Showing papers by "Michael T. Goodrich published in 2018"
27 Jun 2018
TL;DR: In this article, the authors studied C-planarity for flat clustered graphs with a fixed combinatorial embedding whose clusters partition the vertex set and gave a subexponential-time algorithm for these graphs when their face size is bounded.
Abstract: The C-Planarity problem asks for a drawing of a clustered graph, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for embedded flat clustered graphs, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of embedded tree decomposition in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.
10 citations
TL;DR: In this article, the authors generalize the notion of Lombardi drawings to smooth (smooth) $k$-Lombardi drawings, in which each edge may be drawn as a (differentiable) sequence of circular arcs.
Abstract: In Lombardi drawings of graphs, edges are represented as circular arcs and the edges incident on vertices have perfect angular resolution. It is known that not every planar graph has a planar Lombardi drawing. We give an example of a planar 3-tree that has no planar Lombardi drawing and we show that all outerpaths do have a planar Lombardi drawing. Further, we show that there are graphs that do not even have any Lombardi drawing at all. With this in mind, we generalize the notion of Lombardi drawings to that of (smooth) $k$-Lombardi drawings, in which each edge may be drawn as a (differentiable) sequence of $k$ circular arcs; we show that every graph has a smooth $2$-Lombardi drawing and every planar graph has a smooth planar $3$-Lombardi drawing. We further investigate related topics connecting planarity and Lombardi drawings.
7 citations
29 May 2018
TL;DR: This work studies oblivious random access machine (ORAM) simulation, in cloud computing environments where a thin client outsources her data to a server using O(1)-sized messages.
Abstract: We study oblivious random access machine (ORAM) simulation, in cloud computing environments where a thin client outsources her data to a server using O(1)-sized messages.
6 citations
Posted Content•
TL;DR: In this paper, a simple repeated insertion-sort algorithm can maintain an O(n) Kendall tau distance, with high probability, between a maintained list and an underlying total order of n items in an evolving data model where each comparison is followed by a swap between a random consecutive pair of items in the underlying order.
Abstract: We give optimal sorting algorithms in the evolving data framework, where an algorithm's input data is changing while the algorithm is executing. In this framework, instead of producing a final output, an algorithm attempts to maintain an output close to the correct output for the current state of the data, repeatedly updating its best estimate of a correct output over time. We show that a simple repeated insertion-sort algorithm can maintain an O(n) Kendall tau distance, with high probability, between a maintained list and an underlying total order of n items in an evolving data model where each comparison is followed by a swap between a random consecutive pair of items in the underlying total order. This result is asymptotically optpimal, since there is an Omega(n) lower bound for Kendall tau distance for this problem. Our result closes the gap between this lower bound and the previous best algorithm for this problem, which maintains a Kendall tau distance of O(n log log n) with high probability. It also confirms previous experimental results that suggested that insertion sort tends to perform better than quicksort in practice.
4 citations
Posted Content•
TL;DR: This work presents novel reactive proximity data structures for graphs of polynomial expansion, i.e., the class of graphs with small separators, such as planar graphs and road networks, and experimentally compares their data structure to Dijkstra's algorithm.
Abstract: We consider data structures for graphs where we maintain a subset of the nodes called sites, and allow proximity queries, such as asking for the closest site to a query node, and update operations that enable or disable nodes as sites. We refer to a data structure that can efficiently react to such updates as reactive. We present novel reactive proximity data structures for graphs of polynomial expansion, i.e., the class of graphs with small separators, such as planar graphs and road networks. Our data structures can be used directly in several logistical problems and geographic information systems dealing with real-time data, such as emergency dispatching. We experimentally compare our data structure to Dijkstra's algorithm in a system emulating random queries in a real road network.
4 citations
Proceedings Article•
01 Jan 2018TL;DR: In this model, a sorting algorithm maintains an approximation to the sorted order of a list of data items while simultaneously, with each comparison made by the algorithm, an adversary randomly swaps the order of adjacent items in the true sorted order as mentioned in this paper.
Abstract: We empirically study sorting in the evolving data model. In this model, a sorting algorithm maintains an approximation to the sorted order of a list of data items while simultaneously, with each comparison made by the algorithm, an adversary randomly swaps the order of adjacent items in the true sorted order. Previous work studies only two versions of quicksort, and has a gap between the lower bound of Omega(n) and the best upper bound of O(n log log n). The experiments we perform in this paper provide empirical evidence that some quadratic-time algorithms such as insertion sort and bubble sort are asymptotically optimal for any constant rate of random swaps. In fact, these algorithms perform as well as or better than algorithms such as quicksort that are more efficient in the traditional algorithm analysis model.
4 citations
Proceedings Article•
01 Aug 2018TL;DR: In this article, it was shown that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1/δ) and when α = 1/2, trees can be drawn with logarithmic ply number with an area that is polynomial for bounded-degree trees.
Abstract: Ply number is a recently developed graph drawing metric inspired by studying road networks. Informally, for each vertex v, which is associated with a point in the plane, a disk is drawn centered on v with a radius that is alpha times the length of the longest edge incident to v, for some constant alpha in (0, 0.5]. The ply number is the maximum number of disks that overlap at a single point. We show that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1 / Delta). We also show that when alpha = 1/2, trees can be drawn with logarithmic ply number, with an area that is polynomial for bounded-degree trees. Lastly, we show that this logarithmic upper bound does not apply to 2-trees, by giving a lower bound of Omega(sqrt(n / log n)) ply for any value of alpha.
1 citations
Posted Content•
TL;DR: In this article, a geometric approximation algorithm for matching a pattern point set against a background point set, where the points have angular orientations in addition to their positions, is presented.
Abstract: Motivated by the problem of fingerprint matching, we present geometric approximation algorithms for matching a pattern point set against a background point set, where the points have angular orientations in addition to their positions.
1 citations
02 Jul 2018
TL;DR: This work provides efficient algorithms for computing compact representations of Voronoi diagrams using a convex-straight-skeleton distance function when sites are line segments or convex polygons.
Abstract: We provide efficient algorithms for computing compact representations of Voronoi diagrams using a convex-straight-skeleton (i.e., convex polygon offset) distance function when sites are line segments or convex polygons.
1 citations
Posted Content•
TL;DR: This work studies C-Planarity for graphs with a fixed combinatorial embedding whose clusters partition the vertex set, and considers a variation of the notion of $\textit{embedded tree decomposition}$ in which, for each face, including the outer face, there is a bag that contains every vertex of the face.
Abstract: The C-Planarity problem asks for a drawing of a $\textit{clustered graph}$, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for $\textit{embedded flat clustered graphs}$, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of $\textit{embedded tree decomposition}$ in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.
1 citations
Posted Content•
TL;DR: It is shown that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1 / Delta), and that whenalpha = 1/2, trees can be drawn with logarithmic ply number, with an area that is polynomial for bounded-degree trees.
Abstract: Ply number is a recently developed graph drawing metric inspired by studying road networks. Informally, for each vertex v, which is associated with a point in the plane, a disk is drawn centered on v with a radius that is alpha times the length of the longest edge incident to v, for some constant alpha in (0, 0.5]. The ply number is the maximum number of disks that overlap at a single point. We show that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1 / Delta). We also show that when alpha = 1/2, trees can be drawn with logarithmic ply number, with an area that is polynomial for bounded-degree trees. Lastly, we show that this logarithmic upper bound does not apply to 2-trees, by giving a lower bound of Omega(sqrt(n / log n)) ply for any value of alpha.
Proceedings Article•
01 Jan 2018TL;DR: This work presents geometric approximation algorithms for matching a pattern point set against a background point set, where the points have angular orientations in addition to their positions.
Abstract: Motivated by the problem of fingerprint matching, we present geometric approximation algorithms for matching a pattern point set against a background point set, where the points have angular orientations in addition to their positions.
Posted Content•
TL;DR: The experiments performed in this paper provide empirical evidence that some quadratic-time algorithms such as insertion sort and bubble sort are asymptotically optimal for any constant rate of random swaps.
Abstract: We empirically study sorting in the evolving data model. In this model, a sorting algorithm maintains an approximation to the sorted order of a list of data items while simultaneously, with each comparison made by the algorithm, an adversary randomly swaps the order of adjacent items in the true sorted order. Previous work studies only two versions of quicksort, and has a gap between the lower bound of Omega(n) and the best upper bound of O(n log log n). The experiments we perform in this paper provide empirical evidence that some quadratic-time algorithms such as insertion sort and bubble sort are asymptotically optimal for any constant rate of random swaps. In fact, these algorithms perform as well as or better than algorithms such as quicksort that are more efficient in the traditional algorithm analysis model.
16 Apr 2018
TL;DR: In this paper, the authors propose reactive proximity data structures for graphs of polynomial expansion, i.e., the class of graphs with small separators, such as planar graphs and road networks.
Abstract: We consider data structures for graphs where we maintain a subset of the nodes called sites, and allow proximity queries, such as asking for the closest site to a query node, and update operations that enable or disable nodes as sites. We refer to a data structure that can efficiently react to such updates as reactive. We present novel reactive proximity data structures for graphs of polynomial expansion, i.e., the class of graphs with small separators, such as planar graphs and road networks. Our data structures can be used directly in several logistical problems and geographic information systems dealing with real-time data, such as emergency dispatching. We experimentally compare our data structure to Dijkstra’s algorithm in a system emulating random queries in a real road network.
TL;DR: In this article, the authors studied the complexity of stable-matching Voronoi diagrams, where each site has a quota or "appetite" indicating the area of the set of points that can be matched to it.
Abstract: We study algorithms and combinatorial complexity bounds for \emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $\mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or "appetite" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work on the stable-matching Voronoi diagram provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. In this paper, we show that a stable-matching Voronoi diagram of $n$ point sites has $O(n^{2+\varepsilon})$ faces and edges, for any $\varepsilon>0$, and show that this bound is almost tight by giving a family of diagrams with $\Theta(n^2)$ faces and edges. We also provide a discrete algorithm for constructing it in $O(n^3\log n+n^2f(n))$ time in the real-RAM model of computation, where $f(n)$ is the runtime of a geometric primitive (which we define) that can be approximated numerically, but cannot, in general, be performed exactly in an algebraic model of computation. We show, however, how to compute the geometric primitive exactly for polygonal convex distance functions.