On-line list colouring of random graphs

In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number

On-line List Colouring of Random Graphs

<p>In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n−1) tends to infinity faster than (loglogn)1/3(logn)2n2/3. For sparser graphs, we are slightly less successful; we show that if d≥(logn)2+ϵ for some ϵ>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C∈[2,4], depending on the range of d. Also, for d=O(1), the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.</p

Subspace Procrustes Analysis

<p>Procrustes Analysis (PA) has been a popular technique to align and build 2-D statistical models of shapes. Given a set of 2-D shapes PA is applied to remove rigid transformations. Then, a non-rigid 2-D model is computed by modeling (e.g., PCA) the residual. Although PA has been widely used, it has several limitations for modeling 2-D shapes: occluded landmarks and missing data can result in local minima solutions, and there is no guarantee that the 2-D shapes provide a uniform sampling of the 3-D space of rotations for the object. To address previous issues, this paper proposes Subspace PA (SPA). Given several instances of a 3-D object, SPA computes the mean and a 2-D subspace that can simultaneously model all rigid and non-rigid deformations of the 3-D object. We propose a discrete (DSPA) and continuous (CSPA) formulation for SPA, assuming that 3-D samples of an object are provided. DSPA extends the traditional PA, and produces unbiased 2-D models by uniformly sampling different views of the 3-D object. CSPA provides a continuous approach to uniformly sample the space of 3-D rotations, being more efficient in space and time. Experiments using SPA to learn 2-D models of bodies from motion capture data illustrate the benefits of our approach.</p