner problems. Furthermore, -spanners of bounded tree-width have much more structure to exploit algorithmically than sparse -spanners (which have a small number of edges but may lack other nice structural properties).

1.1 Related work

Tree spanners.

Substantial work has been done on the tree -spanner problem on unweighted graphs. Cai and Corneil [17] have shown that, for a given graph , the problem to decide whether has a tree -spanner is NP-complete for any fixed and is linear time solvable for (the status of the case is open for general graphs)222When is an unweighted graph, can be assumed to be an integer.. The NP-completeness result was further strengthened in [15] and [16], where Branstädt et al. showed that the problem remains NP-complete even for the class of chordal graphs (i.e., for graphs where each induced cycle has length 3) and every fixed , and for the class of chordal bipartite graphs (i.e., for bipartite graphs where each induced cycle has length 4) and every fixed .

The tree -spanner problem on planar graphs was studied in [28, 41]. In [41], Fekete and Kremer proved that the tree -spanner problem on planar graphs is NP-complete (when is part of the input) and polynomial time solvable for . For fixed , the complexity of the tree -spanner problem on arbitrary planar graphs was left as an open problem in [41]. This open problem was recently resolved in [28] by Dragan et al., where it was shown that the tree -spanner problem is linear time solvable for every fixed constant on the class of apex-minor-free graphs which includes all planar graphs and all graphs of bounded genus. Note also that a number of particular graph classes (like interval graphs, permutation graphs, asteroidal-triple–free graphs, strongly chordal graphs, dually chordal graphs, and others) admit additive tree -spanners for small values of (we refer reader to [14, 15, 16, 17, 41, 49, 53, 54, 59, 60] and papers cited therein).

The first -approximation algorithm for the minimum value of for the tree -spanner problem was developed by Emek and Peleg in [39] (where is the number of vertices in a graph). Recently, another logarithmic approximation algorithm for the problem was proposed in [30] (we elaborate more on this in Subsection 1.2). Emek and Peleg also established in [39] that unless P = NP, the problem cannot be approximated additively by any term. Hardness of approximation is established also in [49], where it was shown that approximating the minimum value of for the tree -spanner problem within factor better than 2 is NP-hard (see also [54] for an earlier result).

Sparse spanners.

Sparse -spanners were introduced by Peleg, Schäffer and Ullman in [55, 57] and since that time were studied extensively. It was shown by Peleg and Schäffer in [55] that the problem of deciding whether a graph has a -spanner with at most edges is NP-complete. Later, Kortsarz [47] showed that for every there is a constant such that it is NP-hard to approximate the sparsest -spanner within the ratio , where is the number of vertices in the graph. On the other hand, the problem admits a -ratio approximation for [48, 47] and a -ratio approximation for [38]. For some other inapproximability and approximability results for the Sparsest -Spanner problem on general graphs we refer the reader to [6, 12, 22, 23, 36, 37, 38, 62] and papers cited therein. It is interesting to note also that any (even weighted) -vertex graph admits an -spanner with at most edges for any , and such a spanner can be constructed in polynomial time [2, 9, 62].

On planar graphs the Sparsest -Spanner problem was studied as well. Brandes and Handke have shown that the decision version of the problem remains NP-complete on planar graphs for every fixed (the case is open) [13]. Duckworth, Wormald, and Zito [33] have shown that the problem of finding a sparsest -spanner of a -connected planar triangulation admits a polynomial time approximation scheme (PTAS). Dragan et al. [29] proved that the Sparsest -Spanner problem admits PTAS for graph classes of bounded local tree-width (and therefore for planar and bounded genus graphs).

Sparse additive spanners were considered in [8, 24, 35, 50, 63]. It is known that every -vertex graph admits an additive 2-spanner with at most edges [24, 35], an additive 6-spanner with at most edges [8], and an additive -spanner with at most edges for any [8]. All those spanners can be constructed in polynomial time. We refer the reader to paper [63] for a good summary of the state of the art results on the sparsest additive spanner problem in general graphs.

Collective tree spanners.

The problem of finding “small” systems of collective additive tree -spanners for small values of was examined on special classes of graphs in [20, 27, 31, 32, 64]. For example, in [20, 32], sharp results were obtained for unweighted chordal graphs and -chordal graphs (i.e., the graphs where each induced cycle has length at most ): every -chordal graph admits a system of at most collective additive tree –spanners, constructible in polynomial time; no system of constant number of collective additive tree -spanners can exist for chordal graphs (i.e., when ) and , and no system of constant number of collective additive tree -spanners can exist for outerplanar graphs for any constant .

Only papers [31, 44, 64] have investigated collective (multiplicative or additive) tree spanners in weighted graphs. It was shown that any weighted -vertex planar graph admits a system of collective multiplicative tree 1-spanners (equivalently, additive tree 0-spanners) [31, 44] and a system of at most collective multiplicative tree –spanners [44]. Furthermore, any weighted graph with genus at most admits a system of collective additive tree –spanners [31, 44], any weighted graph with tree-width at most admits a system of at most collective additive tree –spanners [31, 44], any weighted graph with clique-width at most admits a system of at most collective additive tree –spanners [31], any weighted -chordal graph admits a system of collective additive tree –spanners [31] (where denotes the maximum edge weight in ).

Collective tree spanners of Unit Disk Graphs (UDGs) (which often model wireless ad hoc networks) were investigated in [64]. It was shown that every -vertex UDG admits a system of at most spanning trees of such that, for any two vertices and of , there exists a tree in with . That is, the distances in any UDG can be approximately represented by the distances in at most of its spanning trees. Based on this result a new compact and low delay routing labeling scheme was proposed for Unit Disk Graphs.

Spanners with bounded tree-width.

The -Tree-width -spanner problem was considered in [28] and [42]. It was shown that the problem is linear time solvable for every fixed constants and on the class of apex-minor-free graphs [28], which includes all planar graphs and all graphs of bounded genus, and on the graphs with bounded degree [42].

1.2 Our results and their place in the context of the previous results.

This paper was inspired by few recent results from [25, 30, 38, 39]. Elkin and Peleg in [38], among other results, described a polynomial time algorithm that, given an -vertex graph admitting a tree -spanner, constructs a -spanner of with edges. Emek and Peleg in [39] presented the first -approximation algorithm for the minimum value of for the tree -spanner problem. They described a polynomial time algorithm that, given an -vertex graph admitting a tree -spanner, constructs a tree -spanner of . Later, a simpler and faster -approximation algorithm for the problem was given by Dragan and Köhler [30]. Their result uses a new necessary condition for a graph to have a tree -spanner: if a graph has a tree -spanner, then admits a Robertson-Seymour’s tree-decomposition with bags of radius at most in .

To describe the results of [25] and to elaborate more on the Dragan-Köhler’s approach, we need to recall definitions of a few graph parameters. They all are based on the notion of tree-decomposition introduced by Robertson and Seymour in their work on graph minors [61].

A tree-decomposition of a graph is a pair where is a collection of subsets of , called bags, and is a tree. The nodes of are the bags satisfying the following three conditions:

  1. ;

  2. for each edge , there is a bag such that ;

  3. for all , if is on the path from to in , then . Equivalently, this condition could be stated as follows: for all vertices , the set of bags induces a connected subtree of .

For simplicity we denote a tree-decomposition of a graph by .

Tree-decompositions were used to define several graph parameters to measure how close a given graph is to some known graph class (e.g., to trees or to chordal graphs) where many algorithmic problems could be solved efficiently. The width of a tree-decomposition is . The tree-width of a graph , denoted by , is the minimum width, over all tree-decompositions of [61]. The trees are exactly the graphs with tree-width 1. The length of a tree-decomposition of a graph is (i.e., each bag has diameter at most in ). The tree-length of , denoted by , is the minimum of the length, over all tree-decompositions of [26]. The chordal graphs are exactly the graphs with tree-length 1. Note that these two graph parameters are not related to each other. For instance, a clique on vertices has tree-length 1 and tree-width , whereas a cycle on vertices has tree-width 2 and tree-length . In [30], yet another graph parameter was introduced, which is very similar to the notion of tree-length and, as it turns out, is related to the tree -spanner problem. The breadth of a tree-decomposition of a graph is the minimum integer such that for every there is a vertex with (i.e., each bag can be covered by a disk of radius at most in ). Note that vertex does not need to belong to . The tree-breadth of , denoted by , is the minimum of the breadth, over all tree-decompositions of . Evidently, for any graph , holds. Hence, if one parameter is bounded by a constant for a graph then the other parameter is bounded for as well.

We say that a family of graphs is of bounded tree-breadth (of bounded tree-width, of bounded tree-length) if there is a constant such that for each graph from , (resp., , ).

It was shown in [30] that if a graph admits a tree -spanner then its tree-breadth is at most and its tree-length is at most . Furthermore, any graph with tree-breadth admits a tree -spanner that can be constructed in polynomial time. Thus, these two results gave a new -approximation algorithm for the tree -spanner problem on general (unweighted) graphs (see [30] for details). The algorithm of [30] is conceptually simpler than the previous -approximation algorithm proposed for the problem by Emek and Peleg [39].

Dourisboure et al. in [25] concerned with the construction of additive spanners with few edges for -vertex graphs having a tree-decomposition into bags of diameter at most , i.e., the tree-length graphs. For such graphs they construct additive -spanners with edges, and additive -spanners with edges. Combining these results with the results of [30], we obtain the following interesting fact (in a sense, turning a multiplicative stretch into an additive surplus without much increase in the number of edges).

Theorem 1.1

(combining [25] and [30]) If a graph admits a (multiplicative) tree -spanner then it has an additive -spanner with edges and an additive -spanner with edges, both constructible in polynomial time.

This fact rises few intriguing questions. Does a polynomial time algorithm exist that, given an -vertex graph admitting a (multiplicative) tree -spanner, constructs an additive -spanner of with or edges (where the number of edges in the spanner is independent of )? Is a result similar to one presented by Elkin and Peleg in [38] possible? Namely, does a polynomial time algorithm exist that, given an -vertex graph admitting a (multiplicative) tree -spanner, constructs an additive -spanner333Recall that any additive -spanner is a multiplicative -spanner. of with edges? If we allow to use more trees (like in collective tree spanners), does a polynomial time algorithm exist that, given an -vertex graph admitting a (multiplicative) tree -spanner, constructs a system of collective additive tree -spanners of (where is similar to Big- notation up to a poly-logarithmic factor)? Note that an interesting question whether a multiplicative tree spanner can be turned into an additive tree spanner with a slight increase in the stretch is (negatively) settled already in [39]: if there exist some and and a polynomial time algorithm that for any graph admitting a tree -spanner constructs a tree -spanner, then P=NP.

We give some partial answers to these questions in Section 3. We investigate there a more general question whether a graph with bounded tree-breadth admits a small system of collective additive tree spanners. We show that any -vertex graph has a system of at most collective additive tree -spanners, where . This settles also an open question from [25] whether a graph with tree-length admits a small system of collective additive tree -spanners.

As a consequence, we obtain that there is a polynomial time algorithm that, given an -vertex graph admitting a (multiplicative) tree -spanner, constructs:

  1. a system of at most collective additive tree -spanners of (compare with [30, 39] where a multiplicative tree -spanner was constructed for in polynomial time; thus, we “have turned” a multiplicative tree -spanner into at most collective additive tree -spanners);

  2. an additive -spanner of with at most edges (compare with Theorem 1.1).

In Section 4 we generalize the method of Section 3. We define a new notion which combines both the tree-width and the tree-breadth of a graph.

The -breadth of a tree-decomposition of a graph is the minimum integer such that for each bag , there is a set of at most vertices such that for each , we have (i.e., each bag can be covered with at most disks of of radius at most each; ). The -tree-breadth of a graph , denoted by , is the minimum of the -breadth, over all tree-decompositions of . We say that a family of graphs is of bounded -tree-breadth, if there is a constant such that for each graph from , . Clearly, for every graph , , and if and only if . Thus, the notions of the tree-width and the tree-breadth are particular cases of the -tree-breadth.

In Section 4, we show that any -vertex graph with has a system of at most collective additive tree -spanners. In Section 5, we extend a result from [30] and show that if a graph admits a (multiplicative) -spanner with then its -tree-breadth is at most . As a consequence, we obtain that, for every fixed , there is a polynomial time algorithm that, given an -vertex graph admitting a (multiplicative) -spanner with tree-width at most , constructs:

  1. a system of at most collective additive tree -spanners of ;

  2. an additive -spanner of with at most edges.

We conclude the paper with few open questions.

2 Preliminaries

All graphs occurring in this paper are connected, finite, unweighted, undirected, loopless and without multiple edges. We call an -vertex -edge graph if and . A clique is a set of pairwise adjacent vertices of . By we denote a subgraph of induced by vertices of . Let also be the graph (which is not necessarily connected). A set is called a separator of if the graph has more than one connected component, and is called a balanced separator of if each connected component of has at most vertices. A set is called a balanced clique-separator of if is both a clique and a balanced separator of . For a vertex of , the sets and are called the open neighborhood and the closed neighborhood of , respectively.

In a graph the length of a path from a vertex to a vertex is the number of edges in the path. The distance between vertices and is the length of a shortest path connecting and in . The diameter in of a set is and its radius in is (in some papers they are called the weak diameter and the weak radius to indicate that the distances are measured in not in ). The disk of of radius centered at vertex is the set of all vertices at distance at most to : A disk is called a balanced disk-separator of if the set is a balanced separator of .

It is easy to see that the -spanners can equivalently be defined as follows.

Proposition 1

Let be a connected graph and be a positive number. A spanning subgraph of is a -spanner of if and only if for every edge of , holds.

This proposition implies that the stretch of a spanning subgraph of a graph is always obtained on a pair of vertices that form an edge in . Consequently, throughout this paper, can be considered as an integer which is greater than 1 (the case is trivial since must be itself).

It is also known that every additive -spanner of is a (multiplicative) -spanner of .

Proposition 2

Every additive -spanner of is a (multiplicative) -spanner of . The converse is generally not true.

3 Collective Additive Tree Spanners and the Tree-Breadth of a Graph

In this section, we show that every -vertex graph has a system of at most collective additive tree -spanners, where . We also discuss consequences of this result. Our method is a generalization of techniques used in [32] and [30]. We will assume that since any connected graph with at most 3 vertices has an additive tree 1-spanner.

Note that we do not assume here that a tree-decomposition of breadth is given for as part of the input. Our method does not need to know , our algorithm works directly on . For a given graph and an integer , even checking whether has a tree-decomposition of breadth could be a hard problem. For example, while graphs with tree-length 1 (as they are exactly the chordal graphs) can be recognized in linear time, the problem of determining whether a given graph has tree-length at most is NP-complete for every fixed (see [52]).

We will need the following results proven in [30].

Lemma 1 ([30])

Every graph has a balanced disk-separator centered at some vertex , where .

Lemma 2 ([30])

For an arbitrary graph with vertices and edges a balanced disk-separator with minimum can be found in time.

3.1 Hierarchical decomposition of a graph with bounded tree-breadth

In this subsection, following [30], we show how to decompose a graph with bounded tree-breadth and build a hierarchical decomposition tree for it. This hierarchical decomposition tree is used later for construction of collective additive tree spanners for such a graph.

Let be an arbitrary connected -vertex -edge graph with a disk-separator . Also, let be the connected components of . Denote by the neighborhood of with respect to . Let also be the graph obtained from component by adding a vertex (representative of ) and making it adjacent to all vertices of , i.e., for a vertex , if and only if there is a vertex with . See Fig. 1 for an illustration. In what follows, we will call vertex a meta vertex representing disk in graph . Given a graph and its disk-separator , the graphs can be constructed in total time . Furthermore, the total number of edges in the graphs does not exceed the number of edges in , and the total number of vertices (including meta vertices) in those graphs does not exceed the number of vertices in plus .

Figure 1: A graph with a disk-separator and the corresponding graphs obtained from . are meta vertices representing the disk in the corresponding graphs.

Denote by the graph obtained from by contracting its edge . Recall that edge contraction is an operation which removes from while simultaneously merging together the two vertices previously connected. If a contraction results in multiple edges, we delete duplicates of an edge to stay within the class of simple graphs. The operation may be performed on a set of edges by contracting each edge (in any order). The following lemma guarantees that the tree-breadths of the graphs , , are no larger than the tree-breadth of .

Lemma 3 ([30])

For any graph and its edge , implies . Consequently, for any graph with , holds for each .

Clearly, one can get from by repeatedly contracting (in any order) edges of that are not incident to vertices of . In other words, is a minor of . Recall that a graph is a minor of if can be obtained from by contracting some edges, deleting some edges, and deleting some isolated vertices. The order in which a sequence of such contractions and deletions is performed on does not affect the resulting graph .

Let be a connected -vertex, -edge graph and assume that . Lemma 1 and Lemma 2 guarantee that has a balanced disk-separator with , which can be found in time by an algorithm that works directly on graph and does not require construction of a tree-decomposition of of breadth . Using these and Lemma 3, we can build a (rooted) hierarchical tree for as follows. If is a connected graph with at most 5 vertices, then is one node tree with root node . Otherwise, find a balanced disk-separator in with minimum (see Lemma  2) and construct the corresponding graphs . For each graph () (by Lemma 3, ), construct a hierarchical tree recursively and build by taking the pair to be the root and connecting the root of each tree as a child of .

The depth of this tree is the smallest integer such that

that is, the depth is at most .

It is also easy to see that, given a graph with vertices and edges, a hierarchical tree can be constructed in total time. There are at most levels in , and one needs to do at most operations per level since the total number of edges in the graphs of each level is at most and the total number of vertices in those graphs can not exceed .

For an internal (i.e., non-leaf) node of , since it is associated with a pair , where , is a minor of and is the center of disk of , it will be convenient, in what follows, to denote by , by , by , and by itself. Thus, in these notations, and we identify node of with the set and associate with this node also the graph . See Fig. 2 for an illustration. Each leaf of , since it corresponds to a pair , we identify with the set and use, for a convenience, the notation for .

Figure 2: a) A graph and its balanced disk-separator . b) A hierarchical tree of . We have , . Meta vertices are shown circled, disk centers are shown in bold. c) The graph with its balanced disk-separator . is a minor of . d) The graph , a minor of and of . is a leaf of .

If now is the path of connecting the root of with a node , then the vertex set of the graph consists of some (original) vertices of plus at most meta vertices representing the disks , . Note also that each (original) vertex of belongs to exactly one node of .

3.2 Construction of collective additive tree spanners

Unfortunately, the class of graphs of bounded tree-breadth is not hereditary, i.e., induced subgraphs of a graph with tree-breath are not necessarily of tree-breadth at most (for example, a cycle of length with one extra vertex adjacent to each vertex of the cycle has tree-breadth 1, but the cycle itself has tree-breadth ). Thus, the method presented in [32], for constructing collective additive tree spanners for hereditary classes of graphs admitting balanced disk-separators, cannot be applied directly to the graphs of bounded tree-breadth. Nevertheless, we will show that, with the help of Lemma 3, the notion of hierarchical tree from previous subsection and a careful analysis of distance changes (see Lemma 4), it is possible to generalize the method of [32] and construct in polynomial time for every -vertex graph a system of at most collective additive tree -spanners, where . Unavoidable presence of meta vertices in the graphs resulting from a hierarchical decomposition of the original graph complicates the construction and the analysis. Recall that, in [32], it was shown that if every induced subgraph of a graph enjoys a balanced disk-separator with radius at most , then admits a system of at most collective additive tree -spanners.

Let be a connected -vertex, -edge graph and assume that . Let be a hierarchical tree of . Consider an arbitrary internal node of , and let be the path of connecting the root of with . Let be the graph obtained from by removing all its meta vertices (note that may be disconnected).

Lemma 4

For any vertex from there exists an index such that the vertices and can be connected in the graph by a path of length at most . In particular, holds.

Proof

Set , , and let be a shortest path of connecting vertices and . We know that this path has at most edges. If does not contain any meta vertices, then this path is a path of and of and therefore holds.

Assume now that does contain meta vertices and let be the closest to meta vertex in . See Fig. 3 for an illustration. Let . By construction of , meta vertex was created at some earlier recursive step to represent disk of graph for some . Hence, there is a path of length at most in with . Again, if does not contain any meta vertices, then this path is a path of and of and therefore holds. If does contain meta vertices then again, “unfolding” a meta vertex of closest to , we obtain a path of length at most in with for some .

By continuing “unfolding” this way meta vertices closest to , after at most steps, we will arrive at the situation when, for some index , a path of length at most will connect vertices and in the graph . ∎

Figure 3: Illustration to the proof of Lemma 4: “unfolding” meta vertices.

Consider two arbitrary vertices and of , and let and be the nodes of containing and , respectively. Let also be the nearest common ancestor of nodes and in and be the path of connecting the root of with (in other words, are the common ancestors of and ).

Lemma 5

Any path connecting vertices and in contains a vertex from .

Let be a shortest path of connecting vertices and , and let be the node of the path with the smallest index such that in . The following lemma holds.

Lemma 6

For each , we have , where .

Let now be the nodes at depth of the tree . For each node that is not a leaf of , consider its (central) vertex . If is an original vertex of (not a meta vertex created during the construction of ), then define a connected graph obtained from by removing all its meta vertices. If removal of those meta vertices produced few connected components, choose as that component which contains the vertex . Denote by a BFS–tree of graph rooted at vertex of . If is a leaf of , then has at most 5 vertices. In this case, remove all meta vertices from and for each connected component of the resulting graph construct an additive tree spanner with optimal surplus . Denote the resulting subtree (forest) by .

The trees (, ), obtained this way, are called local subtrees of . Clearly, the construction of these local subtrees can be incorporated into the procedure of constructing hierarchical tree of and will not increase the overall run-time (see Subsection 3.1).

Lemma 7

For any two vertices , there exists a local subtree such that .

Proof

We know, by Lemma 6, that a shortest path , intersecting and not intersecting any (), lies entirely in . Thus, . If is a leaf of then for a local subtree (it could be a forest) of constructed for the following holds: (since and ).

Assume now that is an internal node of . We have , since the depth of is at most . Let be a vertex on the shortest path . By Lemma 4, there exists an index such that the vertices and can be connected in the graph by a path of length at most . Set and . By Lemma 6, . Let be the local tree constructed for graph , i.e., a BFS–tree of a connected component of the graph and rooted at vertex .

We have and . By the triangle inequality, and That is, Now, using Lemma  6 and inequality , we get

This lemma implies two important results. Let be a graph with vertices and edges having . Also, let be its hierarchical tree and be the family of all its local subtrees (defined above). Consider a graph obtained by taking the union of all local subtrees of (by putting all of them together), i.e.,

Clearly, is a spanning subgraph of , constructible in total time, and, for any two vertices and of , holds. Also, since for every level () of hierarchical tree , the corresponding local subtrees are pairwise vertex-disjoint, their union has at most edges. Therefore, cannot have more than edges in total. Thus, we have proven the following result.

Theorem 3.1

Every graph with vertices and admits an additive –spanner with at most edges. Furthermore, such a sparse additive spanner of can be constructed in polynomial time.

Instead of taking the union of all local subtrees of , one can fix () and consider separately the union of only local subtrees , corresponding to the level of the hierarchical tree , and then extend in linear time that forest to a spanning tree of (using, for example, a variant of the Kruskal’s Spanning Tree algorithm for the unweighted graphs). We call this tree the spanning tree of corresponding to the level of the hierarchical tree . In this way we can obtain at most spanning trees for , one for each level of . Denote the collection of those spanning trees by . Thus, we obtain the following theorem.

Theorem 3.2

Every graph with vertices and admits a system of at most collective additive tree –spanners. Furthermore, such a system of collective additive tree spanners of can be constructed in polynomial time.

3.3 Additive spanners for graphs admitting (multiplicative) tree –spanners

Now we give two implications of the above results for the class of tree –spanner admissible graphs. In [30], the following important (“bridging”) lemma was proven.

Lemma 8 ([30])

If a graph admits a tree -spanner then its tree-breadth is at most .

Note that the tree-breadth bounded by provides only a necessary condition for a graph to have a multiplicative tree -spanner. There are (chordal) graphs which have tree-breadth 1 but any multiplicative tree -spanner of them has [30]. Furthermore, a cycle on vertices has tree-breadth but admits a system of 2 collective additive tree -spanners.

Combining Lemma  8 with Theorem 3.1 and Theorem  3.2, we deduce the following results.

Theorem 3.3

Let be a graph with vertices and <