Computational geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Computational geometry.

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures-we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.

Programming curvature using origami tessellations

Voting based extreme learning machine

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)). A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull.

We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result.

The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

https://drops.dagstuhl.de/opus/volltexte/2016/5907/pdf/LIPIcs-SoCG-2016-15.pdf/

Incremental Voronoi diagrams

We show that, unlike the Yao-Yao graph $YY_6$, the Theta-Theta graph $\Theta\Theta_6$ defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of $\Theta\Theta_6$ is unbounded.

Spanning Properties of Theta-Theta-6

A data structure is presented for the Mergeable Dictionary abstract data type, which supports the following operations on a collection of disjoint sets of totally ordered data: Predecessor-Search, Split and Merge While Predecessor-Search and Split work in the normal way, the novel operation is Merge While in a typical mergeable dictionary (eg 2-4 Trees), the Merge operation can only be performed on sets that span disjoint intervals in keyspace, the structure here has no such limitation, and permits the merging of arbitrarily interleaved sets Tarjan and Brown present a data structure which can handle arbitrary Merge operations in O(log n) amortized time per operation if the set of operations is restricted to exclude the Split operation In the presence of Split operations, the amortized time complexity of their structure becomes \Omega(n) A data structure which supports both Split and Merge operations in O(log^2 n) amortized time per operation was given by Farach and Thorup In contrast, our data structure supports all operations, including Split and Merge, in O(log n) amortized time, thus showing that interleaved Merge operations can be supported at no additional cost vis-a-vis disjoint Merge operations

Mergeable Dictionaries

We present a general transformation for combining a constant number of binary search tree data structures (BSTs) into a single BST whose running time is within a constant factor of the minimum of any "well-behaved" bound on the running time of the given BSTs, for any online access sequence. 
(A BST has a well behaved bound with $f(n)$ overhead if it spends at most \bigoh{f(n)} time per access and its bound satisfies a weak sense of closure under subsequences.) In particular, we obtain a BST data structure that is \bigoh{\log\log n} competitive, satisfies the working set bound (and thus satisfies the static finger bound and the static optimality bound), satisfies the dynamic finger bound, satisfies the unified bound with an additive \bigoh{\log\log n} factor, and performs each access in worst-case \bigoh{\log n} time.

John Iacono

Papers

Incremental Voronoi diagrams

Spanning Properties of Theta-Theta-6

Mergeable Dictionaries

Combining Binary Search Trees

Encoding nearest larger values