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

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 G admits a system of µ collective additive tree r-spanners (resp., multiplicative tree t-spanners) if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r (resp., dT (x, y) ≤ t · dG(x, y)). When µ = 1 one gets the notion of additive tree r-spanner (resp., multiplicative tree t-spanner). It is known that if a graph G has a multiplicative tree t-spanner, then G admits a Robertson-Seymour's tree-decomposition with bags of radius at most ⌈t/2⌉ in G. We use this to demonstrate that there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative tree t-spanner, constructs a system of at most log 2 n collective additive tree O(t log n)-spanners of G. 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 G admits a multiplicative t-spanner with tree-width k − 1, then G admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most k disks of G of radius at most ⌈t/2⌉ each. This is used to demonstrate that, for every fixed k, there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative t-spanner with tree-width k − 1, constructs a system of at most k(1 + log 2 n) collective additive tree O(t log n)-spanners of G. a stretch t [17], and an additive tree r-spanner of G is a spanning tree with a surplus r [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 t-spanner problem asks, given a graph G and a positive number t, whether G admits a tree t-spanner. Note that the problem of finding a tree t-spanner of G minimizing t 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 t-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 spar...

Metric tree‐like structures in real‐world networks: an empirical study

“…Every ‐vertex graph with tree‐length has an additive ‐spanner with edges and an additive ‐spanner with edges, both constructible in polynomial time . Every ‐vertex graph with has a system of at most collective additive tree ‐spanners constructible in polynomial time . Those graphs also enjoy a ‐additive routing labeling scheme with ‐bit labels and time routing protocol , and a ‐additive routing labeling scheme with ‐bit labels and time routing protocol with message initiation time (by combining results of and ).…”

“…Hence, enjoys a ‐multiplicative routing labeling scheme with ‐bit labels and time routing protocol (routing is essentially done in that tree spanner). Another result for graphs with , useful for designing routing labeling schemes, is presented in . It states that every ‐vertex graph with has a system of at most collective additive tree ‐spanners, that is, a system of at most spanning trees of such that for any two vertices of there is a tree in with .…”

“…It states that every ‐vertex graph with has a system of at most collective additive tree ‐spanners, that is, a system of at most spanning trees of such that for any two vertices of there is a tree in with . Furthermore, such a system for can be constructed in polynomial time . By combining this with a result from , we obtain that every ‐vertex graph with enjoys a ‐additive routing labeling scheme with ‐bit labels and time routing protocol with message initiation time.…”

“…They allow for efficient approximate solutions for a number of optimization problems. For example, they admit a PTAS for the Traveling Salesman Problem , have an efficient approximate solution for the problem of covering and packing by balls , admit additive sparse spanners and collective additive tree‐spanners , enjoy efficient and compact approximate distance and routing labeling schemes, and have efficient algorithms for quick and accurate estimations of diameters and radii . We elaborate more on these results in appropriate sections.…”

Easy computation of eccentricity approximating trees

“…For every u, v ∈ V , let dist G (u, v) be the minimum length (number of edges) of a uv-path in G. A spanning tree T of G is a k-additive tree spanner if we have dist T (u, v) ≤ dist G (u, v) + k for every u, v ∈ V . There has been a great deal of research on additive tree spanners (e.g., see [2,7,9,12,13,11,17]). This is in part motivated by their various applications, e.g.…”