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Minimizing width in linear layouts

  • F. S. Makedon
  • I. H. Sudborough
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 154)

Abstract

A (linear) layout of an undirected graph G is a one-to-one function mapping the vertices of G to integers. The cutwidth of G under a linear layout L, denoted by cw(G,L), is the maximum, taken over all possible i, of the number of edges connecting vertices assigned to integers less than i to vertices assigned to integers at least as large as i. The cutwidth of a graph G, denoted by cw(G), is the minimum of cw(G,L), taken over all possible linear layouts L. The problem of determining the cutwidth of a graph, called the Min Cut Linear Arrangement problem, has applications in VLSI, for example in the minimization of interconnection channels in Weinberger arrays [16].

We describe a relationship between the cutwidth of a graph G and its “search number” [10], denoted by s(G). We show that, for all graphs G, s(G) ≤ cw(G) ≤ ≫deg(G)/ 2⌋ · s(G), where deg(G) denotes the maximum degree of any vertex in G. In particular, this means that, for any graph G with maximum vertex degree 3, s(G)=cw(G).

The earlier dynamic programming algorithm of Gurari and Sudborough [5 ] is improved to show that, for any k≥2, the problem of deciding if a given graph has cutwidth at most k can be done in O(nk−1) steps. We also characterize the classes of graphs with cutwidth 2 and cutwidth 3. The latter characterization strongly suggests a linear time algorithm to determine whether a given graph has cutwidth at most three.

Keywords

Incident Edge Linear Time Algorithm Optimal Layout Node Splitting Search Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • F. S. Makedon
    • 1
  • I. H. Sudborough
    • 2
  1. 1.Dept. of Computer ScienceIllinois Institute of Tech.ChicagoUSA
  2. 2.Electrical Eng. and Computer Sci.Northwestern UniversityEvanstonUSA

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