cluster synchrony in systems of coupled phase oscillators with higher order coupling we study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher order modes dominate the coupling between oscillators for the first time we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony cluster asymmetry and switching we use a variation of the recent dimensionality reduction technique of ott and antonsen chaos 18 037113 2008 and find an analytic description of the degree of cluster synchrony valid on a globally attracting manifold shaped by this manifold there is an infinite family of steady state distributions of oscillators resulting in a high degree of multistability in the cluster asymmetry we also show how through external forcing the degree of asymmetry can be controlled and suggest that systems displaying cluster synchrony can be used to encode and store data

the number of shortest paths on the surface of a polyhedron it is proven that if the shortest paths on the surface of a convex polyhedron are grouped into equivalence classes according to the sequences of edges that they cross then the resulting number of equivalence classes is o n 4 where n is the number of vertices of the polyhedron in fact the more general result that any family of pseudosegments a set of open simple curves on the plane such that two curves intersect each other in at most one point lying on a planar subdivision defined by n other pseudosegments can give rise to at most o n 4 edge sequences is also proven this bound is shown to be asymptotically tight by giving an example of a family of polyhedra with omega n 4 shortest path equivalence classes

on shortest paths amidst convex polyhedra let k be a 3 d convex polyhedron having n vertices a sequence xi of edges of k is called a shortest path sequence if there exist two points x y on the surface s of k such that is the sequence of edges crossed by the shortest path from x to y along s we show that the number of shortest path sequences for k is polynomial in n and as a consequence prove that the shortest path between two points in 3 space which must avoid the interiors of a fixed number of disjoint convex polyhedral obstacles can be calculated in time polynomial in the total number of vertices of these obstacles but exponential in the number of obstacles

space time tradeoffs for emptiness queries we develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries our lower bounds apply to a general class of geometric range query data structures called partition graphs informally a partition graph is a directed acyclic graph that describes a recursive decomposition of space we show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the fredman yao semigroup arithmetic model with the same asymptotic space and time bounds thus results of bronnimann chazelle and pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality st d omega n log n d d 1 d 1 using different techniques we improve previous lower bounds for hopcroft s problem given a set of points and hyperplanes does any hyperplane contain a point in dimensions four and higher using this offline result we show that for online hyperplane emptiness queries omega n d mbox polylog n space is required to achieve polylogarithmic query time and omega n d 1 d mbox polylog n query time is required if only o n polylog n space is available these two lower bounds are optimal up to polylogarithmic factors for two dimensional queries we obtain an optimal continuous tradeoff st 2 omega n 2 between these two extremes finally using a lifting argument we show that the same lower bounds hold for both offline and online halfspace emptiness queries in mathbb r d d 3 2

a constrained two dimensional triangulation and the solution of closest node problems in the presence of barriers a delaunay triangulation of a set of nodes is a collection of triangles whose vertices are at the nodes and whose union fills the convex hull of the set of nodes it also has several geometrical properties making it useful for solving closest point problems the generalization presented here allows the triangulation to cover nonconvex regions including those with holes although a variety of such generalizations are possible the one presented here is shown to retain important closest point characteristics thus it is useful for determining shortest paths within planar regions with polygonal boundaries

frequency synchronization in a random oscillator network we study the frequency synchronization of a randomly coupled oscillators by analyzing the continuum limit we obtain a sufficient condition for the mean field type synchronization we especially find that the critical coupling constant k becomes 0 in the random scale free network p k ensuremath propto k ensuremath ensuremath gamma if 2l ensuremath gamma ensuremath leqslant 3 numerical simulations in finite networks are consistent with this analysis

approximating shortest path for the skew lines problem in time doubly logarithmic in 1 epsilon we consider two three dimensional situations when a polytime algorithm for approximating a shortest path can be constructed the main part of the paper treats a well known problem of constructing a shortest path touching lines in r3 given a list of straight lines l l1 ln in r3 and two points s and t find a shortest path that starting from s touches the lines li in the given order and ends at t we remark that such a shortest path is unique we show that it can be length position e approximated i e both its length and its position can be found approximately in time rn d 16 o n2 log log 1 e where d is the minimal distance between consecutive lines of l is the minimum of sines of angles between consecutive lines and r is the radius of a ball where the initial approximation can be placed such a radius can be easily computed from the initial data as computational model we take real ram extended by square and cubic roots extraction this problem of constructing a shortest path touching lines is known for quite some time to be a challenging problem the existing methods for approximating shortest paths based on adding steiner points which form a grid and subsequently applying dijkstra s algorithm for finding a shortest path in the grid provide a complexity bound which depends polynomially on 1 e while our algorithm for the problem under consideration has complexity linear in log log 1 e our algorithm is motivated by the observation that the shortest path in question is a geodesic in a certain length space of non positive curvature in the sense of a d alexandrov and it relies on the elementary theory of cat 0 spaces in the second part of the paper we analyze very simple grid approximations we assume that a parameter a 0 describing separability of obstacles is given and the part of a grid with mesh size a outside the obstacles is built for semi algebraic obstacles all these precalculations are polytime we show that there is an algorithm of time complexity o 1 a 6 which given a separated obstacles in a unit cube finds a path between given vertices s and t of the grid whose length is bounded from above by 84 96a where is the length of a shortest path on the other hand as we show by an example one cannot approximate the length of a shortest path better than 7 if one uses only grid polygons constructed only from grid edges for semi algebraic obstacles our computational model is bitwise for a general type of obstacles the model is bitwise modulo constructing the part of the grid admissible for our paths observe that the existing methods for approximating shortest paths are not directly applicable for semi algebraic obstacles since they usually place the steiner points forming a grid on the edges of polyhedral obstacles

shortest paths in the plane with polygonal obstacles we present a practical algorithm for finding minimum length paths between points in the euclidean plane with not necessarily convex polygonal obstacles prior to this work the best known algorithm for finding the shortest path between two points in the plane required o n 2 log n time and o n 2 space where n denotes the number of obstacle edges assuming that a triangulation or a voronoi diagram for the obstacle space is provided with the input if is not either one can be precomputed in o n log n time we present an o kn time algorithm where k denotes the number of islands connected components in the obstacle space the algorithm uses only o n space and given a source point s produces an o n size data structure such that the distance between s and any other point x in the plane x is not necessarily an obstacle vertex or a point on an obstacle edge can be computed in o 1 time the algorithm can also be used to compute shortest paths for the movement of a disk so that optimal movement for arbitrary objects can be computed to the accuracy of enclosing them with the smallest possible disk

determining approximate shortest paths on weighted polyhedral surfaces in this article we present an approximation algorithm for solving the single source shortest paths problem on weighted polyhedral surfaces we consider a polyhedral surface p as consisting of n triangular faces where each face has an associated positive weight the cost of travel through a face is the euclidean distance traveled multiplied by the face s weight for a given parameter e 0 e 1 the cost of the computed paths is at most 1 p e times the cost of corresponding shortest paths our algorithm is based on a novel way of discretizing polyhedral surfaces and utilizes a generic greedy approach for computing shortest paths in geometric graphs obtained by such discretization its running time is o c p n se log n e log 1 e time where c p captures geometric parameters and the weights of the faces of p

onset of synchronization in large networks of coupled oscillators we study the transition from incoherence to coherence in large networks of coupled phase oscillators we present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength we find that under appropriate conditions the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix we show how with an additional assumption a mean field approximation recently proposed is recovered from our results we test our theory with numerical simulations and find that it describes the transition when our assumptions are satisfied we find that our theory describes the transition well in situations in which the mean field approximation fails we study the finite size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase

dynamics of influence on hierarchical structures dichotomous spin dynamics on a pyramidal hierarchical structure the bethe lattice are studied the system embodies a number of classes where a class comprises nodes that are equidistant from the root head node weighted links exist between nodes from the same and different classes the spin hereafter state of the head node is fixed we solve for the dynamics of the system for different boundary conditions we find necessary conditions so that the classes eventually repudiate or acquiesce in the state imposed by the head node the results indicate that to reach unanimity across the hierarchy it suffices that the bottommost class adopts the same state as the head node then the rest of the hierarchy will inevitably comply this also sheds light on the importance of mass media as a means of synchronization between the topmost and bottommost classes surprisingly in the case of discord between the head node and the bottommost classes the average state over all nodes inclines towards that of the bottommost class regardless of the link weights and intraclass configurations hence the role of the bottommost class is signified

ray shooting and parametric search efficient algorithms for the ray shooting problem are presented given a collection gamma of objects in mathbb r d build a data structure so that for a query ray the first object of gamma hit by the ray can be quickly determined using the parametric search technique this problem is reduced to the segment emptiness problem for various ray shooting problems space query time trade offs of the following type are achieved for some integer b and a parameter m n leqslant m leqslant n b the queries are answered in time o n m 1 b log o 1 n with o m 1 varepsilon space and preprocessing time varepsilon 0 is arbitrarily small but fixed constant b lfloor d 2 rfloor is obtained for ray shooting in a convex d polytope defined as an intersection of n half spaces b d for an arrangement of n hyperplanes in mathbb r d and b 3 for an arrangement of n half planes in mathbb r 3 this approach also yields fast procedures for fin

an optimal algorithm for intersecting line segments in the plane the main contribution of this work is an o n log n k time algorithm for computing all k intersections among n line segments in the plane this time complexity is easily shown to be optimal within the same asymptotic cost our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment if any lies right above or below each intersection and each endpoint the algorithm has been implemented and performs very well the storage requirement is on the order of n k in the worst case but it is considerably lower in practice to analyze the complexity of the algorithm an amortization argument based on a new combinatorial theorem on line arrangements is used

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

a single exponential upper bound for finding shortest paths in three dimensions we derive a single exponential time upper bound for finding the shortest path between two points in 3 dimensional euclidean space with nonnecessarily convex polyhedral obstacles prior to this work the best known algorithm required double exponential time given that the problem is known to be pspace hard the bound we present is essentially the best in the worst case sense that can reasonably be expected

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

approximate polytope membership queries in the polytope membership problem a convex polytope k in mathbb r d is given and the objective is to preprocess k into a data structure so that given any query point q in mathbb r d it is possible to determine efficiently whether q in k we consider this problem in an approximate setting given an approximation parameter varepsilon the query can be answered either way if the distance from q to k s boundary is at most varepsilon times k s diameter we assume that the dimension d is fixed and k is presented as the intersection of n halfspaces previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by dudley approx theory 10 1974 pp 227 236 and bentley faust and preparata commun acm 25 1982 pp 64 68 the former is optimal in the worst case with respect to space and the latter is optimal with respect to query time we present four main results first we show how

on shortest paths in polyhedral spaces we consider the problem of computing the shortest path between two points in two or three dimensional space bounded by polyhedral surfaces in the 2 d case the problem is easily solved in time o n 2 log n in the general 3 d case the problem is quite hard to solve and is not even discrete we present a doubly exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes finally we consider a favorable special case of the 3 d shortest path problem namely that of finding the shortest path between two points along the surface of a convex polyhedron and solve it in time o n 3 log n

the onset of synchronization in systems of globally coupled chaotic and periodic oscillators abstract a general stability analysis is presented for the determination of the transition from incoherent to coherent behavior in an ensemble of globally coupled heterogeneous continuous time dynamical systems the formalism allows for the simultaneous presence of ensemble members exhibiting chaotic and periodic behavior and in a special case yields the kuramoto model for globally coupled periodic oscillators described by a phase numerical experiments using different types of ensembles of lorenz equations with a distribution of parameters are presented

path integral approach to dynamics in a sparse random network we study the dynamics involved in a sparse random network model we extend the standard mean field approximation for the dynamics of a random network by employing the path integral approach the result indicates that the distribution of the variable is essentially identical to that obtained from globally coupled oscillators with random gaussian interaction we present the results of a numerical simulation of the kuramoto transition in a random network which is found to be consistent with this analysis

solution of the voter model by spectral analysis an exact spectral analysis of the markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks using a well defined martingale approximation in diffusion dominated regions of phase space which is almost everywhere for the voter model this method is applied to compute analytically several key quantities such as exact expressions for the m time step propagator of the voter model all moments of consensus times and the local times for each macrostate this spectral method is motivated by a related method for solving the ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model comparisons of the analytical results from the spectral method and numerical results from monte carlo simulations are presented to validate the spectral method

network connectivity during mergers and growth optimizing the addition of a module the principal eigenvalue of a network s adjacency matrix often determines dynamics on the network e g in synchronization and spreading processes and some of its structural properties e g robustness against failure or attack and is therefore a good indicator for how strongly a network is connected we study how is modified by the addition of a module or community which has broad applications ranging from those involving a single modification e g introduction of a drug into a biological process to those involving repeated additions e g power grid and transit development we describe how to optimally connect the module to the network to either maximize or minimize the shift in noting several applications of directing dynamics on networks

curve segmentation under partial occlusion 2 d shape boundary segmentation is required as a fundamental and important stage in the recognition of partially occluded objects we introduce here a new segmentation method capable of extracting a controlled number of segments along a smooth boundary curve this new approach is invariant to similarity transformation and partial occlusion has only marginal influence on the segmentation of the visible part the basic concept is to transform the curve into another one which intersects itself points of intersection of the new curve are re transformed to the original curve and serve as endpoints of segments properties of the transform are discussed and conditions for existence of intersection points are given simulation results of gray level images are presented and advantages of our method over conventional approaches relying on singular points of the curvature are discussed

instance optimal geometric algorithms we prove the existence of an algorithm a for computing 2d or 3d convex hulls that is optimal for every point set in the following sense for every sequence of n points and for every algorithm a in a certain class a the running time of a on input is at most a constant factor times the running time of a on the worst possible permutation of for a in fact we can establish a stronger property for every sequence of points and every algorithm a the running time of a on is at most a constant factor times the average running time of a over all permutations of we call algorithms satisfying these properties instance optimal in the order oblivious and random order setting such instance optimal algorithms simultaneously subsume output sensitive algorithms and distribution dependent average case algorithms and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order the class a under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments to establish an instance specific lower bound we deviate from traditional ben or style proofs and adopt a new adversary argument for 2d convex hulls we prove that a version of the well known algorithm by kirkpatrick and seidel 1986 or chan snoeyink and yap 1995 already attains this lower bound for 3d convex hulls we propose a new algorithm we further obtain instance optimal results for a few other standard problems in computational geometry such as maxima in 2d and 3d orthogonal line segment intersection in 2d finding bichromatic l close pairs in 2d offline orthogonal range searching in 2d offline dominance reporting in 2d and 3d offline half space range reporting in 2d and 3d and offline point location in 2d our framework also reveals a connection to distribution sensitive data structures and yields new results as a byproduct for example on online orthogonal range searching in 2d and online half space range reporting in 2d and 3d

decomposable multi parameter matroid optimization problems a framework for solving certain multi dimensional parametric matroid optimization problems in randomized linear time by prune and search is presented the common feature of these problems which include the multi parameter minimum spanning tree problem on planar and dense graphs is that their fixed parameter versions are solvable by tournament like algorithms whose structure is represented by a balanced decomposition tree

on the continuous fermat weber problem we give the firstexact algorithmic study of facility location problems that deal with finding a median for acontinuum of demand points in particular we consider versions of the continuousk median fermat weber problem where the goal is to select one or more center points that minimize the average distance to a set of points in a demandregion in such problems the average is computed as an integral over the relevant region versus the usual discrete sum of distances the resulting facility location problems are inherently geometric requiring analysis techniques of computational geometry we provide polynomial time algorithms for various versions of thel1 1 median fermat weber problem we also consider the multiple center version of thel1k median problem which we prove is np hard for largek

star unfolding of a polytope with applications we introduce the notion of a star unfolding of the surface cal p of a three dimensional convex polytope with n vertices and use it to solve several problems related to shortest paths on cal p the first algorithm computes the edge sequences traversed by shortest paths on cal p in time o n 6 beta n log n where beta n is an extremely slowly growing function a much simpler o n 6 time algorithm that finds a small superset of all such edge sequences is also sketched the second algorithm is an o n 8 log n time procedure for computing the geodesic diameter of cal p the maximum possible separation of two points on cal p with the distance measured along cal p finally we describe an algorithm that preprocesses cal p into a data structure that can efficiently answer the queries of the following form given two points what is the length of the shortest path connecting them given a parameter 1 le m le n 2 it can preprocess cal p in time o n 6 m 1 delta for any delta 0 into a data structure of size o n 6m 1 delta so that a query can be answered in time o sqrt n m 1 4 log n if one query point always lies on an edge of cal p the algorithm can be improved to use o n 5 m 1 delta preprocessing time and storage and guarantee o n m 1 3 log n query time for any choice of m between 1 and n

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

algebraic properties of location problems with one circular barrier abstract the consideration of barriers to travel plays an increasingly important role in the transportation and location literature in one of the classical papers on location problems with barriers katz and cooper eur j oper res 6 1981 166 considered the weber problem often also referred to as median problem with one circular barrier region considering the same problem we develop new structural results showing that the set of feasible solutions can be subdivided into a polynomial number of cells of algebraic invariance on every convex subset of which the generally non convex objective function is convex these results imply improved exact and heuristic solution procedures based on convex optimization methods

direct multiple shooting method for solving approximate shortest path problems we use the idea of the direct multiple shooting method presented by bock in proceedings of the 9th ifac world congress budapest pergamon press 1984 for solving optimal control problems to introduce an algorithm for solving some approximate shortest path problems in motion planning the algorithm is based on a direct multiple shooting discretization that includes a collinear condition a continuity condition type in the direct multiple shooting method multiple shooting structure and approximation conditions in the case of monotone polygons it is implemented by a c code and a numerical example shows that our algorithm significantly reduces the running time and memory usage of the system

functional approach to data structures and its use in multidimensional searching we establish new upper bounds on the complexity of multidimensional searching our results include in particular linear size data structures for range and rectangle counting in two dimensions with logarithmic query time more generally we give improved data structures for rectangle problems in any dimension in a static as well as a dynamic setting several of the algorithms we give are simple to implement and might be the solutions of choice in practice central to this paper is the nonstandard approach followed to achieve these results at its root we find a redefinition of data structures in terms of functional specifications

multiple shooting approach for computing approximately shortest paths on convex polytopes abstract in this paper we use a multiple shooting approach in solving boundary value problems for ode to introduce a novel iterative algorithm for computing an approximate shortest path between two points on the surface of a convex polytope in 3d namely the polytope is partitioned into subpolytopes shooting points and a straightness condition are established the algorithm specifies how to combine shortest paths between shooting points in subpolytopes to become the required shortest path by the straightness condition in particular the algorithm does not rely on steiner points and graph tools on the entire polytope it is implemented in c and a comparison with agarwal har peled and karia s algorithm on the accurate construction of the shortest path is presented

the many proofs and applications of perron s theorem this paper chronicles the wide dispersal of perron s 1907 result on positive matrices into many fields of science the many proofs given during the last 93 years are categorized and critiqued including perron s original two proofs and a more natural proof is presented this simple to understand result of perron provides a unequaled vehicle for taking students on a tour of many applied areas with some depth

synchronization transition in scale free networks clusters of synchrony we study the synchronization transition in scale free networks that display power law asymptotic behaviors in their degree distributions the critical coupling strength and the order parameter critical exponent derived by the mean field approach depend on the degree exponent ensuremath lambda which implies a close connection between structural organization and the emergence of dynamical order in complex systems we also derive the finite size scaling behavior of the order parameter finding that the giant cluster of synchronized nodes is formed in different ways between scale free networks with 2l ensuremath lambda l3 and those with ensuremath lambda g3

emergence of synchronization in complex networks of interacting dynamical systems abstract we study the emergence of coherence in large complex networks of interacting heterogeneous dynamical systems we show that for a large class of dynamical systems and network topologies there is a critical coupling strength at which the systems undergo a transition from incoherent to coherent behavior we find that the critical coupling strength at which this transition takes place is k c z 1 where z depends only on the uncoupled dynamics of the individual systems on each node while is the largest eigenvalue of the network adjacency matrix thus we achieve a separation of the problem into two parts one depending solely on the node dynamics and one depending solely on network topology

voter models with conserved dynamics we propose a modified voter model with locally conserved magnetization and investigate its phase ordering dynamics in two dimensions in numerical simulations imposing a local constraint on the dynamics has the surprising effect of speeding up the phase ordering process the system is shown to exhibit a scaling regime characterized by algebraic domain growth at odds with the logarithmic coarsening of the standard voter model a phenomenological approach based on cluster diffusion and similar to smoluchowski ripening correctly predicts the observed scaling regime our analysis exposes unexpected complexity in the phase ordering dynamics without thermodynamic potential

sorting jordan sequences in linear time using level linked search trees for a jordan curve c in the plane nowhere tangent to the x axis let x1 x2 xn be the abscissas of the intersection points of c with the x axis listed in the order the points occur on c we call x1 x2 xn a jordan sequence in this paper we describe an o n time algorithm for recognizing and sorting jordan sequences the problem of sorting such sequences arises in computational geometry and computational geography our algorithm is based on a reduction of the recognition and sorting problem to a list splitting problem to solve the list splitting problem we use level linked search trees

tight bounds on the solutions of multidimensional divide and conquer maximin recurrences abstract in this paper upper bounds are presented for the solution of the following multidimensional divide and conquer maximin recurrence g n max n 1 n 2 n p n n 1 n 2 n p 1 i 1 p g n i smin 1 i p k f n i where p 2 1 k f is an arbitrary nondecreasing function and smin 1 i p k f n i denotes the sum of the smallest k numbers among f n 1 f n 2 and f n p all the presented upper bounds are at most log 2 k p times the exact solution of g n the derivation of the upper bounds is based on properties of partition trees for k 1 and k p 1 we obtain respectively two of the recurrences previously studied by alonso et al siam j discrete math 8 1995 428 447 in both of these two cases our results improve theirs

robust plane sweep for intersecting segments in this paper we reexamine in the framework of robust computation the bentley ottmann algorithm for reporting intersecting pairs of segments in the plane this algorithm has been reported as being very sensitive to numerical errors indeed a simple analysis reveals that it involves predicates of degree 5 presumably never evaluated exactly in most implementations within the exact computation paradigm we introduce two models of computation aimed at replacing the conventional model of real number arithmetic the first model predicate arithmetic assumes the exact evaluation of the signs of algebraic expressions of some degree and the second model exact arithmetic assumes the exact computation of the value of such bounded degree expressions we identify the characteristic geometric property enabling the correct report of all intersections by plane sweeps verification of this property involves only predicates of optimal degree 2 but its straightforward implementation appears highly inefficient we then present algorithmic variants that have low degree under these models and achieve the same performance as the original bentley ottmann algorithm the technique is applicable to a more general case of curved segments

semialgebraic range reporting and emptiness searching with applications in a typical range emptiness searching resp reporting problem we are given a set p of n points in mathbb r d and we wish to preprocess it into a data structure that supports efficient range emptiness resp reporting queries in which we specify a range sigma which in general is a semialgebraic set in mathbb r d of constant description complexity and we wish to determine whether p cap sigma emptyset or to report all the points in p cap sigma range emptiness searching and reporting arise in many applications and have been treated by matousek comput geom theory appl 2 1992 pp 169 186 in the special case where the ranges are half spaces bounded by hyperplanes as shown in matousek s work the two problems are closely related and they have solutions for the case of half spaces with similar performance bounds in this paper we extend the analysis to general semialgebraic ranges and show how to adapt matousek s technique without the need to linearize the ranges into a higher dimensional space this yields more efficient solutions to several useful problems and we demonstrate the new technique in four applications with the following results i an algorithm for ray shooting amid balls in mathbb r 3 which uses o n storage and o n preprocessing we use the notation o n gamma to mean an upper bound of the form c varepsilon n gamma varepsilon which holds for any varepsilon 0 where c varepsilon is a constant that depends on varepsilon and answers a query in o n 2 3 time improving the previous bound of o n 3 4 ii an algorithm that preprocesses in o n time a set p of n points in mathbb r 3 into a data structure with o n storage so that for any query line ell or for that matter any simply shaped convex set the point of p farthest from ell can be computed in o n 1 2 time this in turn yields an algorithm that computes the largest area triangle spanned by p in time o n 26 11 as well as nontrivial algorithms for computing the largest perimeter or largest height triangle spanned by p iii an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that for any query alpha fat triangle delta we can determine in o 1 time whether delta cap p is empty alternatively we can report in o 1 o k time the points of delta cap p where k delta cap p iv an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that given any query semidisk c or a circular cap larger than a semidisk we can determine in o 1 time whether c cap p is empty or report the k points in c cap p in o 1 o k time adapting the recent techniques of b aronov and s har peled siam j comput 38 2008 pp 899 921 b aronov s har peled and m sharir on approximate halfspace range counting and relative epsilon approximations in proceedings of the 23rd acm symposium comput geom 2007 pp 327 336 b aronov and m sharir siam j comput 39 2010 pp 2704 2725 we can turn our solutions into efficient algorithms for approximate range counting with small relative error for the cases mentioned above our technique is closely related to the notions of nearest or farthest neighbor generalized voronoi diagrams and of the union or intersection of geometric objects where sharper bounds on the combinatorial complexity of decompositions of complements of these structures yield faster range emptiness searching or reporting algorithms

the discrete geodesic problem we present an algorithm for determining the shortest path between a source and a destination on an arbitrary possibly nonconvex polyhedral surface the path is constrained to lie on the surface and distances are measured according to the euclidean metric our algorithm runs in time o n log n and requires o n2 space where n is the number ofedges ofthe surface afterwe run our algorithm the distance from the source to any other destination may be determined using standard techniques in time o log n by locating the destination in the subdivision created by the algorithm the actual shortest path from the source to a destination can be reported in time o k log n where k is the number of faces crossed by the path the algorithm generalizes to the case of multiple source points to build the voronoi diagram on the surface where n is now the maximum of the number of vertices and the number of sources

generalized percolation in random directed networks we develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges that is edges pointing in both directions simultaneously these two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena equations for the percolation threshold and the sizes of the giant components are derived in the most general case we also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments