The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$

On Universal Point Sets for Planar Graphs

A set P of points in R^2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n>=15. Conversely, we use a computer program to show that there exist universal point sets for all n<=10 and to enumerate all corresponding order types. Finally, we describe a collection G of 7'393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in G.Comment: Fixed incorrect numbers of universal point sets in the last par

The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

Abstract. Given a set P of n labeled points in the plane, the radial system of P describes, for each p ∈ P , the radial ordering of the other points around p. This notion is related to the order type of P , which describes the orientation (clockwise or counterclockwise) of every ordered triple of P . Given only the order type of P , it is easy to reconstruct the radial system of P , but the converse is not true. Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) defined T (R) to be the set of order types with radial system R and showed that sometimes |T (R)| = n − 1. They give polynomial-time algorithms to compute T (R) when only given R. We describe an optimal O(n 2 ) time algorithm for computing T (R). The algorithm constructs the convex hulls of all possible point sets with the given radial system, after which sidedness queries on point triples can be answered in constant time. This set of convex hulls can be found in O(n) time. Our results generalize to abstract order types

Analysing the Control Software of the Compact Muon Solenoid Experiment at the Large Hadron Collider

The control software of the CERN Compact Muon Solenoid experiment contains over 30,000 finite state machines. These state machines are organised hierarchically: commands are sent down the hierarchy and state changes are sent upwards. The sheer size of the system makes it virtually impossible to fully understand the details of its behaviour at the macro level. This is fuelled by unclarities that already exist at the micro level. We have solved the latter problem by formally describing the finite state machines in the mCRL2 process algebra. The translation has been implemented using the ASF+SDF meta-environment, and its correctness was assessed by means of simulations and visualisations of individual finite state machines and through formal verification of subsystems of the control software. Based on the formalised semantics of the finite state machines, we have developed dedicated tooling for checking properties that can be verified on finite state machines in isolation.Comment: To appear in FSEN'11. Extended version with details of the ASF+SDF translation of SML into mCRL

Value of MRI and diffusion-weighted MRI for the diagnosis of locally recurrent rectal cancer

OBJECTIVES: To evaluate the accuracy of standard MRI, diffusion-weighted MRI (DWI) and fusion images for the diagnosis of locally recurrent rectal cancer in patients with a clinical suspicion of recurrence. METHODS: Forty-two patients with a clinical suspicion of recurrence underwent 1.5-T MRI consisting of standard T2-weighted FSE (3 planes) and an axial DWI (b0,500,1000). Two readers (R1,R2) independently scored the likelihood of recurrence; [1] on standard MRI, [2] on standard MRI+DWI, and [3] on T2-weighted+DWI fusion images. RESULTS: 19/42 patients had a local recurrence. R1 achieved an area under the ROC-curve (AUC) of 0.99, sensitivity 100% and specificity 83% on standard MRI versus 0.98, 100% and 91% after addition of DWI (p = 0.78). For R2 these figures were 0.87, 84% and 74% on standard MRI and 0.91, 89% and 83% with DWI (p = 0.09). Fusion images did not significantly improve the performance. Interobserver agreement was kappa0.69 for standard MRI, kappa0.82 for standard MRI+DWI and kappa0.84 for the fusion images. CONCLUSIONS: MRI is accurate for the diagnosis of locally recurrent rectal cancer in patients with a clinical suspicion of recurrence. Addition of DWI does not significantly improve its performance. However, with DWI specificity and interobserver agreement increase. Fusion images do not improve accuracy

An analysis of the control hierarchy modeling of the CMS detector control system

The supervisory level of the Detector Control System (DCS) of the CMS experiment is implemented using Finite State Machines (FSM), which model the behaviours and control the operations of all the sub-detectors and support services. The FSM tree of the whole CMS experiment consists of more than 30.000 nodes. An analysis of a system of such size is a complex task but is a crucial step towards the improvement of the overall performance of the FSM system. This paper presents the analysis of the CMS FSM system using the micro Common Representation Language 2 (mcrl2) methodology. Individual mCRL2 models are obtained for the FSM systems of the CMS sub-detectors using the ASF+SDF automated translation tool. Different mCRL2 operations are applied to the mCRL2 models. A mCRL2 simulation tool is used to closer examine the system. Visualization of a system based on the exploration of its state space is enabled with a mCRL2 tool. Requirements such as command and state propagation are expressed using modal mu-calculus and checked using a model checking algorithm. For checking local requirements such as endless loop freedom, the Bounded Model Checking technique is applied. This paper discusses these analysis techniques and presents the results of their application on the CMS FSM system