Additive Spanners for Circle Graphs and Polygonal Graphs
Graph-Theoretic Concepts in Computer Science
Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Collective additive tree spanners of homogeneously orderable graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Synchronous rendezvous for location-aware agents
DISC'11 Proceedings of the 25th international conference on Distributed computing
Navigating in a Graph by Aid of Its Spanning Tree Metric
SIAM Journal on Discrete Mathematics
Compact and low delay routing labeling scheme for Unit Disk Graphs
Computational Geometry: Theory and Applications
Collective additive tree spanners for circle graphs and polygonal graphs
Discrete Applied Mathematics
Hi-index | 0.00 |
In this paper we introduce a new notion of collective tree spanners. We say that a graph G=(V,E)admits a system of $\mu$ collective additive tree r-spanners if there is a system T(G) of at most $\mu$ spanning trees of G such that for any two vertices x,y of G a spanning tree T\in \cT(G) exists such that d_T(x,y)\leq d_G(x,y)+r. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log2n collective additive tree 2-spanners. These results are complemented by lower bounds, which say that any system of collective additive tree 1-spanners must have $\Omega(\sqrt{n})$ spanning trees for some chordal graphs and $\Omega(n)$ spanning trees for some chordal bipartite graphs and some cocomparability graphs. Furthermore, we show that any c-chordal graph admits a system of at most log2n collective additive tree (2\lfloor c/2\rfloor)-spanners, any circular-arc graph admits a system of two collective additive tree 2-spanners. Towards establishing these results, we present a general property for graphs, called (\al,r)$-decomposition, and show that any $(\al,r)$-decomposable graph $G$ with $n$ vertices admits a system of at most $\log_{1/\al} n$ collective additive tree $2r$-spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs. For any graph on n vertices admitting a system of at most $\mu$ collective additive tree r-spanners, there is a routing scheme of deviation r with addresses and routing tables of size $O(\mu \log^2n/\log \log n)$ bits per vertex. This leads, for example, to a routing scheme of deviation $(2\lfloor c/2\rfloor)$ with addresses and routing tables of size $O(\log^3n/\log \log n)$ bits per vertex on the class of c-chordal graphs.