# Updating the Vertex Separation of a Dynamically Changing Tree

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## Date

2004

## Authors

Olsar, Peter

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## Publisher

University of Waterloo

## Abstract

This thesis presents several algorithms that update the vertex separation of a tree after the tree is modified; the vertex separation of a graph measures the largest number of vertices to the left of and including a vertex that are adjacent to vertices to the right of the vertex, when the vertices in the graph are arranged in the best possible linear ordering. Vertex separation was introduced by Lipton and Tarjan and has since been applied mainly in VLSI design. The tree is modified by either attaching another tree or removing a subtree. The first algorithm handles the special case when another tree is attached to the root, and the second algorithm updates the vertex separation after a subtree of the root is removed. The last two algorithms solve the more general problem when subtrees are attached to or removed from arbitrary vertices; they have good running time performance only in the amortized sense. The running time of all our algorithms is sublinear in the number of vertices in the tree, assuming certain information is precomputed for the tree. This improves upon current algorithms by Skodinis and Ellis, Sudborough, and Turner, both of which have linear running time for this problem. Lower and upper bounds on the vertex separation of a general graph are also derived. Furthermore, analogous bounds are presented for the cutwidth of a general graph, where the cutwidth of a graph equals the maximum number of edges that cross over a vertex, when the vertices in the graph are arranged in the best possible linear ordering.

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## Keywords

Computer Science, algorithm, tree, graph, layout, vertex separation, cutwidth, linear ordering, VLSI design, amortized analysis, bounds