diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

lower bounds for randomized k serverand motion planning algorithms in this paper the authors prove lower bounds on the competitive ratio of randomized algorithms for two on line problems the k server problem suggested by manasse mcgeoch and sleator competitive algorithms for on line problems j algorithms 11 1990 pp 208 230 and an on line motion planning problem due to papadimitriou and yannakakis shortest paths without a map lecture notes in comput sci 372 springer verlag new york 1989 pp 610 620 the authors prove against an oblivious adversary an omega log k lower bound on the competitive ratio of any randomized on line k server algorithm in any sufficiently large metric space an omega log log k lower bound on the competitive ratio of any randomized on line k server algorithm in any metric space with at least k 1 points and an omega log log n lower bound on the competitive ratio of any on line motion planning algorithm for a scene with n obstacles previously no superconstant lower bound on the competitive ratio of randomized on line algorithms was known for any of these problems

design of on line algorithms using hitting times random walks are well known for playing a crucial role in the design of randomized off line as well as on line algorithms in this work we prove some basic identities for ergodic markov chains e g an interesting characterization of reversibility in markov chains is obtained in terms of first passage times besides providing new insight into random walks on weighted graphs we show how these identities give us a way of designing competitive randomized on line algorithms for certain well known problems

competitive exploration of rectilinear polygons exploring a polygon with robots when the robots do not have knowledge of the surroundings can be viewed as an online problem typical for online problems is that decisions must be made based on past events without complete information about the future in our case the robots do not have complete information about the environment competitive analysis can be used to measure the performance of methods solving online problems the competitive ratio of such a method is the ratio between the method s performance and the performance of the best method having full knowledge of the future we prove constant competitive strategies and lower bounds for exploring a simple rectilinear polygon in the l1 metric

a general decomposition theorem for the k server problem the first general decomposition theorem for the k server problem is presented whereas all previous theorems are for the case of a finite metric with k 1 points the theorem given here allows an arbitrary number of points in the underlying metric space this theorem implies o polylog k competitive randomized algorithms for certain metric spaces consisting of a polylogarithmic number of widely separated subspaces and takes a first step toward a general o polylog k competitive algorithm the only other cases for which polylogarithmic competitive randomized algorithms are known are the uniform metric space and the weighted cache metric space with two weights

tree exploration with advice we study the amount of knowledge about the network that is required in order to efficiently solve a task concerning this network the impact of available information on the efficiency of solving network problems such as communication or exploration has been investigated before but assumptions concerned availability of particular items of information about the network such as the size the diameter or a map of the network in contrast our approach is quantitative we investigate the minimum number of bits of information bits of advice that has to be given to an algorithm in order to perform a task with given efficiency we illustrate this quantitative approach to available knowledge by the task of tree exploration a mobile entity robot has to traverse all edges of an unknown tree using as few edge traversals as possible the quality of an exploration algorithm a is measured by its competitive ratio i e by comparing its cost number of edge traversals to the length of the shortest path containing all edges of the tree depth first search has competitive ratio 2 and in the absence of any information about the tree no algorithm can beat this value we determine the minimum number of bits of advice that has to be given to an exploration algorithm in order to achieve competitive ratio strictly smaller than 2 our main result establishes an exact threshold number of bits of advice that turns out to be roughly loglogd where d is the diameter of the tree more precisely for any constant c we construct an exploration algorithm with competitive ratio smaller than 2 using at most loglogd c bits of advice and we show that every algorithm using loglogd g d bits of advice for any function g unbounded from above has competitive ratio at least 2

a polylogarithmic competitive algorithm for the k server problem we give the first polylogarithmic competitive randomized online algorithm for the k server problem on an arbitrary finite metric space in particular our algorithm achieves a competitive ratio of o log3 n log2 k for any metric space on n points our algorithm improves upon the deterministic 2k 1 competitive algorithm of koutsoupias and papadimitriou koutsoupias and papadimitriou 1995 for a wide range of n

a randomized algorithm for two servers in cross polytope spaces it has been a long standing open problem to determine the exact randomized competitiveness of the 2 server problem that is the minimum competitiveness of any randomized online algorithm for the 2 server problem for deterministic algorithms the best competitive ratio that can be obtained is 2 and no randomized algorithm is known that improves this ratio for general spaces for the line bartal et al 1998 2 give a 15578 competitive algorithm but their algorithm is specific to the geometry of the line we consider here the 2 server problem over cross polytope spaces m 2 4 we obtain an algorithm with competitive ratio of 1912 and show that this ratio is best possible this algorithm gives the second non trivial example of metric spaces with better than2 competitive ratio the algorithm uses a design technique called the knowledge state technique a method not specific to m 2 4

convergence from two fluid incompressible navier stokes maxwell system with ohm s law to solenoidal ohm s law classical solutions abstract the asymptotics from the two fluid incompressible navier stokes maxwell system with ohm s law to solenoidal ohm s law was pointed out in arsenio saint raymond s work 3 we justify rigorously this limit in the context of global in time classical solutions the key is to derive the global in time uniform in e energy estimate of the rescaled system with ohm s law by employing the decay properties of both the electric field e e and the wave equation with linear damping of the divergence free magnetic field b e then take the limit as e 0 to obtain the solutions of the system with solenoidal ohm s law

unified algorithms for online learning and competitive analysis online learning and competitive analysis are two widely studied frameworks for online decision making settings despite the frequent similarity of the problems they study there are significant differences in their assumptions goals and techniques hindering a unified analysis and richer interplay between the two in this paper we provide several contributions in this direction we provide a single unified algorithm which by parameter tuning interpolates between optimal regret for learning from experts in online learning and optimal competitive ratio for the metrical task systems problem mts in competitive analysis improving upon previous results the algorithm also allows us to obtain new regret bounds against drifting experts which might be of independent interest moreover our approach allows us to go beyond experts mts obtaining similar unifying results for structured action sets and combinatorial experts whenever the setting has a certain matroid structure

more on randomized on line algorithms for caching we address the tradeoff between the competitive ratio and the resources used by randomized on line algorithms for caching two algorithms reported in the literature that achieve the optimal ratio hk require a lot of memory and perform extensive computation at each step on the other hand a very simple algorithm called rmark has competitive ratio 2hk 1 within a factor of 2 of the optimum a natural question that arises here is whether there is a tradeoff between simplicity and the competitive ratio in particular is it possible to achieve a competitive ratio better than 2hk 1 with a simple algorithm like rmark we first consider marking algorithms that are natural generalizations of rmark and we prove that for any 0 there is no randomized marking algorithm for caching with competitive ratio 2 hk thus rmark is essentially optimal among marking algorithms another model of simple caching algorithms is that of trackless algorithms these are algorithms that do not store any information about items that are not in the cache it is known that for k 2 there is no randomized trackless algorithm for caching with ratio better than 3724 1 5416 the trivial upper bound is 2 achieved even by deterministic algorithms lru and fifo we reduce this gap by giving a trackless randomized algorithm with competitive ratio 14 3 13 1 6514

robust height estimation of moving objects from uncalibrated videos this paper presents an approach for video metrology from videos acquired by an uncalibrated stationary camera we first recover the vanishing line and the vertical point of the scene based upon tracking moving objects that primarily lie on a ground plane using geometric properties of moving objects a probabilistic model is constructed for simultaneously grouping trajectories and estimating vanishing points then we apply a single view mensuration algorithm to each of the frames to obtain height measurements we finally fuse the multiframe measurements using the least median of squares lmeds as a robust cost function and the robbins monro stochastic approximation rmsa technique this method enables less human supervision more flexibility and improved robustness from the uncertainty analysis we conclude that the method with auto calibration is robust in practice results are shown based upon realistic tracking data from a variety of scenes

solutions of navier stokes maxwell systems in large energy spaces large weak solutions to navier stokes maxwell systems are not known to exist in their corresponding energy space in full generality here we mainly focus on the three dimensional setting of a classical incompressible navier stokes maxwell system and in an effort to build solutions in the largest possible functional spaces prove that global solutions exist under the assumption that the initial velocity and electromagnetic fields have finite energy and that the initial electromagnetic field is small in dot h s left mathbb r 3 right with s in left frac 12 frac 32 right we also apply our method to improve known results in two dimensions by providing uniform estimates as the speed of light tends to infinity the method of proof relies on refined energy estimates and a gr onwall like argument along with a new maximal estimate on the heat flow in besov spaces the latter parabolic estimate allows us to bypass the use of the so called chemin lerner spaces altogether which is crucial and could be of independent interest

on list update and work function algorithms the list update problem a well studied problem in dynamic data structures can be described abstractly as a metrical task system in this paper we prove that a generic metrical task system algorithm called the work function algorithm has constant competitive ratio for list update in the process we present a new formulation of the well known list factoring technique in terms of a partial order on the elements of the list this approach leads to a new simple proof that a large class of online algorithms including move to front is 2 1 k competitive for k the list length

extending lipschitz functions via random metric partitions many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture inevitably the difficulty of such a procedure lies at the local boundary where overlapping views of the same locality must somehow be merged it is therefore desirable that the boundaries be smooth allowing a graceful transition from one viewpoint to the next for instance one may point to whitney s use of partitions of unity in studying what is now known as the whitney extension problem 36 37 in the present work we consider what is perhaps the most basic whitney type extension problem that of extending a lipschitz function so that it remains lipschitz often such a map is extended by first producing a cover of the new domain extending the mapping locally and then gluing together the individual pieces our main observation is that in many cases if one chooses a random cover from the right distribution the boundary can be made smooth on average even when the local maps are individually quite coarse this insight leads to the unification generalization and improvement of many known results as well as to new results for many interesting spaces

an alternative proof for the equivalence of searcher and 2 searcher it was conjectured in 1992 that the 2 searcher who has two flashlights has the same searching capability as the searcher who has an omni directional light source park et al proved this conjecture affirmatively in 2001 but their proof is rather complicated and not easy to understand in this paper we present an alternative proof which we believe is conceptually more transparent and easier to understand for this purpose we introduce a tool called the 2 link visibility diagram that represents 2 link visibility which has other applications

piecemeal graph exploration by a mobile robot we study how a mobile robot can learn an unknown environment in a piecemeal manner the robot s goal is to learn a complete map of its environment while satisfying the constraint that it must return every so often to its starting position for refueling say the environment is modeled as an arbitrary undirected graph which is initially unknown to the robot we assume that the robot can distinguish vertices and edges that it has already explored we present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph g v e in which the robot explores every vertex and edge in the graph by traversing at most o e v1 o 1 edges this nearly linear algorithm improves on the best previous algorithm in which the robot traverses at most o e v2 edges we also give an application of piecemeal learning to the problem of searching a graph for a treasure

the polygon exploration problem we present an on line strategy that enables a mobile robot with vision to explore an unknown simple polygon we prove that the resulting tour is less than 26 5 times as long as the shortest watchman tour that could be computed off line our analysis is doubly founded on a novel geometric structure called angle hull let d be a connected region inside a simple polygon p we define the angle hull of d cal ah d to be the set of all points in p that can see two points of d at a right angle we show that the perimeter of cal ah d cannot exceed in length the perimeter of d by more than a factor of 2 this upper bound is tight

a randomized algorithm for two servers on the line abstract in the k server problem we wish to minimize in an online fashion the movement cost of k servers in response to a sequence of requests for two servers it is known that the optimal deterministic algorithm has competitive ratio 2 and it has been a long standing open problem whether it is possible to improve this ratio using randomization we give a positive answer to this problem when the underlying metric space is a real line by providing a randomized online algorithm for this case with competitive ratio at most 155 78 1 987 this is the first algorithm for two servers that achieves a competitive ratio smaller than 2 in a nonuniform metric space with more than three points we consider a more general problem called the k l server problem in which a request is served using l out of k available servers we show that the randomized 2 server problem can be reduced to the deterministic 2l l server problem we prove a lower bound of 2 on the competitive ratio of the 4 2 server problem this implies that one unbiased random bit is not sufficient to improve the ratio of 2 for the two server problem then we give a 155 78 competitive algorithm for the 6 3 server problem on the real line our algorithm is simple and memoryless the solution has been obtained using linear programming techniques that may have applications for other online problems

a derivation of the magnetohydrodynamic system from navier stokes maxwell systems we provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system mhd as the asymptotic limit of navier stokes maxwell systems when the speed of light is infinitely large we work in the physical setting provided by the natural energy bounds and therefore mainly consider leray solutions of fluid dynamical systems our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term the lorentz force neither assuming any strong compactness of the components nor applying standard compensated compactness methods which actually fail in this case

unfair problems and randomized algorithms for metrical task systems borodin linial and saks introduced a general model for online systems calledmetrical task systems 1992 j assoc comput mach 39 4 745 763 in this paper the unfair two state problem a natural generalization of the two state metrical task system problem is studied a randomized algorithm for this problem is presented and it is shown that this algorithm is optimal using the analysis of the unfair two state problem a proof of a decomposition theorem similar to that of blum karloff rabani and saks 1992 proc 33rd symposium on foundations of computer science pp 197 207 is presented this theorem allows one to design divide and conquer algorithms for specific metrical task systems our theorem gives the same bounds asymptotically but it has less restrictive boundary conditions

optimality and competitiveness of exploring polygons by mobile robots a mobile robot represented by a point moving along a polygonal line in the plane has to explore an unknown polygon and return to the starting point the robot has a sensing area which can be a circle or a square centered at the robot this area shifts while the robot moves inside the polygon and at each point of its trajectory the robot sees explores all points for which the segment between the robot and the point is contained in the polygon and in the sensing area we focus on two tasks exploring the entire polygon and exploring only its boundary we consider several scenarios both shapes of the sensing area and the manhattan and the euclidean metrics we focus on two quality benchmarks for exploration performance optimality the length of the trajectory of the robot is equal to that of the optimal robot knowing the polygon and competitiveness the length of the trajectory of the robot is at most a constant multiple of that of the optimal robot knowing the polygon most of our results concern rectilinear polygons we show that optimal exploration is possible in only one scenario that of exploring the boundary by a robot with square sensing area starting at the boundary and using the manhattan metric for this case we give an optimal exploration algorithm and in all other scenarios we prove impossibility of optimal exploration for competitiveness the situation is more optimistic we show a competitive exploration algorithm for rectilinear polygons whenever the sensing area is a square for both tasks regardless of the metric and of the starting point finally we show a competitive exploration algorithm for arbitrary convex polygons for both shapes of the sensing area regardless of the metric and of the starting point

trees and ultrametric spaces a categorical equivalence abstract there is a well known correspondence between infinite trees and ultrametric spaces that comes from considering the end space of the tree the correspondence is interpreted here as an equivalence between two categories one of which encodes the geometry of trees at infinity and the other encodes the micro geometry of complete ultrametric spaces

on metric ramsey type phenomena the main question studied in this article may be viewed as a nonlinear analogue of dvoretzky s theorem in banach space theory or as part of ramsey theory in combinatorics given a finite metric space on n points we seek its subspace of largest cardinality which can be embedded with a given distortion in hilbert space we provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion our main theorem states that for any 0 every n point metric space contains a subset of size at least n1 which is embeddable in hilbert space with o log 1 distortion the bound on the distortion is tight up to the log 1 factor we further include a comprehensive study of various other aspects of this problem

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

on the k server conjecture we prove that the work function algorithm for the k server problem has a competitive ratio at most 2 k 1 manasse et al 1988 conjectured that the competitive ratio for the k server problem is exactly k it is trivially at least k previously the best known upper bound was exponential in k our proof involves three crucial ingredients a quasiconvexity property of work functions a duality lemma that uses quasiconvexity to characterize the configuration that achieve maximum increase of the work function and a potential function that exploits the duality lemma

a lower bound on the competitivity of memoryless algorithms for a generalization of the cnn problem we consider the cnn problem in arbitrary dimension and over any metric space containing the integers we prove that in every dimension at least 2 no memoryless online algorithm can achieve a constant competitive ratio under a weak symmetry constraint on the algorithm this generalizes in several aspects the lower bounds obtained by koutsoupias and taylor the cnn problem and other k server variants theoret comput sci 324 2004 347 359 for the original problem the proof consists in the analysis of carefully selected random walks which appear naturally in the framework of memoryless algorithms

quantitative geometry rather than describing a traditional field of mathematics quantitative geometry refers to a modern way of thinking about geometry and its applications that occur in a wide range of mathematical disciplines the overarching theme here is that certain geometric problems can only be formulated after the introduction of auxiliary quantitative parameters and asking for their asymptotic or approximate behavior this leads to meaningful questions that do not necessarily have counterparts in classical qualitative geometric investigations a viewpoint that allows one to uncover deep and useful structural information that appears only in the quantitative regime quantitative reasoning allows one to tame complicated objects and phenomena that are unwieldy if one insists on exact measurements it is often the case that a rich yet structured picture emerges only if one asks the correct questions by allowing for controlled errors

optimal graph exploration without good maps a robot has to visit all nodes and traverse all edges of an unknown undirected connected graph using as few edge traversals as possible the quality of an exploration algorithm a is measured by comparing its cost number of edge traversals to that of the optimal algorithm having full knowledge of the graph the ratio between these costs maximized over all starting nodes in the graph and over all graphs in a given class u is called the overhead of algorithm a for the class u of graphs we consider three scenarios providing the robot with varying amount of information the robot may either know nothing about the explored graph or have an unlabeled isomorphic copy of it an unanchored map or have such a copy with a marked starting node an anchored map for all of the above scenarios we construct natural exploration algorithms that have smallest or in one case close to smallest overhead while for the class of all graphs depth first search turns out to be an optimal algorithm for all scenarios the situation for trees is much different we show that under the scenario without any knowledge dfs is still optimal for trees but this is not the case if a map is available under the scenario with an unanchored map we show that optimal overhead is at least 3 but strictly below 2 and thus dfs is not optimal under the scenario with an anchored map we construct an optimal algorithm for trees and show that its overhead is 3 2 we also consider exploration of the class of lines simple paths in this case depth first search remains optimal for the scenario without any knowledge with overhead 2 under the scenario with an unanchored map we construct an optimal algorithm and show that its overhead is 3 finally under the scenario with an anchored map we construct an optimal algorithm and show that its overhead is 7 5 an important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms

a decomposition theorem for task systems and bounds for randomized server problems a lower bound of omega sqrt log k log log k is proved for the competitive ratio of randomized algorithms for the k server problem against an oblivious adversary the bound holds for arbitrary metric spaces having at least k 1 points and provides a new lower bound for the metrical task system problem as well this improves the previous best lower bound of omega log log k for arbitrary metric spaces h j karloff y rabani and y ravid siam j comput 23 1994 pp 293 312 and more closely approaches the conjectured lower bound of omega log k for the server problem on k 1 equally spaced points on a line which corresponds to a natural motion planning problem a lower bound of omega frac log k log log k is obtained the results are deduced from a general decomposition theorem for a simpler version of both the k server and the metrical task system problems called the pursuit evasion game it is shown that if a metric space cal m can be decomposed into two spaces cal m l and cal m r such that the distance between them is sufficiently large compared to their diameter then the competitive ratio for this game on cal m can be expressed nearly exactly in terms of the ratios on each of the two subspaces this yields a divide and conquer approach to bounding the competitive ratio of a space

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

the weighted 2 server problem we consider a generalization of the 2 server problem in which servers have different costs we prove that in uniform spaces a version of the work function algorithm is 5 competitive and that no better ratio is possible we also give a 5 competitive randomized memoryless algorithm for uniform spaces and a matching lower bound for arbitrary metric spaces in contrast with the non weighted case we prove that there is no memoryless randomized algorithm with finite competitive ratio we also propose a version of the problem in which a request specifies two points to be covered by the servers and the algorithm must decide which server to move to which point for this version we show a 9 competitive algorithm and we prove that no better ratio is possible

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

vc dimension of exterior visibility in this paper we study the vapnik chervonenkis vc dimension of set systems arising in 2d polygonal and 3d polyhedral configurations where a subset consists of all points visible from one camera in the past it has been shown that the vc dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes here we consider the case of exterior visibility where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary we present results for the cases of cameras lying on a circle containing a polygon vc dimension 2 or lying outside the convex hull of a polygon vc dimension 5 the main result of this paper concerns the 3d case we prove that the vc dimension is unbounded if the cameras lie on a sphere containing the polyhedron hence the term exterior visibility

ultrametric subsets with large hausdorff dimension it is shown that for every e 0 1 every compact metric space x d has a compact subset s x that embeds into an ultrametric space with distortion o 1 e and dim h s geqslant 1 varepsilon dim h x where dimh denotes hausdorff dimension the above o 1 e distortion estimate is shown to be sharp via a construction based on sequences of expander graphs

the generalized work function algorithm is competitive for the generalized 2 server problem the generalized 2 server problem is an online optimization problem where a sequence of requests has to be served at minimal cost requests arrive one by one and need to be served instantly by at least one of two servers we consider the general model where the cost function of the two servers may be different formally each server moves in its own metric space and a request consists of one point in each metric space it is served by moving one of the two servers to its request point requests have to be served without knowledge of future requests the objective is to minimize the total traveled distance the special case where both servers move on the real line is known as the cnn problem we show that the generalized work function algorithm mathrm wfa lambda is constant competitive for the generalized 2 server problem further we give an outline for a possible extension to k geqslant2 servers and discuss the applicability of our techniques and of the work function algorithm in general we co

the power of a pebble exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts many results have focused on finding efficient solutions to restricted versions of the problem in this paper we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting we model the environment by an unknown directed graph g and consider the problem of a robot exploring and mapping g the edges emanating from each vertex are numbered from 1 to d but we do not assume that the vertices of g are labeled since the robot has no way of distinguishing between vertices it has no hope of succeeding unless it is given some means of distinguishing between vertices for this reason we provide the robot with a pebble a device that it can place on a vertex and use to identify the vertex later in this paper we show 1 if the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble 2 if the robot does not know an upper bound on the number of vertices n then log log n pebbles are both necessary and sufficient in both cases our algorithms are deterministic

well posedness of the navier stokes maxwell equations we study the local and global wellposedness of a full system of magneto hydro dynamic equations the system is a coupling of the forced lorentz force incompressible navier stokes equations with the maxwell equa tions through ohm s law for the current we show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for navier stokes these solutions are global if the initial data are small enough our results not only simplify and unify the proofs for the space dimensions two and three but also refine those in 10 the main simplification comes from an a priori l 2 l estimate for solutions of the forced navier stokes equations

the generalized two server problem we consider the generalized on line two server problem in which each server moves in its own metric space requests for service arrive one by one and every request is represented by two points one in each metric space the problem is to move at every request one of the two servers to its request point such that the total distance travelled by the two servers is minimized the special case in which both metric spaces are the real line is known as the cnn problem it has been a well known open question in on line optimization if an algorithm with a constant competitive ratio exists for this problem we answer this question in the affirmative by providing a constant competitive algorithm for the generalized two server problem on any metric space the basic result in this article is a characterization of competitiveness for metrical service systems that seems much easier to use when looking for a competitive algorithm the existence of a competitive algorithm for the generalized two server problem follows rather easily from this result

ultrametric skeletons abstract we prove that for every e 0 1 there exists ce 0 with the following property if x d is a compact metric space and is a borel probability measure on x then there exists a compact subset s x that embeds into an ultrametric space with distortion o 1 e and a probability measure supported on s satisfying bd x r bd x cer 1 e for all x x and r 0 the dependence of the distortion on e is sharp we discuss an extension of this statement to multiple measures as well as how it implies talagrand s majorizing measure theorem

single view metrology we describe how 3d affine measurements may be computed from a single perspective view of a scene given only minimal geometric information determined from the image this minimal information is typically the vanishing line of a reference plane and a vanishing point for a direction not parallel to the plane it is shown that affine scene structure may then be determined from the image without knowledge of the camera s internal calibration e g focal length nor of the explicit relation between camera and world pose in particular we show how to i compute the distance between planes parallel to the reference plane up to a common scale factor s ii compute area and length ratios on any plane parallel to the reference planes iii determine the camera s location simple geometric derivations are given for these results we also develop an algebraic representation which unifies the three types of measurement and amongst other advantages permits a first order error propagation analysis to be performed associating an uncertainty with each measurement we demonstrate the technique for a variety of applications including height measurements in forensic images and 3d graphical modelling from single images

the minimum backlog problem we study the minimum backlog problem mbp this online problem arises e g in the context of sensor networks we focus on two main variants of mbp the discrete mbp is a 2 person game played on a graph g v e the player is initially located at a vertex of the graph in each time step the adversary pours a total of one unit of water into cups that are located on the vertices of the graph arbitrarily distributing the water among the cups the player then moves from her current vertex to an adjacent vertex and empties the cup at that vertex the player s objective is to minimize the backlog i e the maximum amount of water in any cup at any time the geometric mbp is a continuous time version of the mbp the cups are points in the two dimensional plane the adversary pours water continuously at a constant rate and the player moves in the plane with unit speed again the player s objective is to minimize the backlog we show that the competitive ratio of any algorithm for the mbp has a lower bound of d where d is the diameter of the graph for the discrete mbp or the diameter of the point set for the geometric mbp therefore we focus on determining a strategy for the player that guarantees a uniform upper bound on the absolute value of the backlog for the absolute value of the backlog there is a trivial lower bound of d and the deamortization analysis of dietz and sleator gives an upper bound of o d log n for n cups our main result is a tight upper bound for the geometric mbp we show that there is a strategy for the player that guarantees a backlog of o d independently of the number of cups we also study a localized version of the discrete mbp the adversary has a location within the graph and must act locally filling cups with respect to his position just as the player acts locally emptying cups with respect to her position we prove that deciding the value of this game is pspace hard

an optimal on line algorithm for k servers on trees the k server problem is investigated when the metric space is a tree for this case an on line k competitive algorithm for k servers is presented the competitiveness ratio k is optimal the algorithm is memoryless in the sense that it does not use any information from the past

online computation with advice we consider a model for online computation in which the online algorithm receives together with each request some information regarding the future referred to as advice the advice is a function defined by the online algorithm of the whole request sequence the advice provided to the online algorithm may allow an improvement in its performance compared to the classical model of complete lack of information regarding the future we are interested in the impact of such advice on the competitive ratio and in particular in the relation between the size b of the advice measured in terms of bits of information per request and the improved competitive ratio since b 0 corresponds to the classical online model and b log a where a is the algorithm s action space corresponds to the optimal offline one our model spans a spectrum of settings ranging from classical online algorithms to offline ones in this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems namely metrical task systems mts and the k server problem for mts we establish tight up to constant factors upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1 b q logn where n is the number of states in the system we prove that any randomized online algorithm for mts has competitive ratio w log n b and we present a deterministic online algorithm for mts with competitive ratio o log n b for the k server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k o 1 b for any choice of q 1 b logk

appearance modeling using a geometric transform a general transform called the geometric transform get that models the appearance inside a closed contour is proposed the proposed get is a functional of an image intensity function and a region indicator function derived from a closed contour it can be designed to combine the shape and appearance information at different resolutions and to generate models invariant to deformation articulation or occlusion by choosing appropriate functionals and region indicator functions the get unifies radon transform trace transform and a class of image warpings by varying the region indicator and the types of features used for appearance modeling five novel types of gets are introduced and applied to fingerprinting the appearance inside a contour they include the gets based on a level set shape matching feature curves and the get invariant to occlusion and a multiresolution get mrget applications of get to pedestrian identity recognition human body part segmentation and image synthesis are illustrated the proposed approach produces promising results when applied to fingerprinting the appearance of a human and body parts despite the presence of nonrigid deformations and articulated motion

competitive online approximation of the optimal search ratio how efficiently can we search an unknown environment for a goal in an unknown position how much would it help if the environment were known we answer these questions for simple polygons and for undirected graphs by providing online search strategies that are as good as the best offline search algorithms up to a constant factor for other settings we prove that no such online algorithms exist we introduce a natural measure which gives reasonable results and is more realistic than pure pessimistic competitive analysis

the 3 server problem in the plane in the k server problem we wish to minimize in an online fashion the movement cost of k servers in response to a sequence of requests we assume that k 2 the request issued at each step is specified by a point r in a given metric space m to serve this request one of the k servers must move to r it is known that if m has at least k 1 points then no online algorithm for the k server problem in m has competitive ratio smaller than k the best known upper bound on the competitive ratio in arbitrary metric spaces by koutsoupias and papadimitriou j acm 42 1995 971 is 2k 1 there are only a few special cases for which k competitive algorithms are known for k 2 when m is a tree or when m has at most k 2 points we prove that the work function algorithm is 3 competitive for the 3 server problem in the manhattan plane as a corollary we obtain a 4 243 competitive algorithm for 3 servers in the euclidean plane the best previously known competitive ratio for 3 servers in these metric spaces was 5

low mach number limit of the full compressible navier stokes maxwell system abstract this paper deals with the low mach number limit of the full compressible navier stokes maxwell system it is justified rigorously that for the well prepared initial data the solutions of the full compressible navier stokes maxwell system converge to that of the incompressible navier stokes maxwell system as the mach number tends to zero

determining the epipolar geometry and its uncertainty a review two images of a single scene object are related by the epipolar geometry which can be described by a 3 3 singular matrix called the essential matrix if images internal parameters are known or the fundamental matrix otherwise it captures all geometric information contained in two images and its determination is very important in many applications such as scene modeling and vehicle navigation this paper gives an introduction to the epipolar geometry and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty a well founded measure is proposed to compare these techniques projective reconstruction is also reviewed the software which we have developed for this review is available on the internet

how to learn an unknown environment i the rectilinear case we consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it we seek algorithms that achieve a bounded ratio of the worst case distance traversed in order to see all visible points of the environment thus creating a map divided by the optimum distance needed to verify the map if we had it in the beginning the situation is complicated by the fact that the latter off line problem the problem of optimally verifying a map is np hard although we show that there is no such competitive algorithm for general obstacle courses we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it we restrict ourselves to the rectilinear case where each side of the obstacles and the room is parallel to one of the coordinates and the robot must also move either parallel or perpendicular to the sides in a subsequent paper we will discuss the extension to polygons of general shapes we also discuss the off line problem for simple rectilinear polygons and find an optimal solution in the l 1 metric in polynomial time in the case where the entry and the exit are different points

the k resource problem in uniform metric spaces we define a natural generalization of the prominent k server problem the k resource problem it occurs in a metric space with some integer demands given at its points the demands may vary with time but the total demand may never exceed k an online algorithm has k servers at its disposal and its goal is to satisfy demands by moving servers while minimizing the cost of their transport we show asymptotically tight bounds on the competitive ratio of the k resource problem in the uniform metric space of n points we prove that the optimal competitive ratios are between min k n 1 and min k 2 n 1 for deterministic algorithms and between min h k h n 1 and min h k email protected h n 1 for randomized ones this extends known results for k server in such spaces to the more general setting of k resource

an optimal on line algorithm for metrical task system in practice almost all dynamic systems require decisions to be made on line without full knowledge of their future impact on the system a general model for the processing of sequences of tasks is introduced and a general on line decision algorithm is developed it is shown that for an important class of special cases this algorithm is optimal among all on line algorithms specifically a task system s d for processing sequences of tasks consists of a set s of states and a cost matrix d where d i j is the cost of changing from state i to state j we assume that d satisfies the triangle inequality and all diagonal entries are 0 the cost of processing a given task depends on the state of the system a schedule for a sequence t 1 t 2 t k of tasks is a sequence s 1 s 2 s k of states where s i is the state in which t i is processed the cost of a schedule is the sum of all task processing costs and the state transition costs incurred an on line scheduling algorithm is one that chooses s i only knowing t 1 t 2 t i such an algorithm is w competitive if on any input task sequence its cost is within an additive constant of w times the optimal offline schedule cost the competitive ratio w s d is the infimum w for which there is a w competitive on line scheduling algorithm for s d it is shown that w s d 2 s 1 for every task system in which d is symmetric and w s d o s 2 for every task system finally randomized on line scheduling algorithms are introduced it is shown that for the uniform task system in which d i j 1 for all i j the expected competitive ratio w s d o log s

tight bounds for undirected graph exploration with pebbles and multiple agents we study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents the vertices of the graph are unlabeled and cannot be distinguished by the agents but the edges incident to a vertex have locally distinct labels the graph is explored when all vertices have been visited by at least one agent in this setting it is known that for a single agent without pebbles log n bits of memory are necessary and sufficient to explore any graph with at most n vertices we are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph we give tight results for both questions showing that for a single agent with constant memory log log n pebbles are necessary and sufficient for exploration we further prove that using collaborating agents instead of pebbles does not help as log log n agents with constant memory each are necessary and sufficient for exploration for the upper bounds we devise an algorithm for a single agent with constant memory that explores any n vertex graph using o log log n pebbles even when n is not known a priori the algorithm terminates after polynomial time and returns to the starting vertex we further show that the algorithm can be realized with additional constant memory agents rather than pebbles implying that o log log n agents with constant memory can explore any n vertex graph for the lower bound we show that the number of agents needed for exploring any graph with at most n vertices is already log log n when we allow each agent to have at most o log n 1 e bits of memory for any e 0 our argument also implies that a single agent with sublogarithmic memory needs log log n pebbles to explore any n vertex graph

broadcast in the rendezvous model in many large distributed or mobile networks broadcast algorithms are used to update information stored at the nodes in this paper we propose a new model of communication based on rendezvous and analyze a multi hop distributed algorithm to broadcast a message in a synchronous setting in the rendezvous model two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously thus nodes u and v obtain a rendezvous at a meeting point if m is the number of meeting points the network can be modeled by a graph of n vertices and m edges at each round every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities rendezvous enable an exchange of information between the two entities we get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast we show that for any graph the expected number of rounds is between ln n and o n2 for these two bounds we prove that there exist some graphs for which the expected number of rounds is either o ln n or n2 for specific topologies additional bounds are given

on the competitive ratio of the work function algorithm for the k server problem the k server problem is one of the most fundamental online problems the problem is to schedule k mobile servers to visit a sequence of points in a metric space with minimum total mileage the k server conjecture of manasse mcgeogh and sleator states that there exists a k competitive online algorithm the conjecture has been open for over 15 years the top candidate online algorithm for settling this conjecture is the work function algorithm wfa which was shown to have competitive ratio at most 2k 1 in this paper we lend support to the conjecture that wfa is in fact k competitive by proving that it achieves this ratio in several special metric spaces the line the star and all metric spaces with k 2 points

regularity criteria for the 3d density dependent incompressible maxwellnavierstokes system this paper proves some regularity criteria for the 3d density dependent incompressible maxwellnavierstokes system which improves a recent result of kang and lee 2013

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs