on the efficiency of pairing heaps and related data structures the pairing heap is well regarded as an efficient data structure for implementing priority queue operations it is included in the gnu c library strikingly simple in design the pairing heap data structure nonetheless seems difficult to analyze belonging to the genre of self adjusting data structures with its design originating as a self adjusting analogue of the fibonacci heap it has been previously conjectured that the pairing heap provides constrant amortized time decrease key operations and experimental studies have supported this conjecture this paper demonstrates contrary to conjecture that the pairing heap requires more than constant amortized time to perform decrease key operations moreover new experimental findings are presented that reveal detectable growth in the amortized cost of the decrease key operation second a unifying framework is developed that includes both pairing heaps and fibonacci heaps the parameter of interest in this framework is the storage capacity available in the nodes of the data structure for auxiliary balance information fields in this respect fibonacci heaps require log log n bits per node when n items are present this is shown to be asymptotically optimal for data structures that achieve the same asymptotic performance bounds as fibonacci heaps and fall within this framework

approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference a feedback vertex set of an undirected graph is a subset of vertices that intersects with the vertex set of each cycle in the graph given an undirected graph g with n vertices and weights on its vertices polynomial time algorithms are provided for approximating the problem of finding a feedback vertex set of g with smallest weight when the weights of all vertices in g are equal the performance ratio attained by these algorithms is 4 2 n this improves a previous algorithm which achieved an approximation factor of o sqrt log n for this case for general vertex weights the performance ratio becomes min 2 delta 2 4 log 2 n where delta denotes the maximum degree in g for the special case of planar graphs this ratio is reduced to 10 an interesting special case of weighted graphs where a performance ratio of 4 2 n is achieved is the one where a prescribed subset of the vertices so called blackout vertices is not allowed to participate in any feedback vertex set it is shown how these algorithms can improve the search performance for constraint satisfaction problems an application in the area of bayesian inference of graphs with blackout vertices is also presented

global solutions and blow up phenomena to a shallow water equation abstract a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated the local well posedness of solutions for the nonlinear equation in the sobolev space h s r with s 3 2 is developed provided that 1 x 2 u 0 does not change sign u 0 h s s 3 2 and u 0 l 1 r the existence and uniqueness of the global solutions to the equation are shown to be true in u t x c 0 h s r c 1 0 h s 1 r conditions that lead to the development of singularities in finite time for the solutions are also acquired

radially symmetric solutions to a dirichlet problem involving critical exponents in this paper we answer for n 3 4 the question raised in 1 on the number of radially symmetric solutions to the boundary value problem au x xu x u x u x 4 n 2 i x g b x e rn x 0 indeed we prove that if n 3 4 then for any a 0 this problem has only finitely many radial solutions for n 3 4 5 we show that for each x 0 the set of radially symmetric solutions is bounded moreover we establish geometric properties of the branches of solutions bifurcating from zero and from infinity b given j there exists dx j such that if kx ux and k2 u2 are radial solutions to 1 1 with the property that ux and u2 have j nodal surfaces mi 0 dx j and w2 0 dx j then kx w2 0 c for n 3 4 and k 0 the boundary value problem 1 1 has finitely many radially symmetric solutions

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

long time behaviour for a class of low regularity solutions of the camassa holm equation in this paper we investigate the long time behaviour for a class of low regularity solutions of the camasssa holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks

an approximation algorithm for the l pseudoforest deletion problem abstract an l pseudoforest is a graph each of whose connected components is at most l edges removal being a tree the l pseudoforest deletion problem is to delete a vertex set p of minimum weight from a given vertex weighted graph g v e such that the remaining graph g v p is an l pseudoforest the feedback vertex set problem is a special case of the l pseudoforest deletion problem with l 0 in this paper we present a polynomial time 4l approximation algorithm for the l pseudoforest deletion problem with l 1 by using the local ratio technique when l 1 we get a better approximation ratio 2 for the problem by further analyzing the local ratio which matches the current best constant approximation factor for the feedback vertex set problem

the design of a precompensator for multivariable adaptive control a network theoretic approach a network theoretic approach to the design of a dynamic precompensator c s for a multiinput multioutput plant t s is considered the design is based on the relative degree of each element of t s specifically an efficient algorithm is presented for determining whether a given plant t s has a diagonal precompensator c s such that for almost all cases t s c s has a diagonal interactor the algorithm also finds any optimal precompensator in the sense that the total relative degree is minimal the algorithm can be easily modified to work even when a t s represented by a nonsquare matrix is given

a sufficiently fast algorithm for finding close to optimal clique trees abstract we offer an algorithm that finds a clique tree such that the size of the largest clique is at most 2 1 k where k is the size of the largest clique in a clique tree in which this size is minimized and is the approximation ratio of an approximation algorithm for the 3 way vertex cut problem when 4 3 our algorithm s complexity is o 2 4 67k n poly n and it errs by a factor of 3 67 where poly n is the running time of linear programming this algorithm is extended to find clique trees in which the state space of the largest clique is bounded when k o log n our algorithm yields a polynomial inference algorithm for bayesian networks

approximation algorithms for single source unsplittable flow in the single source unsplittable flow problem we are given a network g a source vertex s and k commodities with sinks ti and real valued demands rho i 1 leq i leq k we seek to route the demand rho i of each commodity i along a single s ti flow path so that the total flow routed across any edge e is bounded by the edge capacity ue the conceptual difficulty of this np hard problem arises from combining packing constraints due to the existence of capacities with path selection in a graph of arbitrary topology in this paper we give a generic framework which yields approximation algorithms that are simpler than those previously known and achieve significant improvements upon the approximation ratios our framework with appropriate subroutines applies to all optimization versions previously considered and unlike previous work treats in a unified manner directed and undirected graphs we provide extensions of our algorithms which yield the best possible approximation guarantees for restricted sets of demand values and an associated scheduling problem

fredholm determinants and the camassa holm hierarchy the equation of camassa and holm 2 2 is an approximate description of long waves in shallow water

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

finding the shortest paths by node combination by repeatedly combining the source node s nearest neighbor we propose a node combination nc method to implement the dijkstra s algorithm the nc algorithm finds the shortest paths with three simple iterative steps find the nearest neighbor of the source node combine that node with the source node and modify the weights on edges that connect to the nearest neighbor the nc algorithm is more comprehensible and convenient for programming as there is no need to maintain a set with the nodes distances experimental evaluations on various networks reveal that the nc algorithm is as efficient as dijkstra s algorithm as the whole process of the nc algorithm can be implemented with vectors we also show how to find the shortest paths on a weight matrix

initial boundary value problems for nonlinear dispersive wave equations in this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half line and on a finite interval subject to homogeneous dirichlet boundary conditions we first prove local well posedness of the rod equation and of the b equation for general initial data furthermore we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span in the case of finite time singularities we are able to describe the precise blow up scenario of breaking waves our approach is based on sharp extension results for functions on the half line or on a finite interval and several symmetry preserving properties of the equations under discussion

a polynomial dual simplex algorithm for the generalized circulation problem this paper presents a polynomial time dual simplex algorithm for the generalized circulation problem an efficient implementation of this algorithm is given that has a worst case running time of o m2 m nlogn logb where n is the number of nodes m is the number of arcs and b is the largest integer used to represent the rational gain factors and integral capacities in the network this running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem

relaxed most negative cycle and most positive cut canceling algorithms for minimum cost flow this paper presents two new scaling algorithms for the minimum cost network flow problem one a primal cycle canceling algorithm the other a dual cut canceling algorithm both algorithms scale a relaxed optimality parameter and create a second inner relaxation the primal algorithm uses the inner relaxation to cancel a most negative node disjoint family of cycles w r t the scaled parameter the dual algorithm uses it to cancel most positive cuts w r t the scaled parameter we show that in a network with n nodes and m arcs both algorithms need to cancel only omn objects per scaling phase furthermore we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms our algorithms have potential practical advantages compared to some other canceling algorithms as well along the way we give a comprehensive survey of cycle and cut canceling algorithms for min cost flow we also clarify the formal duality between cycles and cuts

improving space efficiency with path length prediction for finding k shortest simple paths finding mbi k shortest simple paths in a directed graph is a fundamental problem in many engineering applications most existing algorithms such as yen s algorithm and its variants have polynomial worst case time complexity but their average case running time is very high the heuristic algorithm mps can run significantly faster in practice however it requires an excessive amount of memory space in this paper we provide a new heuristic algorithm that achieves high space efficiency while maintaining similar average case running time we first propose a sidetrack representation of path with which a path can be stored in mbi o bf 1 space we then show how to categorize a candidate path as either partial or complete and restrict the number of paths added to the queue in addition we provide an empirical equation that can very accurately predict the mbi k th shortest path length provided that a much smaller number of shortest paths have been found extensive experiments prove that our algorithm can achieve an mbi o n speedup in practice over yen s algorithm in comparison with mps it runs up to three times faster and uses less space by an order of magnitude

stability of peakons for the degasperis procesi equation the degasperis procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the euler equations with the same asymptotic accuracy as that of the camassaholm equation in this paper we study the orbital stability problem of the peaked solitons to the degasperis procesi equation on the line by constructing a liapunov function we prove that the shapes of these peakon solitons are stable under small perturbations

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

a shortest augmenting path algorithm for the semi assignment problem the objective of this study is to develop a shortest augmenting path algorithm for solving the semi assignment problem and conduct an extensive computational comparison with the best alternative approaches the algorithm maintains dual feasibility and complementary slackness and works toward satisfying primal feasibility effective heuristics arc used to achieve an excellent advanced start and convergence is assured via the use of the shortest augmenting path procedure using reduced costs for arc lengths unlike other algorithms such as the primal simplex or the auction algorithm each iteration during the final phase of the procedure also known as the end game achieves one additional assignment the software implementations of our algorithm are fastest for the semi assignment problems that we tested our dense code is also faster than the best software for assignment problems in addition the algorithm has the best polynomial worst case running time bound that we have seen and the memory requirements are relatively small

on existence and uniqueness of the global weak solution for a shallow water equation abstract a shallow water equation including the camassa holm and degasperis procesi equations as special cases is investigated provided that initial value u 0 h 1 r u 0 l 1 r and 1 x 2 u 0 does not change sign the existence and uniqueness of global weak solution for the equation are established

on the camassa holm equation and a direct method of solution iii n soliton solutions in the concluding study designated iii we modify the direct bilinear transformation method for solving the camassa holm ch equation that was set down in part ii of this work we demonstrate its efficacy for finding analytic multisoliton solutions of the equation and give explicit expressions for the first few solitons it is shown that at each order n the n soliton has a non standard representation that is characterized by an extra parameter the stucture of this parameter is investigated and a procedure for constructing the general n soliton solution of the ch equation is presented

conservation laws of the camassa holm equation we use the bi hamiltonian structure of the camassa holm equation to show that its conservation laws h n m are homogeneous with respect to the scaling m lambdam moreover a direct argument is presented proving that h 1 h 2 are of local character finally simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the h n s

on solutions of the camassa holm equation the techniques that have been developed for the application of the inverse scattering transform method to the solution of the camassa holm equation principally by constantin are implemented we use this approach first to represent the known solitary wave solution in a simple parametric form and then second to obtain the general two and three soliton solutions these latter two solutions require rather extensive use of mathematical packages mathematica and maple in order to complete the construction of solutions a number of examples are presented the phase shifts evident after an interaction are found and the special limit that recovers the peakon solutions is discussed

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

faster scaling algorithms for network problems this paper presents algorithms for the assignment problem the transportation problem and the minimum cost flow problem of operations research the algorithms find a minimum cost solution yet run in time close to the best known bounds for the corresponding problems without costs for example the assignment problem equivalently minimum cost matching in a bipartite graph can be solved in o sqrt nm log nn time where n m and n denote the number of vertices number of edges and largest magnitude of a cost costs are assumed to be integral the algorithms work by scaling as in the work of goldberg and tarjan in each scaled problem an approximate optimum solution is found rather than an exact optimum

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

controllability and asymptotic stabilization of the camassa holm equation we investigate the problems of exact controllability and asymptotic stabilization of the camassa holm equation on the circle by means of a distributed control the results are global and in particular the control prevents the solution from blowing up

optimization of pearl s method of conditioning and greedy like approximation algorithms for the vertex feedback set problem abstract we show how to find a small loop cutset in a bayesian network finding such a loop cutset is the first step in the method of conditioning for inference our algorithm for finding a loop cutset called mga finds a loop cutset which is guaranteed in the worst case to contain less than twice the number of variables contained in a minimum loop cutset the algorithm is based on a reduction to the weighted vertex feedback set problem and a 2 approximation of the latter problem the complexity of mga is o m n log n where m and n are the number of edges and vertices respectively a greedy algorithm called ga for the weighted vertex feedback set problem is also analyzed and bounds on its performance are given we test mga on randomly generated graphs and find that the average ratio between the number of instances associated with the algorithm s output and the number of instances associated with an optimum solution is far better than the worst case bound

enhancement schemes for constraint processing backjumping learning and cutset decomposition abstract researchers in the areas of constraint satisfaction problems logic programming and truth maintenance systems have suggested various schemes for enhancing the performance of the backtracking algorithm this paper defines and compares the performance of three such schemes backjumping learning and cycle cutset the backjumping and the cycle cutset methods work best when the constraint graph is sparse while the learning scheme mostly benefits problem instances with dense constraint graphs an integrated strategy is described which utilizes the distinct advantages of each scheme experiments show that in hard problems the average improvement realized by the integrated scheme is 20 25 higher than any of the individual schemes

min a cyclic feedback vertex sets and min ones monotone 3 sat abstract in directed graphs we investigate the problems of finding 1 a minimum feedback vertex set also called the feedback vertex set problem or mfvs 2 a feedback vertex set inducing an acyclic graph also called the vertex 2 coloring without monochromatic cycles problem or acyclic fvs and 3 a minimum feedback vertex set inducing an acyclic graph acyclic mfvs we show that these problems are strongly related to variants of monotone 3 sat and monotone nae 3 sat where monotone means that all literals are in positive form as a consequence we deduce several np completeness results on restricted versions of these problems in particular we define the 2 choice version of an optimization problem to be its restriction where the optimum value is known to be either d or d 1 for some integer d and the problem is reduced to decide which of d or d 1 is the optimum value we show that the 2 choice versions of mfvs acyclic mfvs min ones monotone 3 sat and min ones monotone nae 3 sat are np complete the two latter problems are the variants of monotone 3 sat and respectively monotone nae 3 sat requiring that the truth assignment minimize the number of variables set to true finally we propose two classes of directed graphs for which acyclic fvs is polynomially solvable namely flow reducible graphs for which mfvs is already known to be polynomially solvable and c1p digraphs defined by an adjacency matrix with the consecutive ones property

local search is a ptas for feedback vertex set in minor free graphs abstract we show that an extremely simple local search gives a ptas for the feedback vertex set fvs problem in minor free graphs it keeps exchanging a constant number of vertices to improve the current solution until a local optimum is reached the previous ptas by fomin lokshtanov raman and saurabh 1 despite theoretical efficiency is much more complicated in the sense that it combines many advanced algorithmic tools such as contraction decomposition framework by demaine and hajiaghayi 2 courcelle s theorem 3 and the robertson and seymour decomposition 4 our main technical contribution is to show that the local optimum only differs the global optimum by a 1 factor we do so by introducing steiner points those who are not in the local and optimal solutions to the standard analysis of local search we believe that our technique is of independent interest

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

a weighted kt t free t factor algorithm for bipartite graphs for a simple bipartite graph and an integer t 2 we consider the problem of finding a minimum weight kt t free t factor which is a t factor containing no complete bipartite graph kt t as a subgraph when t 2 this problem amounts to the square free 2 factor problem in a bipartite graph for the unweighted square free 2 factor problem a combinatorial algorithm is given by hartvigsen and the weighted version of the problem is np hard for general t pap designed a combinatorial algorithm for the unweighted version and makai gave a dual integral description of kt t free t matchings for a certain case where the weight vector is vertex induced on any subgraph isomorphic to kt t for this class of weight vectors we propose a strongly polynomial algorithm to find a minimum weight kt t free t factor the algorithm adapts the unweighted algorithms of hartvigsen and pap and a primal dual approach to the minimum cost flow problem the algorithm is fully combinatorial and thus provides a dual integrality theorem which is tantamount to makai s one

personalized pagerank clustering a graph clustering algorithm based on random walks abstract graph clustering has been an essential part in many methods and thus its accuracy has a significant effect on many applications in addition exponential growth of real world graphs such as social networks biological networks and electrical circuits demands clustering algorithms with nearly linear time and space complexity in this paper we propose personalized pagerank clustering ppc that employs the inherent cluster exploratory property of random walks to reveal the clusters of a given graph we combine random walks and modularity to precisely and efficiently reveal the clusters of a graph ppc is a top down algorithm so it can reveal inherent clusters of a graph more accurately than other nearly linear approaches that are mainly bottom up it also gives a hierarchy of clusters that is useful in many applications ppc has a linear time and space complexity and has been superior to most of the available clustering algorithms on many datasets furthermore its top down approach makes it a flexible solution for clustering problems with different requirements

evolutions of the momentum density deformation tensor and the nonlocal term of the camassa holm equation abstract methods originally developed to study the finite time blow up problem of the regular solutions of the three dimensional incompressible euler equations are used to investigate the regular solutions of the camassa holm equation we obtain results on the relative behaviors of the momentum density the deformation tensor and the nonlocal term along the trajectories in terms of these behaviors we get new types of asymptotic properties of global solutions blow up criterion and blow up time estimate for local solutions more precisely certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions finite time blow up of the accumulated momentum density is necessary and sufficient for the finite time blow up of the solution an upper estimate of the blow up time and a blow up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term

simulation relations for pattern matching in directed graphs we consider the problem of finding the occurrences of a pattern tree t in a directed graph g and propose two algorithms one for preprocessing and one for searching for t in g it is assumed that

fast approximate graph partitioning algorithms we study graph partitioning problems on graphs with edge capacities and vertex weights the problems of b balanced cuts and k balanced partitions are unified into a new problem called minimum capacity rho separators a rho separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most rho times the weight of the graph we present a new and simple o log n approximation algorithm for minimum capacity rho separators which is based on spreading metrics yielding an o log n approximation algorithm both for b balanced cuts and k balanced partitions in particular this result improves the previous best known approximation factor for k balanced partitions in undirected graphs by a factor of o log k we enhance these results by presenting a version of the algorithm that obtains an o log opt approximation factor the algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity rho separator problem as an integer program we also introduce a generalization called the simultaneous separator problem where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously we extend our results to directed graphs for values of rho geq 1 2 we conclude with an efficient algorithm for computing an optimal spreading metric for rho separators this yields more efficient algorithms for computing b balanced cuts than were previously known

the primal dual method for approximation algorithms in this survey we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to np hard problems in combinatorial optimization because of parallels with the primal dual method commonly used in combinatorial optimization we call it the primal dual method for approximation algorithms we show how this technique can be used to derive approximation algorithms for a number of different problems including network design problems feedback vertex set problems and facility location problems

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

infinite propagation speed for the degasperis procesi equation abstract we prove that any nontrivial classical solution of the degasperis procesi equation will not have compact support if its initial data has this property

global existence of weak solutions for a shallow water equation a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated provided that initial value u email protected h s email protected email protected 32 u email protected l 1 r and email protected x 2 u 0 does not change sign it is shown that there exists a unique global weak solution to the equation

efficient implementation of graph algorithms using contraction the component merging problem is a new graph problem versions of this problem appear as bottlenecks in various graph algorithms a new data structure solves this problem efficiently and two special cases of the problem have even more efficient solutions based on other data structures the performance of the data structures is sped up by introducing a new algorithmic tool called packets the algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures specifically trees can be used simultaneously as a priority queue and a concatenable queue similarly f heaps support some kinds of split operations with no loss of efficiency an immediate application of the solution to the simplest version of the merging problem is an o t m n algorithm for finding minimum spanning trees in undirected graphs without using f heaps where t m n m log 2 log 2 log d n the graph has n vertices and m edges and d max m n 2 packets also improve the f heap minimum spanning tree algorithm giving the fastest algorithm currently known for this problem the efficient solutions to the merging problem and the new observation about f heaps lead to an o n t m n n log n algorithm for finding a maximum weighted matching in general graphs this settles an open problem posed by tarjan 15 p 123 where the weaker bound of o nm log n 2 m was conjectured

time version of the shortest path problem in a stochastic flow network many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network a time version of the shortest path problem thus arises to find a quickest path which sends a given amount of data from the unique source to the unique sink with minimum transmission time more specifically the capacity of each arc in the flow network is assumed to be deterministic however in many real life networks such as computer systems telecommunication systems etc the capacity of each arc should be stochastic due to failure maintenance etc such a network is named a stochastic flow network hence the minimum transmission time is not a fixed number we extend the quickest path problem to evaluating the probability that d units of data can be sent under the time constraint t such a probability is named the system reliability in particular the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time a simple algorithm is proposed to generate all d t mps and the system reliability can then be computed in terms of d t mps moreover the optimal pair of minimal paths with highest system reliability could be obtained

global weak solutions to the weakly dissipative degasperis procesi equation abstract the global weak solutions for the degasperis procesi equation with weakly dissipative term are investigated provided that u 0 h 1 r y 0 1 x 2 u 0 m r supp y 0 x 0 and supp y 0 x 0 the existence and uniqueness of global weak solutions for the equation are shown to be true in the space w l o c 1 r r l l o c r h 1 r

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

worst case analysis of set union algorithms this paper analyzes the asymptotic worst case running time of a number of variants of the well known method of path compression for maintaining a collection of disjoint sets under union we show that two one pass methods proposed by van leeuwen and van der weide are asymptotically optimal whereas several other methods including one proposed by rein and advocated by dijkstra are slower than the best methods

all pairs shortest paths in o n 2 time with high probability we present an all pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from 0 1 is o n2 in expectation and with high probability this resolves a long standing open problem the algorithm is a variant of the dynamic all pairs shortest paths algorithm of demetrescu and italiano 2006 the analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is o n2 in expectation and with high probability we also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in o log2n expected time

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

efficient approximation algorithms for bandwidth consecutive multicolorings of graphs let g be a graph in which each vertex v has a positive integer weight b v and each edge v w has a nonnegative integer weight b v w a bandwidth consecutive multicoloring simply called a b coloring of g assigns each vertex v a specified number b v of consecutive positive integers as colors of v so that for each edge v w all integers assigned to vertex v differ from all integers assigned to vertex w by more than b v w the maximum integer assigned to vertices is called the span of the coloring the b coloring problem asks to find a b coloring of a given graph g with the minimum span in the paper we present four efficient approximation algorithms for the problem which have theoretical performance guarantees for the computation time the span of a found b coloring and the approximation ratio we also obtain several upper bounds on the minimum span expressed in terms of the maximum b degrees one of which is an extension of brooks theorem on an ordinary coloring

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

a polynomial algorithm for the multicriteria cent dian location problem given a network with several weights per node and several lengths per edge we address the problem of locating a facility on the network such that the convex combinations of the center and median objective functions are minimized since we consider several weights and several lengths various objective functions should be minimized and hence we have to solve a multicriteria cent dian location problem a polynomial algorithm to characterize the efficient location point set on the network is developed furthermore this model can generalize other problems such as the multicriteria center problem and the multicriteria median problem computing time results on random planar networks considering different combinations of weights and lengths are reported which strengthen the polynomial complexity of the procedure

technical note a note on parametric analysis in linear assignment a classic application of the linear assignment problem is the assignment of people to jobs or jobs to people in this context it is interesting to measure competition for jobs and to generate a suitable list of jobs from which a person can choose the length of the list is a parameter a known list generation procedure is based on an interior point method followed by a parametric analysis we describe a more efficient procedure exploiting linear assignment theory and shortest path computations further we propose an alternative list generation procedure based on a special type of dual values for the linear assignment problem

optimal paths in weighted timed automata we consider the optimal reachability problem for a timed automaton with respect to a linear cost function which results in a weighted timed automaton our solution to this optimization problem consists of reducing it to computing parametric shortest paths in a finite weighted directed graph we call this graph a parametric sub region graph it refines the region graph a standard tool for the analysis of timed automata by adding the information which is relevant to solving the optimal teachability problem we present an algorithm to solve the optimal reachability problem for weighted timed automata that takes time exponential in o n a w max where n is the number of clocks a is the size of the clock constraints and w max is the size of the largest weight we show that this algorithm can be improved if we restrict to weighted timed automata with a single clock in case we consider a single starting state for the optimal reachability problem our approach yields an algorithm that takes exponential time only in the length of clock constraints

self adjusting heaps in this paper we explore two themes in data structure design amortized computational complexity and self adjustment we are motivated by the following observations in most applications of data structures we wish to perform not just a single operation but a sequence of operations possibly having correlated behavior by averaging the running time per operation over a worst case sequence of operations we can sometimes obtain an overall time bound much smaller than the worst case time per operation multiplied by the number of operations we call this kind of averaging amortization standard kinds of data structures such as the many varieties of balanced trees are specifically designed so that the worst case time per operation is small such efficiency is achieved by imposing an explicit structural constraint that must be maintained during updates at a cost of both running time and storage space however if amortized running time is the complexity measure of interest we can guarantee efficiency withou

on the solutions of a model equation for shallow water waves of moderate amplitude abstract this paper is concerned with the cauchy problem of a model equation for shallow water waves of moderate amplitude which was proposed by a constantin and d lannes the hydrodynamical relevance of the camassa holmand degasperis procesi equations arch ration mech anal 192 2009 165 186 first the local well posedness of the model equation is obtained in besov spaces b p r s p r 1 s max 3 2 1 1 p which generalize the sobolev spaces h s by using littlewood paley decomposition and transport equation theory second the local well posedness in critical case with s 3 2 p 2 r 1 is considered moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time finally persistence properties on strong solutions are also investigated

the periodic b equation and euler equations on the circle in this note we show that the periodic b equation can only be realized as a euler equation on the lie group diff s1 of all smooth and orientation preserving diffeomorphisms on the circle if b 2 i e for the camassa holm equation in this case the inertia operator generating the metric on diff s1 is given by a 1 x2 in contrast the degasperis procesi equation for which b 3 is not a euler equation on diff s1 for any inertia operator our result generalizes a recent result of kolev some geometric investigations on the degasperis procesi shallow water equation wave motion 46 412 419 2009

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

feedback vertex sets on restricted bipartite graphs a feedback vertex set fvs in a graph is a subset of vertices whose complement induces a forest finding a minimum fvs is np complete on bipartite graphs but tractable on convex bipartite graphs and on chordal bipartite graphs a bipartite graph is called tree convex if a tree is defined on one part of the vertices such that for every vertex in the other part its neighborhood induces a subtree when the tree is a path a triad or a star the bipartite graph is called convex bipartite triad convex bipartite or star convex bipartite respectively we show that 1 fvs is tractable on triad convex bipartite graphs 2 fvs is np complete on star convex bipartite graphs and on tree convex bipartite graphs where the maximum degree of vertices on the tree is at most three

global existence and blow up phenomena for the weakly dissipative camassa holm equation abstract this paper is concerned with global existence and blow up phenomena for the weakly dissipative camassa holm equation a new global existence result and a new blow up result for strong solutions to the equation with certain profiles are presented the obtained results improve considerable the previous results