Low Stretch Spanning Trees

Peleg, David

doi:10.1007/3-540-45687-2_5

Cited by 16 publications

(15 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem is NP-complete even on chordal and chordal bipartite graphs [37]. This is a well-studied problem with various applications in distributed systems and communication networks [41]. Several pairs of congestion and dilation problems are known [42].…”

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of the Spanning Tree Congestion Problem

et al. 2011

View full text Add to dashboard Cite

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k ≤ 3 the problem becomes polynomially time solvable.Extended abstract of some results in this paper appeared in the proceedings of WG 2010 [40]. Y. Otachi belongs to JSPS Research Fellow.

show abstract

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of the Spanning Tree Congestion Problem

et al. 2011

View full text Add to dashboard Cite

show abstract

“…The Sparsest t-Spanner problem asks, for a given graph G and a number t, to find a t-spanner of G 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 spanners 1 were introduced. This notion slightly weaker than the one of a tree spanner and slightly stronger than the notion of a sparse spanner.…”

mentioning

confidence: 99%

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

Dragan

Abu-Ata

2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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...

show abstract

“…Minimum Routing Cost Trees are also known as uniform Minimum Communication Cost Spanning Trees [20,21] and shortest Total Path Length Spanning Trees [25]. Furthermore the MRCT problem is a special case of the Optimal Network Problem, first studied in the 1960s by [22] and later by [8].…”

Section: Related Workmentioning

confidence: 99%

Distributed Approximation of Minimum Routing Cost Trees

Hochuli

Holzer

Wattenhofer

2014

Structural Information and Communication Complexity

View full text Add to dashboard Cite

We study the NP-hard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to find a spanning tree of a graph G over n nodes that minimizes the sum of distances between all pairs of nodes. In the considered model every node can transmit a different (but short) message to each of its neighbors in each synchronous round. We provide a randomized (2 + ε)-approximation with runtime O(D + log n ε ) for unweighted graphs. Here, D is the diameter of G. This improves over both, the (expected) approximation factor O(log n) and the runtime O(D log 2 n) stated in [17]. Due to stating our results in a very general way, we also derive an (optimal) runtime of O(D) when considering O(log n)-approximations as in [17]. In addition we derive a deterministic 2-approximation.

show abstract

Low Stretch Spanning Trees

Cited by 16 publications

References 33 publications

Parameterized Complexity of the Spanning Tree Congestion Problem

Parameterized Complexity of the Spanning Tree Congestion Problem

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

Distributed Approximation of Minimum Routing Cost Trees

Contact Info

Product

Resources

About