Research

Curve Similarity under the Fréchet Distance

The Fréchet distance is considered an appropriate metric to capture the similarity between the curves. It is, intuitively, the minimum length of the leash connecting a man and dog walking along two curves each without going backward.  It roughly takes quadratic time to compute the Fréchet distance between two piecewise linear curves.
I have a nice proof in my IJCGA paper demonstrating that the runtime can be significantly improved when the two curves have relatively long edges. This induces nice structural properties in the product space of point-to-point distance between curves, allowing a greedy procedure to work prevailing the classic dynamic program. Curve similarity has applications in molecular biology, shape analysis, GIS, image processing, and others. 

Proximity Search Data Structures for Curves 

Similarity search is harnessed in pattern recognition, computer vision, anomaly detection, DNA sequencing, databases, GIS, etc. In the IJCGA paper, I looked into the Distance Oracle Queries in which a single curve is given and the aim is to preprocess the curve into a data structure so that it for any query curve it quickly decides whether the Fréchet distance between between the two curves is small or not.

In the Nearest-Neighbor variants, the task is to preprocess a "bunch" of curves into a data structure to quickly find the closest curve in the database for any query curve. In my paper I show how to build a data structure that handles the nearest-neighbor queries under (continuous) Frechet distance among curves in a linear query time within the approximation factor of (1+ε). 

Equitable Facility Location and Clustering

It introduces a new paradigm in clustering and facility location: balancing efficiency with equity. Traditional models minimize distance but often overlook who is served at each facility, leading to demographic imbalances. I define the Fair-Coverage Facility Location Problem (FC-FLP), where each facility’s coverage must be color-spanning, i.e., locally fair by including all demographic groups, while still ensuring global coverage and minimizing the worst-case travel distance. (i) Theoretical contribution: We design a novel monotone hill–valley search algorithm that computes the optimal fair radius vector in near-linear time. Unlike prior fairness models that allow approximation or radius inflation, my method achieves exact distance optimality without sacrificing local diversity. Using real EMS incident data from Manhattan, my method reduced the maximum emergency response distance to 1.2 km, significantly better than both Jung et al.’s fairness-greedy baseline (1.55 km) and Lloyd’s k-means (1.60 km), while guaranteeing that every hospital catchment is demographically balanced. It also ran 5× faster than k-means. This research demonstrates that fairness and efficiency need not be a trade-off. It provides a scalable, equitable solution for critical applications such as emergency services, healthcare access, and public infrastructure planning.

Progressive Simplification of Maps in ArcGIS:  I designed an ArcPy toolbox to simplify a network representing the river of the Austin, TX area. The interesting feature of this toolbox is that given a 'zoom threshold', it gives you a simplification based on the resolution you desire faster than the existing methods. I used a shapefile and applied a heuristic algorithm which was a combination of the Douglas-Peucker (POINT-REMOVE tool) and Wang-Muller algorithms (BEND-SIMPLIFY tool). My invented tool produced a nice map in a faster execution time. Besides, the GIS course itself taught us how to use different features of the network analysis tool to solve the problems of interest on our end.  Here's the GitHub link to the toolbox I created using ArcPy: https://github.com/arminia/Simplification