Burkay Genç

A layout algorithm for signaling pathways

Visualization is crucial to the effective analysis of biological pathways. A poorly laid out pathway confuses the user, while a well laid out one improves the users comprehension of the underlying biological phenomenon. We present a new, elegant algorithm for layout of biological signaling pathways. Our algorithm uses a force-directed layout scheme, taking into account directional and rectangular regional constraints enforced by different molecular interaction types and subcellular locations in a cell. The algorithm has been successfully implemented as part of a pathway visualization and analysis toolkit named Patika, and results with respect to computational complexity and quality of the layout have been found satisfactory. The algorithm may be easily adapted to be used in other applications with similar conventions and constraints as well. Patika version 1.0 beta is available upon request at http://www.patika.org.

Reconstructing orthogonal polyhedra from putative vertex sets

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. This is well-studied in 2D; we mostly focus on 3D, and on the case where the given set of points may be rotated beforehand. We obtain fast algorithms for reconstruction in the case where the answer must be orthogonally convex.

Technical Section: A multi-graph approach to complexity management in interactive graph visualization

In this paper we describe a new, multi-graph approach for development of a comprehensive set of complexity management techniques for interactive graph visualization tools. This framework facilitates efficient implementation of management of multiple associated graphs with navigation links and nesting of graphs as well as ghosting, folding and hiding of unwanted graph elements. The theoretical analyses show that the involved data structures and operations on them are quite efficient, and an implementation in a graph drawing tool has proven to be successful.

Covering oriented points in the plane with orthogonal polygons is NP-complete

We address the problem of covering points with orthogonal polygons. Specifically, given a set of n grid-points in the plane each designated in advance with either a horizontal or vertical reading, we investigate the existence of an orthogonal polygon covering these n points in such a way that each edge of the polygon covers exactly one point and each point is covered by exactly one edge with the additional requirement that the reading associated with each point dictates whether the edge covering it is to be horizontal or vertical. We show that this problem is NP-complete.

Covering points with orthogonally convex polygons

In this paper, we address the problem of covering points with orthogonally convex polygons. In particular, given a point set of size n on the plane, we aim at finding if there exists an orthogonally convex polygon such that each edge of the polygon covers exactly one point and each point is covered by exactly one edge. We show that if such a polygon exists, it may not be unique. We propose an O(nlogn) algorithm to construct such a polygon if it exists, or else report the non-existence in the same time bound. We also extend our algorithm to count all such polygons without hindering the overall time complexity. Finally, we show how to construct all k such polygons in O(nlogn+kn) time. All the proposed algorithms are fast and practical.

Cauchy’s Theorem for Orthogonal Polyhedra of Genus 0

A famous theorem by Cauchy states that the dihedral angles of a convex polyhedron are determined by the incidence structure and face-polygons alone. In this paper, we prove the same for orthogonal polyhedra of genus 0 as long as no face has a hole. Our proof yields a linear-time algorithm to find the dihedral angles.

STOKER'S THEOREM FOR ORTHOGONAL POLYHEDRA

Stokers theorem states that in a convex polyhedron, the dihedral angles and edge lengths determine the facial angles if the graph is fixed. In this paper, we study under what conditions Stokers theorem holds for orthogonal polyhedra, obtaining uniqueness and a linear-time algorithm in some cases, and NP-hardness in others.

Covering points with orthogonal polygons

We address the problem of covering points with orthogonal polygons. Specifically, given a set of n points in the plane, we investigate the existence of an orthogonal polygon such that there is a one-to-one correspondence between the points and the edges of the polygon. In an earlier paper, we have shown that constructing such a covering with an orthogonally convex polygon, if any, can be done in O(nlogn) time. In case an orthogonally convex polygon cannot cover the point set, we show in this paper that the problem of deciding whether such a point set can be covered with any orthogonal polygon is NP-complete. The problem remains NP-complete even if the orientations of the edges covering each point are specified in advance as part of the input.