This paper is concerned with the problem of partitioning a threedimensional nonconvex polytope into a small number of elementary convex parts. The need for such decompositions arises in tool design, computer-aided manufacturing, finite-element methods, and robotics. Our main result is an algorithm for decomposing a nonconvex polytope of zero genus with n vertices and r reflex edges into O(n + r 2) tetrahedra. This bound is asymptotically tight in the worst case. The algorithm requires O(n + r z) space and runs in O((n + r 2) log r) time.
Abstract. In this paper we study the problems of detecting holes and antiholes in general undirected graphs, and we present algorithms for these problems. For an input graph G on n vertices and m edges, our algorithms run in O(n + m 2 ) time and require O(nm) space; we thus provide a solution to the open problem posed by Hayward et al. in [17] asking for an O(n 4 )-time algorithm for finding holes in arbitrary graphs. The key element of the algorithms is the use of the depth-first-search traversal on appropriate auxiliary graphs in which moving between any two adjacent vertices is equivalent to walking along a P 4 (i.e., a chordless path on four vertices) of the input graph or on its complement, respectively. The approach can be generalized so that for a fixed constant k ≥ 5 we obtain an O(n k−1 )-time algorithm for the detection of a hole (antihole resp.) on at least k vertices. Additionally, we describe a different approach which allows us to detect antiholes in graphs that do not contain chordless cycles on five vertices in O(n + m 2 ) time requiring O(n + m) space. Again, for a fixed constant k ≥ 6, the approach can be extended to yield O(n k−2 )-time and O(n 2 )-space algorithms for detecting holes (antiholes resp.) on at least k vertices in graphs which do not contain holes (antiholes resp.) on k − 1 vertices. Our algorithms are simple and can be easily used in practice. Finally, we also show how our detection algorithms can be augmented so that they return a hole or an antihole whenever such a structure is detected in the input graph; the augmentation takes O(n + m) time and space.
We consider rationally parameterized plane curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of the implicit equation's Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations of point sets defined by Cayley's trick. We consider polynomial and rational parameterizations, where the latter may have the same or different denominators; the implicit polygon is shown to have, respectively, up to four, five, or six vertices.
In this paper, we consider the problem of computing the connected components of the complement of a given graph. We describe a simple sequential algorithm for this problem, which works on the input graph and not on its complement, and which for a graph on n vertices and m edges runs in optimal O(n + m) time. Moreover, unlike previous linear co-connectivity algorithms, this algorithm admits efficient parallelization, leading to an optimal O(log n)-time and O((n + m)/ log n)-processor algorithm on the EREW PRAM model of computation. It is worth noting that, for the related problem of computing the connected components of a graph, no optimal deterministic parallel algorithm is currently available. The co-connectivity algorithms find applications in a number of problems. In fact, we also include a parallel recognition algorithm for weakly triangulated graphs, which takes advantage of the parallel co-connectivity algorithm and achieves an O(log 2 n) time complexity using O((n + m 2 )/ log n) processors on the EREW PRAM model of computation.
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