On Reversible Cascades in Scale-Free and Erdős-Rényi Random Graphs

Consider the following cascading process on a simple undirected graph G(V,E) with diameter Δ. In round zero, a set S?V of vertices, called the seeds, are active. In round i+1, i∈?, a non-isolated vertex is activated if at least a ρ∈(0,1] fraction of its neighbors are active in round i; it is deactivated otherwise. For k∈?, let min-seed^(k)(G,ρ) be the minimum number of seeds needed to activate all vertices in or before round k. This paper derives upper bounds on min-seed^(k)(G,ρ). In particular, if G is connected and there exist constants C>0 and γ>2 such that the fraction of degree-k vertices in G is at most C/k ^γ for all k∈?⁺, then min-seed^(Δ)(G,ρ)=O(?ρ ^γ?1|V|?). Furthermore, for n∈?⁺, p=Ω((ln(e/ρ))/(ρn)) and with probability 1?exp(?n ^Ω(1)) over the Erd?s-Rényi random graphs G(n,p), min-seed⁽¹⁾(G(n,p),ρ)=O(ρn).     

Constrained delaunay triangulations

Given a set ofn vertices in the plane together with a set of noncrossing, straight-line edges, theconstrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimalO(n logn) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (non-Delaunay) triagulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.     

Computing correct Delaunay triangulations

In recent years the practical computation of Delaunay triangulations, resp. Voronoi diagrams has received a lot of attention in the literature. While the Delaunay triangulation is an important basic tool in geometric optimization algorithms, it is nontrivial to achieve a numerically stable computer implementation. In this technical note we assume that all generating points are grid points of a regularM byM lattice in the plane. Depending onM we derive the necessary word length a binary computer must have for integer representation in order to obtain exact Delaunay triangulations. This analysis is carried out for theL ₁-,L ₂- andL _∞-metric.     

AnO(n logn) plane-sweep algorithm forL 1 andL ∞ Delaunay triangulations

TheDelaunay diagram on a set of points in the plane, calledsites, is the straight-line dual graph of the Voronoi diagram. When no degeneracies are present, the Delaunay diagram is a triangulation of the sites, called theDelaunay triangulation. When degeneracies are present, edges must be added to the Delaunay diagram to obtain a Delaunay triangulation. In this paper we describe an optimalO(n logn) plane-sweep algorithm for computing a Delaunay triangulation on a possibly degenerate set of sites in the plane under theL ₁ metric or theL _∞ metric.     

A faster divide-and-conquer algorithm for constructing delaunay triangulations

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its Θ(n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn≤2¹⁶, the range of the experiments. It is conjectured that the average number of edges it creates—a good measure of its efficiency—is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theL _p metric for 1<p≤∞.     

Relative neighborhood graphs in the Li-metric

The relative neighborhood graph of a set of n points in the plane under the L₁-metric is considered. An algorithm that runs in O(nlog n) time for constructing the relative neighborhood graph based on the Delaunay triangulation is presented, improving a previously known algorithm that runs in O(n²log n) time.     

Splitting a Delaunay Triangulation in Linear Time

Abstract. Computing the Delaunay triangulation of n points requires usually a minimum of Ω(n log n) operations, but in some special cases where some additional knowledge is provided, faster algorithms can be designed. Given two sets of points, we prove that, if the Delaunay triangulation of all the points is known, the Delaunay triangulation of each set can be computed in randomized expected linear time.     

Order-k voronoi diagrams of sites with additive weights in the plane

It is shown that the order-k Voronoi diagram of n sites with additive weights in the plane has at most (4k?2)(n?k) vertices, (6k?3)(n?k) edges, and (2k?1)(n?itk) + 1 regions. These bounds are approximately the same as the ones known for unweighted order-k Voronoi diagrams. Furthermore, tight upper bounds on the number of edges and vertices are given for the case that every weighted site has a nonempty region in the order-1 diagram. The proof is based on a new algorithm for the construction of these diagrams which generalizes a plane-sweep algorithm for order-1 diagrams developed by Steven Fortune. The new algorithm has time-complexityO(k ² n logn) and space-complexityO(kn). It is the only nontrivial algorithm known for constructing order-kc Voronoi diagrams of sites withadditive weights. It is fairly simple and of practical interest also in the special case of unweighted sites.     

Parallel divide‐and‐conquer scheme for 2D Delaunay triangulation

This work describes a parallel divide‐and‐conquer Delaunay triangulation scheme. This algorithm finds the affected zone, which covers the triangulation and may be modified when two sub‐block triangulations are merged. Finding the affected zone can reduce the amount of data required to be transmitted between processors. The time complexity of the divide‐and‐conquer scheme remains O(n log n), and the affected region can be located in O(n) time steps, where n denotes the number of points. The code was implemented with C, FORTRAN and MPI, making it portable to many computer systems. Experimental results on an IBM SP2 show that a parallel efficiency of 44–95% for general distributions can be attained on a 16‐node distributed memory system. Copyright © 2006 John Wiley & Sons, Ltd.     

Efficient and good Delaunay meshes from random points

We present a Conforming Delaunay Triangulation (CDT) algorithm based on maximal Poisson disk sampling. Points are unbiased, meaning the probability of introducing a vertex in a disk-free subregion is proportional to its area, except in a neighborhood of the domain boundary. In contrast, Delaunay refinement CDT algorithms place points dependent on the geometry of empty circles in intermediate triangulations, usually near the circle centers. Unconstrained angles in our mesh are between 30° and 120°, matching some biased CDT methods. Points are placed on the boundary using a one-dimensional maximal Poisson disk sampling. Any triangulation method producing angles bounded away from 0° and 180° must have some bias near the domain boundary to avoid placing vertices infinitesimally close to the boundary.Random meshes are preferred for some simulations, such as fracture simulations where cracks must follow mesh edges, because deterministic meshes may introduce non-physical phenomena. An ensemble of random meshes aids simulation validation. Poisson-disk triangulations also avoid some graphics rendering artifacts, and have the blue-noise property.We mesh two-dimensional domains that may be non-convex with holes, required points, and multiple regions in contact. Our algorithm is also fast and uses little memory. We have recently developed a method for generating a maximal Poisson distribution of n output points, where n=Θ(Area/r²) and r is the sampling radius. It takes O(n) memory and O(nlogn) expected time; in practice the time is nearly linear. This, or a similar subroutine, generates our random points. Except for this subroutine, we provably use O(n) time and space. The subroutine gives the location of points in a square background mesh. Given this, the neighborhood of each point can be meshed independently in constant time. These features facilitate parallel and GPU implementations. Our implementation works well in practice as illustrated by several examples and comparison to Triangle.     

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property A_k if, given k pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) of V(G), there are k mutually node-disjoint paths, one connecting s_i and t_i for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy A_k is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A_⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈log n⌉+1 in O(n² log* n) time.     

Numerische Lösung von Anfangswertaufgaben gewöhnlicher Differentialgleichungen n-ter Ordnung

In the present paper a new method is given for the numerical treatment of the initial problemsy ⁽ⁿ⁾=f(x,y,y′, ...,y ^(n?1),y ⁽ⁱ⁾ (x _o )=y _o ⁽ⁱ⁾ , i=0, 1, ...,n?1. This method is an one-step process of order four. For a class of linear differential equations the exact solution is obtained. Moreover some numerical results are presented.     

Constrained delaunay triangulations

Given a set ofn vertices in the plane together with a set of noncrossing, straight-line edges, theconstrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimalO(n logn) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (non-Delaunay) triagulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.An earlier version of the results presented here appeared in theProceedings of the Third Annual Symposium on Computational Geometry (1987).     

Birth and growth of multicyclic components in random hypergraphs

Define an ?-component to be a connected b-uniform hypergraph with k edges and k(b−1)−? vertices. In this paper, we investigate the growth of size and complexity of connected components of a random hypergraph process. We prove that the expected number of creations of ?-components during a random hypergraph process tends to 1 as b is fixed and ? tends to infinity with the total number of vertices n while remaining ?=o(n^1/3). We also show that the expected number of vertices that ever belong to an ?-component is ∼12^1/3?^1/3n^2/3(b−1)^−1/3. We prove that the expected number of times hypertrees are swallowed by ?-components is ∼2^1/33^−1/3n^1/3?^−1/3(b−1)^−5/3. It follows that with high probability the largest ?-component during the process is of size of order O(?^1/3n^2/3(b−1)^−1/3). Our results give insight into the size of giant components inside the phase transition of random hypergraphs and generalize previous results about graphs.     

The Space Complexity of Approximating the Frequency Moments

The frequency moments of a sequence containingm_ielements of typei, 1⩽i⩽n, are the numbersF_k=∑ⁿ_i=1 m^k_i. We consider the space complexity of randomized algorithms that approximate the numbersF_k, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbersF₀,F₁, andF₂can be approximated in logarithmic space, whereas the approximation ofF_kfork⩾6 requiresn^Ω(1)space. Applications to data bases are mentioned as well.     

Order-k voronoi diagrams of sites with additive weights in the plane

It is shown that the order-k Voronoi diagram of n sites with additive weights in the plane has at most (4k–2)(n–k) vertices, (6k–3)(n–k) edges, and (2k–1)(n–itk) + 1 regions. These bounds are approximately the same as the ones known for unweighted order-k Voronoi diagrams. Furthermore, tight upper bounds on the number of edges and vertices are given for the case that every weighted site has a nonempty region in the order-1 diagram. The proof is based on a new algorithm for the construction of these diagrams which generalizes a plane-sweep algorithm for order-1 diagrams developed by Steven Fortune. The new algorithm has time-complexityO(k ² n logn) and space-complexityO(kn). It is the only nontrivial algorithm known for constructing order-kc Voronoi diagrams of sites withadditive weights. It is fairly simple and of practical interest also in the special case of unweighted sites.Work on this paper has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862.     

Improved deterministic algorithms for weighted matching and packing problems

Based on the method of (n,k)-universal sets, we present a deterministic parameterized algorithm for the weighted rd-matching problem with time complexity O^∗(4^{(r−1)k+o(k)}), improving the previous best upper bound O^∗(4^rk+o(k)). In particular, the algorithm applied to the unweighted 3d-matching problem results in a deterministic algorithm with time O^∗(16^k+o(k)), improving the previous best result O^∗(21.26^k). For the weighted r-set packing problem, we present a deterministic parameterized algorithm with time complexity O^∗(2^{(2r−1)k+o(k)}), improving the previous best result O^∗(2^2rk+o(k)). The algorithm, when applied to the unweighted 3-set packing problem, has running time O^∗(32^k+o(k)), improving the previous best result O^∗(43.62^k+o(k)). Moreover, for the weighted r-set packing and weighted rd-matching problems, we give a kernel of size O(k^r), which is the first kernelization algorithm for the problems on weighted versions.     

Asymptotic relation of M- and P-estimators of location

Let T_n and M_n be the P-estimator (Pitman-like estimator) and M_n the M-estimator of the location parameter θ, respectively, both generated by function ρ with the derivative ψ=ρ: It is demonstrated that, under some assumptions on the underlying distribution function F, the difference M_n-T_n is of the order o_p(n^-1/2) in the case of Huber's function ψ. It is further shown that M_n-T_n=o_p(n^-1) if ψ is sufficiently smooth.     

Higher-order Voronoi diagrams on triangulated surfaces

We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order-j Voronoi diagrams of m sites, for j=1,…,k, is O(k²n²+k²m+knm), which is asymptotically tight in the worst case.     

On triangulation axes of polygons

We propose the triangulation axis as an alternative skeletal structure for a simple polygon P. This axis is a straight-line tree that can be interpreted as an anisotropic medial axis of P, where inscribed disks are line segments or triangles. The underlying triangulation that specifies the anisotropy can be varied, to adapt the axis so as to reflect predominant geometrical and topological features of P. Triangulation axes typically have much fewer edges and branchings than the Euclidean medial axis or the straight skeleton of P. Still, they retain important properties, as for example the reconstructability of P   from its skeleton. Triangulation axes can be computed from their defining triangulations in O(n)$O(n)$ time. We investigate the effect of using several optimal triangulations for P  . In particular, careful edge flipping in the constrained Delaunay triangulation leads, in O(nlog?n)$O(n\log ?n)$ overall time, to an axis competitive to ‘high quality axes’ requiring Θ(n³)$\Theta (n^3)$ time for optimization via dynamic programming.