Implementation an Experimental Evaluation of Graph Connectivity Algorithms Using LEDA

In this paper we describe robust and eecient implementations of two graph connectivity algorithms. The implementations are based on the LEDA library of eecient data types and algorithms 18]. Moreover, we provide experimental evaluations of the implemented algorithms and we compare their performance to other graph connectivity algorithms currently implemented in LEDA. The rst algorithm is the Karp and Tarjan algorithm 16] for nding the connected components of an undirected graph. The algorithm achieves to nd the connected components of a graph G = hV; Ei in O(jV j) expected time. This is the rst expected-time algorithm for the static graph connectivity problem implemented in LEDA. The experimental evaluation of the algorithm proves that the algorithm performs well in practice, and establishes that theoretical and experimental results converge. The standard procedure provided by LEDA for nding the connected components of a graph, called COMPONENTS has running time O(jV j + jEj). We have compared the performance of Karp and Tarjan's algorithm to the one of COMPONENTS and we have proved that there exists a wide class of graphs (those that they are dense) that the performance of the rst algorithm dramatically improves upon the one of the second. The second implemented algorithm is the Nikoletseas, Reif, Spirakis and Yung polylog-arithmic algorithm 20] for dynamic graph connectivity. The algorithm can cope with any random sequence of three kinds of operations: (1)Property-Query(parameter): returns true if and only if the property holds; (2) Insert(x; y): inserts a new edge joining vertices x to y; (3) Delete(x; y): deletes the edge (x; y). The experimental evaluation of the algorithm proves that it is very eecient for particular classes of graphs. Comparing the performance of the implemented algorithm to the one of other dynamic connectivity algorithms implemented in LEDA, we conclude that the algorithm always performs better than all these algorithms for dense random graphs and random sequences of operations. Moreover, it works very eeciently even for sparse random graphs when the sequence of operations is long.