The one-dimensional on-line bin-packing problem is considered, A simple
O
(1)-space and
O
(
n
)-time algorithm, called HARMONIC
M
, is presented. It is shown that this algorithm can achieve a worst-case performance ratio of less than 1.692, which is better than that of the
O
(
n
)-space and
O
(
n
log
n
)-time algorithm FIRST FIT. Also shown is that 1.691 … is a lower bound for
all
0
(1)-space on-line bin-packing algorithms. Finally a revised version of HARMONIC
M
, an
O
(
n
)-space and
O
(
n
)- time algorithm, is presented and is shown to have a worst-case performance ratio of less than 1.636.
The medial axis transformation is a means first proposed by Blum to describe a shape. In this paper we present a 0(n log n) algorithm for computing the medial axis of a planar shape represented by an n-edge simple polygon. The algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in Fortran. Some computer-plotted output of the program are also shown in the paper
Abstract.This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen among n points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocess n points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimal O(k+logn) time algorithm for answering such queries, where k is the number of points to be reported. The algorithm requires O(n) space and O(n log n) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.
We introduce the notion of generalized Delaunay triangulation of a planar straight-line graph G = (V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs in O(I VI 2) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-andconquer algorithm based on the polygon cutting theorem of Chazelle is given that runs in O(I V I logl VI) time.
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