The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30

Libbie & Grove Urban Design Plan

This plan was created for the City of Richmond Department of Planning and Development Review to serve as a recommendation for urban design improvements and suggested changes to zoning ordinances for the Libbie and Grove commercial area located in the Westhampton neighborhood. To begin, an in-depth demographic analysis was conducted for the Westhampton neighborhood. Special attention was paid to socioeconomic factors and trends in census tracts directly surrounding the Libbie and Grove commercial corridor. Based on these analyses and new development occurring in the Libbie and Grove commercial corridor, we were able to allocate six sites or “study areas” as candidates for redevelopment. All of these sites represent valuable areas within the Libbie and Grove commercial corridor. The sites were selected and designed with different intentions, but aim to create a complete streetscape for the commercial area. Based on this analysis and study, it is our recommendation that a new zoning code be implemented for the Libbie and Grove commercial area in order to codify form based design requirements in order to preserve and enhance a village feel at Grove and Libbie and promote compatible future development

Locally Self-Adjusting Skip Graphs

We present a distributed self-adjusting algorithm for skip graphs that minimizes the average routing costs between arbitrary communication pairs by performing topological adaptation to the communication pattern. Our algorithm is fully decentralized, conforms to the $\mathcal{CONGEST}$ model (i.e. uses $O(\log n)$ bit messages), and requires $O(\log n)$ bits of memory for each node, where $n$ is the total number of nodes. Upon each communication request, our algorithm first establishes communication by using the standard skip graph routing, and then locally and partially reconstructs the skip graph topology to perform topological adaptation. We propose a computational model for such algorithms, as well as a yardstick (working set property) to evaluate them. Our working set property can also be used to evaluate self-adjusting algorithms for other graph classes where multiple tree-like subgraphs overlap (e.g. hypercube networks). We derive a lower bound of the amortized routing cost for any algorithm that follows our model and serves an unknown sequence of communication requests. We show that the routing cost of our algorithm is at most a constant factor more than the amortized routing cost of any algorithm conforming to our computational model. We also show that the expected transformation cost for our algorithm is at most a logarithmic factor more than the amortized routing cost of any algorithm conforming to our computational model