Design, analysis and implementation of discrete algorithms for graph and computational geometry problems

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The proposed research is in the field of design, analysis and implementation of algorithms, a subfield within theoretical computer science. We focus on  problems arising in computational geometry and graph theory. The problems include (a) Geometric path problems - how to find a path between two points in a geometric domain (surface, planar region, three dimensional space amidst obstacles) under various constraints (shortest, monotone descending, minimizing the number of links, multiple criteria) (b) Graph separators - computing vertex/edge separators of planar (or planar like) graphs (c) Visibility optimization problems (with applications in radiation therapy) - need to see a polyhedra (tumor), avoiding/minimizing a set a polyhedra (healthy organs), from a minimum set of locations (possible position of laser guns placed outside the body) (d) Developing generic external memory algorithmic techniques for solving graph and geometric problems (e) Developing algorithmic technqiues for solving problems in variety of computational models - external memory model, parallel computing model, priced information model, streaming and models with very limited extra space (f) Approximation algorithms based on discretization - we have proposed discretization based schemes to solve geometric path problems. Most often spatial problems arising in practical settings are solved by heuristics using some form of discretization. We propose to study these problems, in light of our results, and hope to provide approximation algorithms with provable bounds in terms of their accuracy and complexity (g) Analyzing complexity of geometric algorithms using geometric parameters - traditionally the complexity of geometric algorithms is analyzed with respect to input parameters (number of segments, vertices, faces, etc.), but often in practice, the run-time is very sensitive to geometric parameters - angle, distance, fatness. So the design of the algorithm needs to take these parameters into account; this came up in our work on shortest paths. We propose to expand the design and analysis of geometric algorithms to include geometric parameters (h) Finding ways to make geometrical concepts presentable on the web in an interactive manner.