Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths

Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from the microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified. Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n 1+Θ(1/k). While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k > 1. Specifically, up to lower order polylogarithmic factors, the upper bound on the cover time is a polynomial in n of exponent 1+ 1/(2k−1). For k = 2, 3, 4 and 5 the exponent is thus 4/3 , 6/5 , 8/7 , and 10/9 , respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k

Weighted simplicial complex reconstruction from mobile laser scanning using sensor topology

We propose a new method for the reconstruction of simplicial complexes (combining points, edges and triangles) from 3D point clouds from Mobile Laser Scanning (MLS). Our method uses the inherent topology of the MLS sensor to define a spatial adjacency relationship between points. We then investigate each possible connexion between adjacent points, weighted according to its distance to the sensor, and filter them by searching collinear structures in the scene, or structures perpendicular to the laser beams. Next, we create and filter triangles for each triplet of self-connected edges and according to their local planarity. We compare our results to an unweighted simplicial complex reconstruction.Comment: 8 pages, 11 figures, CFPT 2018. arXiv admin note: substantial text overlap with arXiv:1802.0748

Alien Registration- Guinard, Frances M. (Portland, Cumberland County)

https://digitalmaine.com/alien_docs/24267/thumbnail.jp

Alien Registration- Guinard, Frances M. (Portland, Cumberland County)

https://digitalmaine.com/alien_docs/24267/thumbnail.jp

Alien Registration- Guinard, Muriel E. (Brunswick, Cumberland County)

https://digitalmaine.com/alien_docs/31493/thumbnail.jp