Facility Location on a Polyhedral Surface

Given an orientable genus-0 polyhedral surface defined by n triangles, and a set of m point sites} on it, we would like to identify its 1-center, i.e., the location on the surface that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the surface. To compute the 1-center, we compute the furthest-site Voronoi diagram of the sites on the polyhedral surface. We show that the diagram has maximum combinatorial complexity (m n ²), and present an algorithm that computes the diagram in O(m n ²log mlog n) expected time. The 1-center can then be identified in time linear in the size of the diagram.     

Voronoi Diagrams of Lines in 3-Space Under Polyhedral Convex Distance Functions

The combinatorial complexity of the Voronoi diagram ofnlines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to beO(n²α(n)log n), where α(n) is a slowly growing inverse of the Ackermann function. There are arrangements ofnlines where this complexity can be as large as Ω(n²α(n)).     

Computing generalized higher-order Voronoi diagrams on triangulated surfaces

We present an algorithm for computing exact shortest paths, and consequently distance functions, from a generalized source (point, segment, polygonal chain or polygonal region) on a possibly non-convex triangulated polyhedral surface. The algorithm is generalized to the case when a set of generalized sites is considered, providing their distance field that implicitly represents the Voronoi diagram of the sites. Next, we present an algorithm to compute a discrete representation of the distance function and the distance field. Then, by using the discrete distance field, we obtain the Voronoi diagram of a set of generalized sites (points, segments, polygonal chains or polygons) and visualize it on the triangulated surface. We also provide algorithms that, by using the discrete distance functions, provide the closest, furthest and k-order Voronoi diagrams and an approximate 1-Center and 1-Median.     

A new duality result concerning voronoi diagrams

A new duality between order-k Voronoi diagrams inE ^d and convex hulls inE ^d+1 is established. It implies a reasonably simple algorithm for computing the order-k diagram forn points in the plane inO(k ² n logn) time and optimalO(k(n–k)) space.Research was supported by the Austrian Fond zur Foerderung der wissenschaftlichen Forschung.     

On the construction of abstract voronoi diagrams

We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7]. This work was supported by the DFG, Me 620/6, and ESPRIT P3075 ALCOM. A preliminary version of this paper has been presented at STACS '90, Rouen, France.     

Jones polynomial of knots formed by repeated tangle replacement operations

In this paper, we prove that the Jones polynomial of a link diagram obtained through repeated tangle replacement operations can be computed by a sequence of suitable variable substitutions in simpler polynomials. For the case that all the tangles involved in the construction of the link diagram have at most k crossings (where k is a constant independent of the total number n of crossings in the link diagram), we show that the computation time needed to calculate the Jones polynomial of the link diagram is bounded above by O(nk). In particular, we show that the Jones polynomial of any Conway algebraic link diagram with n crossings can be computed in O(n²) time. A consequence of this result is that the Jones polynomial of any Montesinos link and two bridge knot or link of n crossings can be computed in O(n²) time.     

A compact piecewise-linear voronoi diagram for convex sites in the plane

In the plane the post-office problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites arek disjoint convex sets we give a compact representation of the Voronoi diagram, usingO (k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons withn total vertices given in standard representations, we compute this diagram optimally in Θ (k logn) deterministic time for the Euclidean metric and inO (k logn logm) deterministic time for the convex distance function defined by a convexm-gon. This work was supported by NSERC in the form of a Graduate Scholarship and two Research Grants.     

The furthest-site geodesic voronoi diagram

We present anO((n+k) log(n+k))-time,O(n+k)-space algorithm for computing the furthest-site Voronoi diagram ofk point sites with respect to the geodesic metric within a simplen-sided polygon.A preliminary version of this paper appeared in theProceedings of the Fourth ACM Symposium on Computational Geometry, 1988. The work of Boris Aronov was supported by an AT&T Bell Laboratories Ph.D. Scholarship. Part of the work was performed while he was at AT&T Bell Laboratories, Murray Hill, NJ, USA. His current address is Computer Science Department, Polytechnic University, Brooklyn, NY 11201, USA.     

Modelling gateway placement in wireless networks: Geometric k-centres of unit disc graphs

Motivated by the gateway placement problem in wireless networks, we consider the geometric k-centre problem on unit disc graphs: given a set of points P in the plane, find a set F of k points in the plane that minimizes the maximum graph distance from any vertex in P to the nearest vertex in F in the unit disc graph induced by P∪F. We show that the vertex 1-centre provides a 7-approximation of the geometric 1-centre and that a vertex k-centre provides a 13-approximation of the geometric k-centre, resulting in an O(kn)-time 26-approximation algorithm. We describe O(n²m)-time and O(n³)-time algorithms, respectively, for finding exact and approximate geometric 1-centres, and an O(mn2k)-time algorithm for finding a geometric k-centre for any fixed k. We show that the problem is NP-hard when k is an arbitrary input parameter. Finally, we describe an O(n)-time algorithm for finding a geometric k-centre in one dimension.     

A “retraction” method for planning the motion of a disc

A new approach to certain motion-planning problems in robotics is introduced. This approach is based on the use of a generalized Voronoi diagram, and reduces the search for a collision-free continuous motion to a search for a connected path along the edges of such a diagram. This approach yields an O(n log n) algorithm for planning an obstacle-avoiding motion of a single circular disc amid polygonal obstacles. Later papers will show that extensions of the approach can solve other motion-planning problems, including those of moving a straight line segment or several coordinated discs in the plane amid polygonal obstacles.     

Flip Graphs of Bounded Degree Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant k. In particular, we consider triangulations of sets of n points in convex position in the plane and prove that their flip graph is connected if and only if k > 6; the diameter of the flip graph is O(n ²). We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for k ≤ 9, and flip graphs of triangulations can be disconnected for any k. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound k by a small constant. Any two triangulations with maximum degree at most k of a convex point set are connected in the flip graph by a path of length O(n log n), where every intermediate triangulation has maximum degree at most k + 4.     

Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions

The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R ^d , the maximum complexity of its Voronoi diagram under the L _∞ metric, and also under a simplicial distance function, are both shown to be . The upper bound for the case of the L _∞ metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R ^d . This complexity is , for d ≥ 1 , and it improves to , for d ≥ 2 , if all the hypercubes have the same size. Under the L ₁ metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R ³ is shown to be . We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations. (This increase does not occur with an appropriate modification of the diagram definition.) Finally, on-line algorithms are proposed for computing the Voronoi diagram of n points in R ^d under a simplicial or L _∞ distance function. Their expected randomized complexities are for simplicial diagrams and for L _∞ -diagrams. Received July 31, 1995, and in revised form September 9, 1997.     

On some algorithmic problems regarding the hairpin completion

We consider a few algorithmic problems regarding the hairpin completion. The first problem we consider is the membership problem of the hairpin and iterated hairpin completion of a language. We propose an O(nf(n)) and O(n²f(n)) time recognition algorithm for the hairpin completion and iterated hairpin completion, respectively, of a language recognizable in O(f(n)) time. We show that the n factor is not needed in the case of hairpin completion of regular and context-free languages. The n² factor is not needed in the case of iterated hairpin completion of context-free languages, but it is reduced to n in the case of iterated hairpin completion of regular languages. We then define the hairpin completion distance between two words and present a cubic time algorithm for computing this distance. A linear time algorithm for deciding whether or not the hairpin completion distance with respect to a given word is connected is also discussed. Finally, we give a short list of open problems which appear attractive to us.     

Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions

We show that the complexity of the Voronoi diagram of a collection of disjoint polyhedra in general position in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n ²⁺), for any > 0. We also show that when the sites are n segments in 3-space, this complexity is O(n ² (n) log n). This generalizes previous results by Chew et al. and by Aronov and Sharir, and solves an open problem put forward by Agarwal and Sharir. Specific distance functions for which our results hold are the L ₁ and L _\infty metrics. These results imply that we can preprocess a collection of polyhedra as above into a near-quadratic data structure that can answer -approximate Euclidean nearest-neighbor queries amidst the polyhedra in time O(log (n/) ), for an arbitrarily small > 0.     

Dynamic construction of abstract Voronoi diagrams

The abstract Voronoi diagram (AVD) introduced by R. Klein is a generalization of various concrete Voronoi diagrams—data structures actively used in the last decades for solving theoretical and practical geometric problems. This paper presents a fully dynamic algorithm for AVD construction based on Klein's incremental approach. It needs O(n) worst-case time for a new site insertion in an AVD with n sites. For the first time a possibility of effective site deletion without full reconstruction of AVD is proved. The proposed method for site deletion requires O(mn) expected time, where m is the number of invisible sites, and O(n) if invisible sites are not allowed. The dynamic algorithm consumes O(n) memory at any moment. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 133–146, 2007.     

Extended Formulations for Polygons

The extension complexity of a polytope?P is the smallest integer?k such that?P is the projection of a polytope?Q with?k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193?C205, 2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on an O(n)×O(n ²) integer grid with extension complexity $\varOmega (\sqrt{n}/\sqrt{\log n})$ .     

Interpretation and optimization of the k-means algorithm

The paper gives a new interpretation and a possible optimization of the wellknown k-means algorithm for searching for a locally optimal partition of the set A = {a _i ∈ ? ⁿ : i = 1, …, m} which consists of k disjoint nonempty subsets π₁, …, π _k , 1 ? k ? m. For this purpose, a new divided k-means algorithm was constructed as a limit case of the known smoothed k-means algorithm. It is shown that the algorithm constructed in this way coincides with the k-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram. If in the partition obtained by applying the divided k-means algorithm there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.     

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver.The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ(m,n,k)=O(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.     

Approximation Algorithms for Steiner and Directed Multicuts

In this paper we consider theSteiner multicutproblem. This is a generalization of the minimum multicut problem where instead of separating nodepairs, the goal is to find a minimum weight set of edges that separates all givensetsof nodes. A set is considered separated if it is not contained in a single connected component. We show anO(log³(kt)) approximation algorithm for the Steiner multicut problem, wherekis the number of sets andtis the maximum cardinality of a set. This improves theO(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimum-weight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecifiedknode pairs. In this paper we describe anO(log² k) approximation algorithm for this directed multicut problem. Ifk ? n, this represents an improvement over theO(log n log log n) approximation algorithm that is implied by the technique of Seymour.