Graph Spanners in the Streaming Model: An Experimental Study

Abstract

This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (α, β)-spanner of a graph G is a subgraph S ⊆ G such that for each pair of vertices the distance in S is at most α times the distance in G plus β. 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: Algorithmica (2009) 55: 346-374 347 for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: ), 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: 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.

Real‐time monitoring of undirected networks: Articulation points, bridges, and connected and biconnected components

“…The restrictions imposed by classical streaming proved to be too strict to allow efficient solution for basic graph problems such as connectivity and shortest paths [15], and Feigenbaum et al [12], exploiting the idea originally introduced by Muthukrishnan [20], proposed the semi‐streaming model, in which the working memory size is O ( n p o l y l o g ( n )), where n is the number of nodes of the streaming graph: as in the semi‐external memory model [1,26], the main memory allows one to store data related to the nodes but not to the edges; Muthukrishnan [20] defines this memory requirement as a “sweet spot” for graph problems, and in this model several results appeared recently, including: connected components, bipartiteness, bipartite matching, minimum spanning tree [12,11], triangle counting [3], matching [17], and t‐spanners [2,9,11]. In particular, in the work of Feigenbaum et al [12], the authors present also an algorithm to compute the articulation points of a graph, but, since it uses a disjoint set data structure for each node of the input graph (to store the node's neighbors), its memory space requirements are not apparently within the bounds of the semi‐streaming model.…”

“…triangle counting [3], matching [17], and t-spanners [2,9,11]. In particular, in the work of Feigenbaum et al [12], the authors present also an algorithm to compute the articulation points of a graph, but, since it uses a disjoint set data structure for each node of the input graph (to store the node's neighbors), its memory space requirements are not apparently within the bounds of the semi-streaming model.…”

Real-time anomalies detection and analysis of network structure, with application to the Autonomous System network

“…The restrictions imposed by classical streaming proved to be too strict to allow efficient solution for basic graph problems such as connectivity and shortest paths [2], and Feigenbaum et al [14], exploiting the idea originally introduced by Muthukrishnan [3], proposed the semi-streaming model, in which the working memory size is O(n polylog (n)), where n is the number of vertices of the streaming graph: like in the semiexternal memory model [15], [16], the main memory allows to store data related to the nodes but not to the edges; in this model several results appeared recently, including: connected components, bipartiteness, bipartite matching, minimum spanning tree [14], [17], triangle counting [18], matching [19], and t-spanners [17], [20], [21].…”

On Resilient Graph Spanners

“…From a theoretical perspective, our σ-resilient 3-spanners have the same asymptotic size as their non-resilient counterparts. From a practical perspective, there is empirical evidence [3] that small stretch spanners provide the best performance in terms of stretch/size trade-offs, and that spanners of larger stretch are not likely to be of practical value. Table 1 puts our results in perspective with the fragility and the size of previously known spanners.…”