J. Mark Keil

Clustering algorithms based on minimum and maximum spanning trees

We consider clustering problems under two different optimization criteria. One is to minimize the maximum intracluster distance (diameter), and the other is to maximize the minimum intercluster distance. In particular, we present an algorithm which partitions a set <italic>S</italic> of <italic>n</italic> points in the plane into two subsets so that their larger diameter is minimized in time <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) and space <italic>&Ogr;</italic>(<italic>n</italic>). Another algorithm with the same bounds computes a <italic>k</italic>-partition of <italic>S</italic> for any <italic>k</italic> so that the minimum intercluster distance is maximized. In both instances it is first shown that an optimal parition is determined by either a maximum or minimum spanning tree of <italic>S</italic>.

Finding Hamiltonian circuits in interval graphs

A simple circuit in a graph G = ( V , E ) is a sequence (v 1, v2, . . . , v k) of distinct vertices from V such that { v , , v i + l } ~ E for l ~ < i < k 1 and {v k, v I } ~ E. A Hamiltonian circuit in G is a simple circuit that includes all the vertices of G. The problem of deciding whether a graph has a Hamiltonian circuit has long been known to be NP-complete [5,11]. The Hamiltonian circuit problem remains NP-complete for planar 3-connected graphs [6], bipartite graphs [12], split graphs [8], edge graphs [1], planar bipartite graphs [10], and grid graphs [10]. Thus far, polynomial time algorithms for the problem have only been developed for 4-connected planar graphs [9] and proper interval graphs [2]. This paper presents a linear time algorithm for the Hamiltonian circuit problem in interval graphs. A graph is an intersection graph if there exists a one-to-one correspondence between its vertices and a family F of sets such that two vertices are adjacent in the graph if and only if their two corresponding sets intersect. If F is a family of intervals of the real line, then G is called an interval graph [8,13] and the family F is called the interval model for G.

Domination in permutation graphs

Abstract O ( n 3 ) algorithms to solve the weighted domination and weighted independent domination problems in permutation graphs, and an O ( n 2 ) algorithm to solve the cardinality domination problem in permutation graphs are presented.

Computing a subgraph of the minimum weight triangulation

Abstract Given a set S of n points in the plane, it is shown that the 2 -skeleton of S is a subgraph of the minimum weight triangulation of S. The β-skeletons are polynomially computable Euclidean graphs introduced by Kirkpatrick and Radke [8]. The 2 -skeleton of S is the β-skeleton of S for β = 2 .

ON THE TIME BOUND FOR CONVEX DECOMPOSITION OF SIMPLE POLYGONS

We show that a decomposition of a simple polygon having n vertices, r of which are reflex, into a minimum number of convex regions without the addition of Steiner vertices can be computed in O(n + r2min{r2, n}) time and space. A Java demo is available at .

The Delauney Triangulation Closely Approximates the Complete Euclidean Graph

Let p and q be a pair of points in a set S of N points in the plane. Let d(p,q) be the Euclidean distance between p and q and let DT(p,q) be the length of the shortest path from p to q in the Delaunay triangulation of S. We show that that the ratio

Minimum Decompositions of Polygonal Objects

\frac{{DT(p,q)}}{{d(p,q)}} \leqslant \frac{{2\pi }}{{3\cos (\frac{\pi }{6})}} \approx 2.42

POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM

independent of S and N.