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

A static binary search tree where every search starts from where the previous one ends (lazy finger) is considered. Such a search method is more powerful than that of the classic optimal static trees, where every search starts from the root (root finger), and less powerful than when rotations are allowed—where finding the best rotation based tree is the topic of the dynamic optimality conjecture of Sleator and Tarjan. The runtime of the classic root-finger tree can be expressed in terms of the entropy of the distribution of the searches, but we show that this is not the case for the optimal lazy finger tree. A non-entropy based asymptotically-tight expression for the runtime of the optimal lazy finger trees is derived, and a dynamic programming-based method is presented to compute the optimal tree.

The power and limitations of static binary search trees with lazy finger

A lower bound is presented which shows that a class of heap algorithms in the pointer model with only heap pointers must spend $\Omega \left( \frac{\log \log n}{\log \log \log n} \right)$ amortized time on the Decrease-Key operation (given O(logn) amortized-time Extract-Min). Intuitively, this bound shows the key to having O(1)-time Decrease-Key is the ability to sort O(logn) items in O(logn) time; Fibonacci heaps [M. .L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this through the use of bucket sort. Our lower bound also holds no matter how much data is augmented; this is in contrast to the lower bound of Fredman [J. ACM 46(4):473-501 (1999)] who showed a tradeoff between the number of augmented bits and the amortized cost of Decrease-Key. A new heap data structure, the sort heap, is presented. This heap is a simplification of the heap of Elmasry [SODA 2009: 471-476] and shares with it a O(loglogn) amortized-time Decrease-Key, but with a straightforward implementation such that our lower bound holds. Thus a natural model is presented for a pointer-based heap such that the amortized runtime of a self-adjusting structure and amortized lower asymptotic bounds for Decrease-Key differ by but a O(logloglogn) factor.

Why Some Heaps Support Constant-Amortized-Time Decrease-Key Operations, and Others Do Not

A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.

Entropy, Triangulation, and Point Location in Planar Subdivisions

We consider preprocessing a set S of n points in convex position in the plane into a data structure supporting queries of the following form: given a point q and a directed line 
 $$\ell $$
 in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q among all points to the left of line 
 $$\ell $$
 . We present two data structures for this problem. The first data structure uses $$O(n^{1+\varepsilon })$$
 space and preprocessing time, and answers queries in $$O(2^{1/\varepsilon }\log n)$$
 time, for any $$0< \varepsilon < 1$$
 . The second data structure uses $$O(n \log ^3 n)$$
 space and polynomial preprocessing time, and answers queries in $$O(\log n)$$
 time. These are the first solutions to the problem with $$O(\log n)$$
 query time and $$o(n^2)$$
 space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $$O(\log n)$$
 amortized pointer changes, in addition to $$O(\log n)$$
 -time point-location queries, even though every such update may make $$\Theta (n)$$
 combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) amortized pointer changes per operation while keeping $$O(\log n)$$
 -time point-location queries.

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

The modular subset sum problem consists of deciding, given a modulus $m$, a multiset $S$ of $n$ integers in $0..m$, and a target integer $t$, whether there exists a subset of $S$ with elements summing to $t \pmod{m}$, and to report such a set if it exists. We give a simple $O(m \log m)$-time with high probability (w.h.p.) algorithm for the modular subset sum problem. This builds on and improves on a previous $\tilde{O}(m)$ w.h.p. algorithm from Axiotis, Backurs, Jin, Tzamos, and Wu (SODA 19). Our method utilizes the ADT of the dynamic strings structure of Gawrychowski et. al (SODA 18). However, as this structure is rather complicated we present a much simpler alternative which we call the Data Dependent Tree. As an application, we consider the computational version of a fundamental theorem in zero-sum Ramsey theory. The Erdős-Ginzburg-Ziv Theorem states that a multiset of $2n - 1$ integers always contains a subset of cardinality exactly $n$ whose values sum to a multiple of $n$. We give an algorithm for finding such a subset in time $O(n \log n)$ w.h.p. which improves on an $O(n^2)$ algorithm due to Del Lungo, Marini, and Mori (Disc. Math. 09).

John Iacono

Papers

The power and limitations of static binary search trees with lazy finger

Why Some Heaps Support Constant-Amortized-Time Decrease-Key Operations, and Others Do Not

Entropy, Triangulation, and Point Location in Planar Subdivisions

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

Modular Subset Sum, Dynamic Strings, and Zero-Sum Sets