Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees

We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC

Transferring Unit Cell Based Tissue Scaffold Design to Solid Freeform Fabrication

Designing for the freeform fabrication of heterogeneous tissue scaffold is always a challenge in Computer Aided Tissue Engineering. The difficulties stem from two major sources: 1) limitations in current CAD systems to assembly unit cells as building blocks to form complex tissue scaffolds, and 2) the inability to generate tool paths for freeform fabrication of unit cell assemblies. To overcome these difficulties, we have developed an abstract model based on skeletal representation and associated computational methods to assemble unit cells into an optimized structure. Additionally we have developed a process planning technique based on internal architecture pattern of unit cells to generate tool paths for freeform fabrication of tissue scaffold. By modifying our optimization process, we are able to transfer an optimized design to our fabrication system via our process planning technique.Mechanical Engineerin

Delay Minimizing User Association in Cellular Networks via Hierarchically Well-Separated Trees

We study downlink delay minimization within the context of cellular user association policies that map mobile users to base stations. We note the delay minimum user association problem fits within a broader class of network utility maximization and can be posed as a non-convex quadratic program. This non-convexity motivates a split quadratic objective function that captures the original problem's inherent tradeoff: association with a station that provides the highest signal-to-interference-plus-noise ratio (SINR) vs. a station that is least congested. We find the split-term formulation is amenable to linearization by embedding the base stations in a hierarchically well-separated tree (HST), which offers a linear approximation with constant distortion. We provide a numerical comparison of several problem formulations and find that with appropriate optimization parameter selection, the quadratic reformulation produces association policies with sum delays that are close to that of the original network utility maximization. We also comment on the more difficult problem when idle base stations (those without associated users) are deactivated.Comment: 6 pages, 5 figures. Submitted on 2013-10-03 to the 2015 IEEE International Conference on Communications (ICC). Accepted on 2015-01-09 to the 2015 IEEE International Conference on Communications (ICC

On a tiling conjecture of Komlós for 3-chromatic graphs

AbstractGiven two graphs G and H, an H-matching of G (or a tiling of G with H) is a subgraph of G consisting of vertex-disjoint copies of H. For an r-chromatic graph H on h vertices, we write u=u(H) for the smallest possible color-class size in any r-coloring of H. The critical chromatic number of H is the number χcr(H)=(r−1)h/(h−u). A conjecture of Komlós states that for every graph H, there is a constant K such that if G is any n-vertex graph of minimum degree at least (1−(1/χcr(H)))n, then G contains an H-matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all sufficiently large values of n when H is a 3-chromatic graph

Heterogeneous Skeleton for Summarizing Continuously Distributed Demand in a Region

There has long been interest in the skeleton of a spatial object in GIScience. The reasons for this are many, as it has proven to be an extremely useful summary and explanatory representation of complex objects. While much research has focused on issues of computational complexity and efficiency in extracting the skeletal and medial axis representations as well as interpreting the final product, little attention has been paid to fundamental assumptions about the underlying object. This paper discusses the implied assumption of homogeneity associated with methods for deriving a skeleton. Further, it is demonstrated that addressing heterogeneity complicates both the interpretation and identification of a meaningful skeleton. The heterogeneous skeleton is introduced and formalized, along with a method for its identification. Application results are presented to illustrate the heterogeneous skeleton and provides comparative contrast to homogeneity assumptions