Reducing the bandwidth of sparse symmetric matrices
ACM '69 Proceedings of the 1969 24th national conference
Bandwidth and probabilistic complexity
Bandwidth and probabilistic complexity
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We study the probabilistic performance of heuristic algorithms for the NP-complete bandwidth minimization problem. Let (equation) be a graph with (equation). Define the bandwidth of G by (equation) where &tgr; ranges over all permutations on V. Let A be a bandwidth minimization algorithm and let A (G) denote the bandwidth of the layout produced by A on the graph G. We say that A is a level algorithm if for all graphs (equation) the layout &tgr; produced by A on G satisfies (equation) The level algorithms were first introduced by Cuthill and McKee [1] and have proved quite successful in practice. However, it is easy to construct examples that cause the level algorithms to perform poorly. Consequently worst-case analysis provides no insight to their practical success. In this paper we use probabilistic analysis in order to gain an understanding of these algorithms and to help us design better algorithms. Let (equation) be the graph defined by (equation) and let G be a random spanning subgraph of Bn↓ in which the vertices have been randomly re-labelled. We show that if A, is a level algorithm and (equation) then (equation) almost always holds, where &egr; is any positive constant. We also introduce a class of algorithms called the modified level algorithms and show that if A ' is a modified level algorithm and (equation) then (equation) almost always holds. A particular modified level algorithm MLA1 is analyzed and we show that when (equation). We also study several other properties of random subgraphs of Bn↓.