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.

Abstract Let $$n \in \mathbb {N}$$ n ∈ N and $$k \in \mathbb {N}_0$$ k ∈ N 0 . Given a set P of n points in the plane, a pair $$\{p,q\}$$ { 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 $$ \max \{1,\lfloor \frac{n}{k+1} \rfloor \}$$ 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 $$n-\lceil \frac{3k}{2} \rceil $$ n - ⌈ 3 k 2 ⌉ ; this is tight for $$k \le \frac{n}{3}$$ k ≤ n 3 . Moreover, for $$k \le \lfloor \frac{n}{2} \rfloor - 1$$ k ≤ ⌊ n 2 ⌋ - 1 , a k-deep clique cannot have size exceeding $$2\sqrt{n(\lfloor \frac{n}{2} \rfloor -k)}$$ 2 n ( ⌊ n 2 ⌋ - k ) ; this is tight within a constant factor. We also pay special attention to $$(\frac{n}{2}-1)$$ ( 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 $$\lfloor \frac{(m-1)(m+3)}{2}\rfloor $$ ⌊ ( m - 1 ) ( m + 3 ) 2 ⌋ points. If Q is in convex position, the set P must have size at least $$m(m-1)$$ m ( m - 1 ) . This is tight, i.e., there are sets $$Q_m$$ Q m of m points in convex position which can be extended to a set of $$m(m-1)$$ m ( m - 1 )

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.

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 $\Theta(\log k)$, 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 $k$th smallest element has expected fragile complexity $O(\log\log k)$ for the element selected. Deterministically finding the minimum element has fragile complexity $\Theta(\log(\INV))$ and $\Theta(\log(\RUNS))$, 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 $\Theta(\log (\INV))$. Deterministic sorting has fragile complexity $\Theta(\log (\INV))$ but it has fragile complexity $\Theta(\log n)$ regardless of the number of runs.

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, matching 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 which 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 secondly from these trees into paths.

For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.

The beacon model is a recent paradigm for guiding the trajectory of messages or small robotic agents in complex environments. A beacon is a fixed point with an attraction pull that can move points within a given polygon. Points move greedily towards a beacon: if unobstructed, they move along a straight line to the beacon, and otherwise they slide on the edges of the polygon. The Euclidean distance from a moving point to a beacon is monotonically decreasing. A given beacon attracts a point if the point eventually reaches the beacon. The problem of attracting all points within a polygon with a set of beacons can be viewed as a variation of the art gallery problem. Unlike most variations, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. It is connected and can be computed in linear time for simple polygons. By contrast, it is known that the inverse attraction region of a point---the set of beacon positions that attract it---could have $Ω(n)$ disjoint connected components. In this paper, we prove that, in spite of this, the total complexity of the inverse attraction region of a point in a simple polygon is linear, and present a $O(n \log n)$ time algorithm to construct it. This improves upon the best previous algorithm which required $O(n^3)$ time and $O(n^2)$ space. Furthermore we prove a matching $Ω(n\log n)$ lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone.

An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost. In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element $e_i$ does not exceed $\min(\log(W/w_i),\log n)+O(1)$ where $w_i$ is the number of times $e_i$ was accessed and $W$ is the total length of the access sequence. Our techniques can also be used to encode a sequence of $m$ symbols with a dynamic alphabetic code in $O(m)$ time so that the encoding length is bounded by $m(H+O(1))$, where $H$ is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol.

We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points 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 the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a 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 a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order. Lastly, we provide an 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.

In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any $$d>0$$ , the first algorithm maintains a proper $$O(\mathcal {C} dN ^{1/d})$$ -coloring while recoloring at most O(d) vertices per update, where $$\mathcal {C} $$ and $$N $$ are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an $$O(\mathcal {C} d)$$ -coloring with $$O(dN ^{1/d})$$ recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on $$N $$ vertices must recolor at least $$\varOmega (N ^\frac{2}{c(c-1)})$$ vertices per update, for any constant $$c \ge 2$$ .

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.

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a $O(n\log^2{n})$-time algorithm for deciding if a polyomino with $n$ edges can tile the plane isohedrally. This improves on the $O(n^{18})$-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.

It is shown that the online binary search tree data structure GreedyASS performs asymptotically as well on a sufficiently long sequence of searches as any static binary search tree where each search begins from the previous search (rather than the root). This bound is known to be equivalent to assigning each item i in the search tree a positive weight wi and bounding the search cost of an item in the search sequence s1, …, sm This result is the strongest finger-type bound to be proven for binary search trees. By setting the weights to be equal, one observes that our bound implies the dynamic finger bound. Compared to the previous proof of the dynamic finger bound for Splay trees, our result is significantly shorter, stronger, simpler, and has reasonable constants.

For a polyhedron P in ℝd, denote by |P| its combinatorial complexity, i.e., the number of faces of all dimensions of the polyhedra. In this paper, we revisit the classic problem of preprocessing polyhedra independently so that given two preprocessed polyhedra P and Q in ℝd, each translated and rotated, their intersection can be tested rapidly. For d = 3 we show how to perform such a test in O(log |P| + log |Q|) time after linear preprocessing time and space. This running time is the best possible and improves upon the last best known query time of O(log |P| log |Q|) by Dobkin and Kirkpatrick (1990). We then generalize our method to any constant dimension d, achieving the same optimal O(log |P| + log |Q|) query time using a representation of size O(|P| ⌊d/2⌋+ε) for any ε > 0 arbitrarily small. This answers an even older question posed by Dobkin and Kirkpatrick 30 years ago. In addition, we provide an alternative O(log |P| + log |Q|) algorithm to test the intersection of two convex polygons P and Q in the plane.

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.

For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BR-matchings are compatible if their union is also non-crossing. We prove that, for any two distinct BR-matchings M and M', there exists a sequence of BR-matchings M = M1, ..., Mk = M' such that Mi-1 is compatible with Mi. This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in their paper "Compatible matchings for bichromatic plane straight-line graphs".

In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memory-constrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Theta(n), we can modify it so that it runs in O(n^2/2^s) time using a workspace of O(s) variables (for any s \in o(log n)) or O(n log n/log p)$ time using O(p log n/log p) variables (for any 2 <= p <= n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives a trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms.

It is shown how to enhance any data structure in the pointer model to make it confluently persistent, with efficient query and update times and limited space overhead. Updates are performed in O(log n) amortized time, and following a pointer takes O(log c log n) time where c is the in-degree of a node in the data structure. In particular, this proves that confluent persistence can be achieved at a logarithmic cost in the bounded in-degree model used widely in previous work. This is a O(n/ log n)-factor improvement over the previous known transform to make a data structure confluently persistent.

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˜ 1/2 +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.

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are √n lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are √n/3 of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs.

Several existing cache-oblivious dynamic dictionaries achieve O(logB N) (or slightly better memory transfers per operation, where N is the number of items stored, M is the memory size, and B is the block size, which matches the classic B-tree data structure. One recent structure achieves the same query bound and a sometimes-better amortized update bound of memory transfers. This paper presents a new data structure, the xDict, implementing predecessor queries in worst-case memory transfers and insertions and deletions in amortized memory transfers, for any constant ε with 0 < ε < 1. For example, the xDict achieves subconstant amortized update cost when N = M B°(B1−∊), whereas the B-tree's is subconstant only when N = o(MB), and the previously obtained is subconstant only when . The xDict attains the optimal tradeoff between insertions and queries, even in the broader external-memory model, for the range where inserts cost between and O(1/lg3 N) memory transfers.

Object versioning refers to how an application can have access to previous states of its objects. Implementing this mechanism is hard because it needs to be efficient in space and time, and well integrated with the programming language. This paper presents HistOOry, an object versioning system that uses an efficient data structure to store and retrieve past states. It needs only three primitives, and existing code does not need to be modified to be versioned. It provides fine-grained control over what parts of objects are versioned and when. It stores all states, past and present, in memory. Code can be executed in the past of the system and will see the complete system at that point in time. We have implemented our model in Smalltalk and used it for three applications that need versioning: checked postconditions, stateful execution tracing and a planar point location implementation. Benchmarks are provided to asses the practical complexity of our implementation.

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.

A partially persistent data structure is a data structure which preserves previous versions of itself when it is modified. General theoretical schemes are known (e.g. the fat node method) for making any data structure partially persistent. To our knowledge however no general implementation of these theoretical methods exists to date. This paper evaluates different methods to achieve this goal and presents the first working implementation of partial persistence in the object-oriented language Java. Our approach is transparent, i.e., it allows any existing data structures to become persistent without changing its implementation where all previous solutions require an extensive modification of the code by hand. This transparent property is important in view of the large number of algorithmic results that rely on persistence. Our implementation uses aspect-oriented programming, a modularization technique which allows us to instrument the existing code with the needed hooks for the persistence implementation. The implementation is then validated by running benchmarks to analyze both the cost of persistence and of the aspect oriented approach. We also illustrate its applicability by implementing a random binary search tree and making it persistent, and then using the resulting structure to implement a point location data structure in just a few lines.

We introduce a new data structuring paradigm in which operations can be performed on a data structure not only in the present, but also in the past. In this new paradigm, called retroactive data structures , the historical sequence of operations performed on the data structure is not fixed. The data structure allows arbitrary insertion and deletion of operations at arbitrary times, subject only to consistency requirements. We initiate the study of retroactive data structures by formally defining the model and its variants. We prove that, unlike persistence, efficient retroactivity is not always achievable. Thus, we present efficient retroactive data structures for queues, doubly ended queues, priority queues, union-find, and decomposable search structures.

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the familes of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the inward normal from any point on the shape's boundary should intersect the line segment connecting the two incident hinges. In constrast, we show that isosceles triangles with any desired apex angle <90° admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.

Let P be a simple polygon with m vertices, k of which are reflex, and which contains r red points and b blue points in its interior. Let n=m+r+b. A ham-sandwich geodesic is a shortest path in P between any two points on the boundary of P that simultaneously bisects the red points and the blue points. We present an O (n log k)-time algorithm for finding a ham-sandwich geodesic. We also show that this algorithm is optimal in thealgebraic computation tree model when parameterizing the running time with respect to n and k.

info:eu-repo/semantics/published

A new data structure is presented for planar point location that executes a point location query quickly if it is spatially near the previous query. Given a triangulation T of size n and a sequence of point location queries A=q1, qm, the structure presented executes qi in time O(log d(qi-1,qi)). The distance function, d, that is used is a two dimensional generalization of rank distance that counts the number of triangles in a region from qi-1 to qi. The data structure uses O(n log log n) space.

We advance the study of collections of open linkages in 3-space that may be interlocked in the sense that the linkages cannot be separated without one bar crossing through another. We consider chains of bars connected with rigid joints, revolute joints, or universal joints and explore the smallest number of chains and bars needed to achieve interlock. Whereas previous work used topological invariants that applied to single or to closed chains, this work relies on geometric invariants and concentrates on open chains.

Given a non convex simple polygon $P$, is it possible to construct a d ata structure which after preprocessing can answer halfspace area queries (i.e. given a line, determine the area of the portion of the polygon above the line) in $o(n)$ time? We answer negatively, proving a $\Omega(n)$ lower bound on the query time of any data structure performing this task. We then consider the batched version of the same problem: given a polygon $P$ with $n$ vertices, and $k$ query lines, we present an algorithm that computes the area of $P$ on both sides of each line in $O^{*}(n^{3/5}k^{4/5}+n+k)$ time. Variants of our method allow the query of a collection of weighted polygons with or without holes, and solve several other related problems within the same time bounds.