6378032
Adding random edges to dense graphs
This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and k-connectivity.
17312370
A note on short cycles in a hypercube
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdos about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,C"l) be the largest number of edges in a subgraph of a hypercube Q"n containing no cycle of length l. It is known that f(n,C"l)=o(|E(Q"n)|), when l=4k, k>=2 and that f(n,C"6)>=13|E(Q"n)|. It is an open question to determine f(n,C"l) for l=4k+2, k>=2. Here, we give a general upper bound for f(n,C"l) when l=4k+2 and provide a coloring of E(Q"n) by four colors containing no induced monochromatic C"1"0.
119154847
How many random edges make a dense graph hamiltonian
This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $\Theta(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.
5439870
Incremental Eigenanalysis for Classification
Eigenspace models are a convenient way to represent sets of observations with widespread applications, including classification. In this paper we describe a new constructive method for incrementally adding observations to an eigenspace model. Our contribution is to explicitly account for a change in origin as well as a change in the number of eigenvectors needed in the basis set. No other method we have seen considers change of origin, yet both are needed if an eigenspace model is to be used for classification purposes. We empirically compare our incremental method with two alternatives from the literature and show our method is the more useful for classification because it computes the smaller eigenspace model representing the observations.
37238502
Computing Volume Properties Using Low-Discrepancy Sequences
This paper considers the use of low-discrepancy sequences for computing volume integrals in solid modelling. An introduction to low-discrepancy point sequences is presented which explains how they can be used to replace random points in Monte Carlo methods. The relative advantages of using low-discrepancy methods compared to random point sequences are discussed theoretically, and then practical results are given for a series of test objects which clearly demonstrate the superiority of the low-discrepancy method when used in a simple approach. Finally, the performance of such methods is assessed when used in conjunction with spatial subdivision in the svLis geometric modeller.
10050571
Adding and Subtracting Eigenspaces
This paper provides two algorithms; one for adding eigenspaces, another for subtracting them, thus allowing for incremental updating and downdating of data models. Importantly, and unlike previous work, we keep an accurate track of the mean of the data, which allows our methods to be used in classification applications. The result of adding eigenspaces, each made from a set of data, is an approximation to that which would obtain were the sets of data taken together. Subtracting eigenspaces yields a result approximating that which would obtain were a subset of data used. Using our algorithms it is possible to perform “arithmetic” on eigenspaces without reference to the original data. We illustrate the use of our algorithms in three generic applications, including the dynamic construction of Gaussian mixture models.
1385691
Skewed Mirror Symmetry for Depth Estimation in 3D Line-Drawings
We aim to reconstruct three-dimensional polyhedral solids from axonometric-like line drawings. A new approach is proposed to make use of planes of mirror symmetry detected in such sketches. Taking account of mirror symmetry of such polyhedra can significantly improve the reconstruction process. Applying symmetry as a regularity in optimisation-based reconstruction is shown to be adequate by itself, without the need for other inflation techniques or regularities. Furthermore, symmetry can be used to reduce the size of the reconstruction problem, leading to a reduction in computing time.
1359860
Skewed Mirror Symmetry in the 3D Reconstruction of Polyhedral Models
We aim to reconstruct three-dimensional polyhedra from axonometric line drawings. Existence of mirror symmetry in polyhedra can assist the reconstruction process. We present a new approach for determining planes of mirror symmetry of such polyhedral objects based on prior detection of their planar faces and any axes of symmetry of these faces. The axes are obtained from skewed facial symmetries, for which we also give a new method of determination.