Finding Small Node Separators
نویسندگان
چکیده
We present a new algorithm to compute small node separators on large, undirected graphs ensuring a user defined balance on the induced connected components. Our algorithm uses edge separators computed by the open source graph partitioning package KaHIP [42] and a new technique to transform them into node separators. Given an undirected graph G, our algorithm constructs a flow problem on a subgraph of G containing the edge separator so that the induced minimum cut in this subgraph corresponds to a node separator in G. Experiments show that our algorithm finds significantly better node separators than other popular tools. The speed of our algorithm is some orders of magnitude slower than other tools, however, parallelizing the algorithm and optimizing some crucial parts should decrease the running time without too much effort. To further evaluate the quality of our node separators we implemented a node ordering algorithm based on nested dissection [12, 13, 25] which uses our separators to compute high quality node orderings. Zusammenfassung Wir stellen einen neuen Algorithmus zum Finden von Knotenseparatoren auf großen, ungerichteten Graphen vor, der eine vom Benutzer geforderte Balance zwischen den induzierten Partitionen sicherstellt. Unser Algorithmus verwendet vom Open Source Graphpartitionierungspaket KaHIP [42] berechnete Kantenseparatoren und ein neues Verfahren um diese in Knotenseparatoren umzuwandeln. Für einen ungerichteten Graphen G konstruiert unser Algorithmus ein Flussproblem auf einem Subgraphen von G der den Kantenseparator enthält, sodass der induzierte minimale Schnitt in diesem Subgraphen einem Knotenseparator in G entspricht. Experimente zeigen, dass unser Algorithmus im Vergleich zu anderen bekannten Algorithmen für dieses Problem signifikant bessere Knotenseparatoren berechnet. Die Laufzeit unseres Algorithmus ist im Vergleich zu den anderen Algorithmen um einige Größenordnungen höher, jedoch sollte sich durch das Parallelisieren und Optimieren einiger kritischer Stellen die Laufzeit ohne großen Aufwand verringern lassen. Um die Qualität der gefundenen Knotenseparatoren noch besser zu beurteilen, haben wir einen Algorithmus zur Berechnung von Knotenordnungen implementiert, der auf dem nested dissection Algorithmus basiert [12, 13, 25] und unsere Knotenseparatoren verwendet, um Knotenordnungen hoher Qualität zu berechnen.
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