Mashhood Ishaque

Augmenting the Edge Connectivity of Planar Straight Line Graphs to Three

We characterize the planar straight line graphs (Pslgs) that can be augmented to 3-connected and 3-edge-connected Pslgs, respectively. We show that if a Pslg with n vertices can be augmented to a 3-edge-connected Pslg, then at most 2n−2 new edges are always sufficient and sometimes necessary for the augmentation. If the input Pslg is, in addition, already 2-edge-connected, then n−2 new edges are always sufficient and sometimes necessary for the augmentation to a 3-edge-connected Pslg.

Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues

We introduce staged self-assembly of Wang tiles, where tiles can be added dynamically in sequence and where intermediate constructions can be stored for later mixing. This model and its various constraints and performance measures are motivated by a practical nanofabrication scenario through protein-based bioengineering. Staging allows us to break through the traditional lower bounds in tile self-assembly by encoding the shape in the staging algorithm instead of the tiles. All of our results are based on the practical assumption that only a constant number of glues, and thus only a constant number of tiles, can be engineered, as each new glue type requires significant biochemical research and experiments. Under this assumption, traditional tile self-assembly cannot even manufacture an n×n square; in contrast, we show how staged assembly enables manufacture of arbitrary orthogonal shapes in a variety of precise formulations of the model.

Disjoint compatible geometric matchings

We prove that for every even set of

One-dimensional staged self-assembly

n

Tri-Edge-Connectivity Augmentation for Planar Straight Line Graphs

pairwise disjoint line segments in the plane in general position, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzer et al. [ABD08]. The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.

Constrained tri-connected planar straight line graphs ⁄

We introduce the problem of staged self-assembly of one-dimensional nanostructures, which becomes interesting when the elements are labeled (e.g., representing functional units that must be placed at specific locations). In a restricted model in which each operation has a single terminal assembly, we prove that assembling a given string of labels with the fewest stages is equivalent, up to constant factors, to compressing the string to be uniquely derived from the smallest possible context-free grammar (a well-studied O(log n)-approximable problem). Without this restriction, we show that the optimal assembly can be substantially smaller than the optimal context-free grammar, by a factor of Ω(√n/ log n) even for binary strings of length n. Fortunately, we can bound this separation in model power by a quadratic function in the number of distinct glues or tiles allowed in the assembly, which is typically small in practice.

Relative Convex Hulls in Semi-Dynamic Arrangements

It is shown that if a planar straight line graph (Pslg) with n vertices in general position in the plane can be augmented to a 3-edge-connected Pslg, then 2n ? 2 new edges are enough for the augmentation. This bound is tight: there are Pslgs with n ? 4 vertices such that any augmentation to a 3-edge-connected Pslg requires 2n ? 2 new edges.

Shooting permanent rays among disjoint polygons in the plane

It is known that for any set V of n ≥ 4 points in the plane, not in convex position, there is a 3-connected planar straight line graph G = (V, E) with at most 2n − 2 edges, and this bound is the best possible. We show that the upper bound | E | ≤ 2n continues to hold if G = (V, E) is constrained to contain a given graph G 0 = (V, E 0), which is either a 1-factor (i.e., disjoint line segments) or a 2-factor (i.e., a collection of simple polygons), but no edge in E 0 is a proper diagonal of the convex hull of V. Since there are 1- and 2-factors with n vertices for which any 3-connected augmentation has at least 2n − 2 edges, our bound is nearly tight in these cases. We also examine possible conditions under which this bound may be improved, such as when G 0 is a collection of interior-disjoint convex polygons in a triangular container.

Conflict-Free graph orientations with parity constraints

We present a data structure for maintaining the geodesic hull of a set of points (sites) in the presence of pairwise noncrossing line segments (barriers) that subdivide a bounding box into simply connected faces. For m barriers and n sites, our data structure has O((m+n)logn) size. It supports a mixed sequence of O(m) barrier insertions and O(n) site deletions in

Bounded-degree polyhedronization of point sets

O((m+n) \operatorname{polylog}(mn))