Top-Down Skiplists

We describe todolists (top-down skiplists), a variant of skiplists (Pugh 1990) that can execute searches using at most $\log_{2-\varepsilon} n + O(1)$ binary comparisons per search and that have amortized update time $O(\varepsilon^{-1}\log n)$. A variant of todolists, called working-todolists, can execute a search for any element $x$ using $\log_{2-\varepsilon} w(x) + o(\log w(x))$ binary comparisons and have amortized search time $O(\varepsilon^{-1}\log w(w))$. Here, $w(x)$ is the "working-set number" of $x$. No previous data structure is known to achieve a bound better than $4\log_2 w(x)$ comparisons. We show through experiments that, if implemented carefully, todolists are comparable to other common dictionary implementations in terms of insertion times and outperform them in terms of search times.Comment: 18 pages, 5 figure

Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees

In the limited-workspace model, we assume that the input of size $n$ lies in a random access read-only memory. The output has to be reported sequentially, and it cannot be accessed or modified. In addition, there is a read-write workspace of $O(s)$ words, where $s \in \{1, \dots, n\}$ is a given parameter. In a time-space trade-off, we are interested in how the running time of an algorithm improves as $s$ varies from $1$ to $n$. We present a time-space trade-off for computing the Euclidean minimum spanning tree (EMST) of a set $V$ of $n$ sites in the plane. We present an algorithm that computes EMST$(V)$ using $O(n^3\log s /s^2)$ time and $O(s)$ words of workspace. Our algorithm uses the fact that EMST$(V)$ is a subgraph of the bounded-degree relative neighborhood graph of $V$, and applies Kruskal's MST algorithm on it. To achieve this with limited workspace, we introduce a compact representation of planar graphs, called an $s$-net which allows us to manipulate its component structure during the execution of the algorithm

Optimal Algorithm for Geodesic Farthest-Point Voronoi Diagrams

Let P be a simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. Given a set S of m sites being a subset of the vertices of P, we present the first randomized algorithm to compute the geodesic farthest-point Voronoi diagram of S in P running in expected O(n + m) time. That is, a partition of P into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic distance. This algorithm can be extended to run in expected O(n + m log m) time when S is an arbitrary set of m sites contained in P

Linear transformation distance for bichromatic matchings

Let $P=B\cup R$ be a set of $2n$ points in general position, where $B$ is a set of $n$ blue points and $R$ a set of $n$ red points. A \emph{$BR$-matching} is a plane geometric perfect matching on $P$ such that each edge has one red endpoint and one blue endpoint. Two $BR$-matchings are compatible if their union is also plane. The \emph{transformation graph of $BR$-matchings} contains one node for each $BR$-matching and an edge joining two such nodes if and only if the corresponding two $BR$-matchings are compatible. In SoCG 2013 it has been shown by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is always connected, but its diameter remained an open question. In this paper we provide an alternative proof for the connectivity of the transformation graph and prove an upper bound of $2n$ for its diameter, which is asymptotically tight

Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant

Space-Time Trade-offs for Stack-Based Algorithms

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 \alg\ that runs in O(n) time using $\Theta(n)$ variables, we can modify it so that it runs in $O(n^2/s)$ 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 $2\leq p\leq n$). 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 the first 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