Research on Dürer's problem focuses on edge unfoldings of convex polyhedra that avoid overlap. We invert the goal and find unfoldings that overlap at some point to any given thickness t. We have two main results. The first is that, if we allow unfolding cuts that do not follow polyhedron edges, then there is a convex polyhedron that can unfold with overlap of any given thickness. The second result is that for any given thickness, there is a convex polyhedron with an edge unfolding that overlaps to that thickness.

A reversible transformation (reversibility) is defined as a generalization of the hinged-transformation between a square and an equilateral triangle in the famous Haberdasher's puzzle by Henry E. Dudeney. In the article we trace various achievements relating to the research on reversibibity. The hidden relations between reversibility and other topics such as tile-makers, appropriate superimposition of two tilings, foldability for Conway tiles are unveiled.

Given a periodic placement of copies of a tromino (either L or I), we prove co-RE-completeness (and hence undecidability) of deciding whether it can be completed to a plane tiling. By contrast, the problem becomes decidable if the initial placement is finite, or if the tile is a domino instead of a tromino (in any dimension). As a consequence, tiling a given periodic subset of the plane with a given tromino (L or I) is co-RE-complete. We also prove co-RE-completeness of tiling the entire plane with two polyominoes (one of which is disconnected and the other of which has constant size), and of tiling 3D space with two connected polycubes (one of which has constant size). If we restrict to tiling by translation only (no rotation), then we obtain co-RE-completeness with one more tile: two trominoes for a periodic subset of 2D, three polyominoes for the 2D plane, and three connected polycubes for 3D space. Along the way, we prove several new complexity and algorithmic results about periodic (infinite) graphs. Notably, we prove that Periodic Planar (1-in-)3SAT-3, 3DM, and Graph Orientation are co-RE-complete in 2D and PSPACE-complete in 1D; we extend basic results in graph drawing to 2D periodic graphs; and we give a polynomial-time algorithm for perfect matching in bipartite periodic graphs.

We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding reflections)? This result improves on the best previous construction which requires five polygons.

A reversible transformation (reversibility) is defined as a generalization of the hinged-transformation between a square and an equilateral triangle in the famous Haberdasher’s puzzle by Henry E. Dudeney. In the article we trace various achievements relating to the research on reversibibity. The hidden relations between reversibility and other topics such as tile-makers, appropriate superimposition of two tilings, foldability for Conway tiles are unveiled.

The order type of a point set in Rd maps each (d+1)-tuple of points to its orientation (e.g., clockwise or counterclockwise in R2). Two point sets X and Y have the same order type if there exists a mapping f from X to Y for which every (d+1)-tuple (a1, a2, ..., ad+1) of X and the corresponding tuple (f(a1), f(a2), ..., f(ad+1)) in Y have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an O(nd) algorithm for this task, thereby improving upon the O(n⌊3d/2⌋) algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.

A planar shape S is a k -fold tile if there is an indexed family T of planar shapes congruent to S that is a k -fold tiling: any point in R-2 that is not on the boundary of any shape in T is covered by exactly k shapes in T. Since a 1 -fold tile is clearly a k -fold tile for any positive integer k, the subjects of our research are nontrivial k -fold tiles, that is, plane shapes with property "not a 1 -fold tile, but a k(>= 2) -fold tile." In this paper, we prove some interesting properties about nontrivial k -fold tiles. First, we show that, for any integer k >= 2, there exists a polyomino with property "not an h -fold tile for any positive integer h < k, but a k -fold tile." We also find, for any integer k >= 2, polyominoes with the minimum number of cells among ones that are nontrivial k -fold tiles. Next, we prove that, for any integer k = 5 or k >= 7, there exists a convex unit -lattice polygon that is a nontrivial k -fold tile whose area is k, and for k = 2 and k = 3, no such convex unit -lattice polygon exists.

Let n ∈ℕ and k ∈ℕ_0 . Given a set P of n points in the plane, a pair {p,q} of points in P is called k - deep , if there are at least k points from P strictly on each side of the line spanned by p and q . A k - deep clique is a subset of P with all its pairs k - deep . We show that if P is in general position (i.e., no three points on a line), there is a k -deep clique of size at least max{1,⌊n/k+1⌋} ; this is tight, for example in convex position. A k -deep clique in any set P of n points cannot have size exceeding n-⌈3k/2⌉ ; this is tight for k ≤n/3 . Moreover, for k ≤⌊n/2⌋ - 1 , a k -deep clique cannot have size exceeding 2√(n(⌊n/2⌋ -k)) ; this is tight within a constant factor. We also pay special attention to (n/2-1) -deep cliques (for n even), which are called halving cliques . These have been considered in the literature by Khovanova and Yang, 2012, and they play a role in the latter bound above. Every set P in general position with a halving clique Q of size m must have at least ⌊(m-1)(m+3)/2⌋ points. If Q is in convex position, the set P must have size at least m(m-1) . This is tight, i.e., there are sets Q_m of m points in convex position which can be extended to a set of m(m-1) points where Q_m is a halving clique. Interestingly, this is not the case for all sets Q in convex position (even if parallel connecting lines among point pairs in Q are excluded).

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O (log log n )-competitive search tree data structure in this model, where n is the number of vertices. This matches the best-known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one that we call Steiner-closed search trees, which may be of independent interest. Moreover, our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees and, second, from these trees into paths.

22 We study the possible moves and reachable space by rolling a 3D convex polyhedron on a 2D periodic 23 tessellation in the xy -plane, where at every step a face of the polyhedron must coincide exactly with 24 a tile of the tessellation it rests upon. We topple the polyhedron around one of the edges of the 25 grounded face toward a neighboring face until it hits the xy -plane on a neighboring tile, only if the 26 new face and the new tile also coincide. We observe the space that can be reached by succession of 27 such rolling moves. If the whole plane can be reached, we call the polyhedron a plane roller for 28 the given tessellation. We further classify polyhedra that only reach a limited strip or a bounded 29 area on given tessellations as band rollers and bounded rollers respectively. We present a 30 polynomial-time algorithm to determine the set of tiles reachable from a given starting position, 31 which in particular determines the roller type of the given polyhedron and periodic tessellation. 32 Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on any 33 regular-tiled ( ≤ 4)-uniform tessellation. Finally, we suggest how to employ these findings in puzzle 34 games. 35

We present nonoverlapping general unfoldings of two infinite families of nonconvex polyhedra, or more specifically, zero-volume polyhedra formed by double-covering an n -pointed star polygon whose triangular points have base angle α . Specifically, we construct general unfoldings when n ∈ { 3 , 4 , 5 , 6 , 8 , 9 , 10 , 12 } (no matter the value of α ), and we construct general unfoldings when α < 60 ∘ ( 1 + 1 / n ) (i.e., when the points are shorter than equilateral, no matter the value of n , or slightly larger than equilateral, especially when n is small). Whether all doubly covered star polygons, or more broadly arbitrary nonconvex polyhedra, have general unfoldings remains open.

The fragile complexity of a comparison-based algorithm is f(n) if each input element participates in O(f(n)) comparisons. In this paper, we explore the fragile complexity of algorithms adaptive to various restrictions on the input, i.e., algorithms with a fragile complexity parameterized by a quantity other than the input size n. We show that searching for the predecessor in a sorted array has fragile complexity Θ(log⁡k), where k is the rank of the query element, both in a randomized and a deterministic setting. For predecessor searches, we also show how to optimally reduce the amortized fragile complexity of the elements in the array. We also prove the following results: Selecting the kth smallest element has expected fragile complexity O(log⁡log⁡k) for the element selected. Deterministically finding the minimum element has fragile complexity Θ(log⁡(Inv)) and Θ(log⁡(Runs)), where Inv is the number of inversions in a sequence and Runs is the number of increasing runs in a sequence. Deterministically finding the median has fragile complexity O(log⁡(Runs)+log⁡log⁡n) and Θ(log⁡(Inv)). Deterministic sorting has fragile complexity Θ(log⁡(Inv)) but it has fragile complexity Θ(log⁡n) regardless of the number of runs.

We present an algorithm that enumerates and classifies all edge-to-edge gluings of unit squares that correspond to convex polyhedra. We show that the number of such gluings of n squares is polynomial in n, and the algorithm runs in time polynomial in n (pseudopolynomial if n is considered the only input). Our technique can be applied in several similar settings, including gluings of regular hexagons and triangles.

We prove that two polygons $A$ and $B$ have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between $A$ and $B$) if and only if $A$ and $B$ are two noncrossing nets of a common polyhedron. Furthermore, monotone reversible hinged dissections (where all hinges rotate in the same direction when changing from $A$ to $B$) correspond exactly to noncrossing nets of a common convex polyhedron. By envelope/parcel magic, it becomes easy to design many hinged dissections.

Cookie Clicker is a popular online incremental game where the goal of the game is to generate as many cookies as possible. In the game you start with an initial cookie generation rate, and you can use cookies as currency to purchase various items that increase your cookie generation rate. In this paper, we analyze strategies for playing Cookie Clicker optimally. While simple to state, the game gives rise to interesting analysis involving ideas from NP-hardness, approximation algorithms, and dynamic programming.

We give a complete description of all convex polyhedra whose surface can be constructed from several congruent regular pentagons by folding and gluing them edge to edge. Our method of determining the graph structure of the polyhedra from a gluing is of independent interest and can be used in other similar settings.

Which convex 3D polyhedra can be obtained by gluing several regular hexagons edge-to-edge? It turns out that there are only 15 possible types of shapes, 5 of which are doubly-covered 2D polygons. We give examples for most of them, including all simplicial and all flat shapes, and give a characterization for the latter ones. It is open whether the remaining can be realized.

We study the problem of connecting two points in a simple polygon with a self-approaching path. A self-approaching path is a directed curve such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac|≥|bc|. We analyze properties of self-approaching paths inside simple polygons, and characterize shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points in a simple polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find the shortest self-approaching path under a model of computation which assumes that we can compute involute curves of high order. Lastly, we provide an efficient algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.

A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure supports insertions in $\tilde O (N^{3/4})$ expected amortized time, where $\tilde O$ suppresses polylogarithmic terms. This is the first result to achieve sublinear time insertions; previously it was shown by Allen et al. that $\Theta(\sqrt{N})$ amortized combinatorial changes per insertion could occur in the Voronoi diagram but a sublinear-time algorithm was only presented for the special case of points in convex position.

Every pair of points lying on a polygonal path P in the plane has a detour associated with it, which is the ratio between their distance along the path and their Euclidean distance. Given a set S of points along the path, this information can be encoded in a weighted complete graph on S. Among all spanning trees on this graph, a bottleneck spanning tree is one whose maximum edge weight is minimum. We refer to such a tree as a bottleneck detour tree of S. In other words, a bottleneck detour tree of S is a spanning tree in which the maximum detour (with respect to the original path) between pairs of adjacent points is minimum. We show how to find a bottleneck detour tree in expected O(nlog3⁡n+m) time, where P consists of m edges and |S|=n.