Joondong Kim

The Art Gallery Theorem for Polyominoes

We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. In particular, we show that

Sensitivity of Capacity Estimation Results subject to Convective Weather Forecast Errors

\lfloor\frac{m+1}{3} \rfloor

Optimizing restriction site placement for synthetic genomes

point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When

Routing a maximum number of disks through a scene of moving obstacles

m \leq\frac{3n}{4} - 4

Flexible, Performance-based Route Planning for Super-Dense Operations

, the sufficiency condition yields a strictly lower guard number than

Capacity Estimation for Super-Dense Operations

\lfloor\frac{n}{4}\rfloor

Scheduling Aircraft to Reduce Controller Workload

, given by the art gallery theorem for orthogonal polygons.

Routing multi-class traffic flows in the plane

In this paper, we investigate the sensitivity of capacity estimation results subject to convective weather forecast errors. Our goal is to examine how throughput estimates vary with different types of weather forecast errors, including time shifting, coverage/intensity, and position shifting. We investigate capacity estimates in both en route airspace and transition airspace for Next Generation Air Transportation System (NextGen) routing scenarios. We consider capacity estimation for a variety of weather forecast look-ahead times, ranging from 5 minutes to 2 hours.

Flexible Airlane Generation to Maximize Flow Under Hard and Soft Constraints

Restriction enzymes are the workhorses of molecular biology. We introduce a new problem which arises in the course of our project to design virus variants to serve as potential vaccines: we wish to modify virus-length genomes to introduce large numbers of unique restriction enzyme recognition sites while preserving wild-type function by substitution of synonymous codons. We show that the resulting problem is NP-Complete, give an exponential-time algorithm, as well as well-performing heuristics, and give excellent results for five sample viral genomes. Our resulting modified genomes have several times more unique restriction sites and reduce the maximum gap between adjacent sites by three to nine-fold.

Guarding polyominoes

This video illustrates an algorithm for computing a maximum number of disjoint paths for unit disks moving among a set of dynamic obstacles in the plane. The problem is motivated by applications in air traffic management: aircraft must be routed while avoiding no-fly zones and weather constraints and while maintaining at least a specified horizontal separation distance between themselves. Given a polygonal domain with moving obstacles, our goal is to determine the maximum number of unit disks (aircraft with safety zones) that can be routed safely through the domain, entering/exiting through specified edges of the domain. The video is meant to accompany the paper [1], which gives details of the algorithm and its analysis.