Graph spanners in the streaming model: an experimental study

This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (alpha, beta)-spanner of a graph G is a subgraph S subset of G such that for each pair of vertices the distance in S is at most a times the distance in G plus beta. To the best of our knowledge, this is the first computational study of graph spanner algorithms in a streaming setting. We compare experimentally the randomized algorithms proposed by Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and by Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716-727, 9-13 July 2007) for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007), Engineering and Applications Track, Eilat, Israel, 8-10 October 2007. LNCS, vol. 4698, pp. 605-617, 2007), designed for building small stretch spanners. All the algorithms we implemented work in a data streaming model where the input graph is given as a stream of edges in arbitrary order, and all of them need a single pass over the data. Differently from the algorithm in Ausiello et al., the algorithms in Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716-727, 9-13 July 2007) need to know in advance the number of vertices in the graph. The results of our experimental investigation on several input families confirm that all these algorithms are very efficient in practice, finding spanners with stretch and size much smaller than the theoretical bounds and comparable to those obtainable by off-line algorithms. Moreover, our experimental findings confirm that small values of the stretch factor are the case of interest in practice, and that the algorithm by Ausiello et al. tends to produce spanners of better quality than the algorithms by Baswana and Elkin, while still using a comparable amount of time and space resources

Collective tree spanners of graphs

In this paper we introduce a new notion of collective tree spanners. We say that a graph G =(V,E) admits a system of µ collective additive tree r-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. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log 2 n collective additive tree 2–spanners and any c-chordal graph admits a system of at most log 2 n collective additive tree (2⌊c/2⌋)–spanners. Towards establishing these results, we present a general property for graphs, called (α, r)– decomposition, and show that any (α, r)–decomposable graph G with n vertices admits a system of at most log 1/α 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