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

logarithmic lower bounds in the cell probe model we develop a new technique for proving cell probe lower bounds on dynamic data structures this technique enables us to prove an amortized randomized omega lg n lower bound per operation for several data structural problems on n elements including partial sums dynamic connectivity among disjoint paths or a forest or a graph and several other dynamic graph problems by simple reductions such a lower bound breaks a long standing barrier of omega lg n lg lg n for any dynamic language membership problem it also establishes the optimality of several existing data structures such as sleator and tarjan s dynamic trees we also prove the first omega log b n lower bound in the external memory model without assumptions on the data structure such as the comparison model our lower bounds also give a query update trade off curve matched e g by several data structures for dynamic connectivity in graphs we also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update

on topological indices of certain interconnection networks abstract in qsar qspr study physico chemical properties and topological indices such as randic atom bond connectivity abc and geometric arithmetic ga index are used to predict the bioactivity of chemical compounds a topological index is actually designed by transforming a chemical structure into a numeric number these topological indices correlate certain physico chemical properties like boiling point stability strain energy etc of chemical compounds graph theory has found a considerable use in this area of research the topological properties of certain networks are studied recently in 13 by hayat and imran 2014 in this paper we extend this study to interconnection networks and derive analytical closed results of general randic index r g for different values of for butterfly and benes networks we also compute first zagreb abc and ga indices for these important classes of networks moreover we construct two new classes of mesh derived networks by using some basic operations of graphs on m n mesh networks and then study certain topological indices for these classes of networks

use of dynamic trees in a network simplex algorithm for the maximum flow problem goldfarb and hao 1990 have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann vertex m arc network in at mostnm pivots and o n 2 m time in this paper we describe how to extend the dynamic tree data structure of sleator and tarjan 1983 1985 to reduce the running time of this algorithm to o nm logn this bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem our extension of dynamic trees is interesting in its own right and may well have additional applications

explicitly stable fundamental measure theory models for classical density functional theory the derivation of the state of the art tensorial versions of fundamental measure theory a form of classical density functional theory for hard spheres is reexamined in the light of the recently introduced concept of global stability of the density functional based on its boundedness lutsko and lam phys rev e 98 012604 2018 2470 004510 1103 physreve 98 012604 it is shown that within the present paradigm explicit stability of the functional can be achieved only at the cost of giving up accuracy at low densities it is argued that this is an acceptable trade off since the main value of dft lies in the study of dense systems explicit calculations for a wide variety of systems show that a proposed explicitly stable functional is competitive in all ways with the popular white bear models while sharing some of their weaknesses when applied to non close packed solids

randomized algorithms for metrical task systems abstract borodin et al 1992 introduce a general model for online systems in 3 called task systems and show a deterministic algorithm which achieves a competitive ratio of 2n 1 for any metrical task system with n states we present a randomized algorithm which achieves a competitive ratio of e e 1 n 1 e 1 1 5820n 0 5820 for this same problem for the uniform metric space borodin et al present an algorithm which achieves a competitive ratio of 2hn and they show a lower bound of hn for any randomized algorithm we improve their upper bound for the uniform metric space by showing a randomized algorithm which is hn o log n competitive

on the dynamic finger conjecture for splay trees part ii the proof the following result is shown on an n node splay tree the amortized cost of an access at distance d from the preceding access is o log d 1 in addition there is an o n initialization cost the accesses include searches insertions and deletions

better analysis of binary search tree on decomposable sequences abstract in their seminal paper sleator and tarjan 1985 15 the authors conjectured that the splay tree is a dynamically optimal binary search tree bst in spite of decades of intensive research the problem remains open perhaps a more basic question which has also attracted much attention is if there exists any dynamically optimal bst algorithm one such candidate is which is a simple and intuitive bst algorithm lucas 1988 11 munro 2000 12 demaine et al 2009 6 demaine et al 2009 6 showed a novel connection between a geometric problem and the binary search tree problem related to the above conjecture however there has been little progress in solving this geometric problem too since dynamic optimality conjecture in its most general form remains elusive despite much effort researchers have studied this problem on special sequences recently chalermsook et al 2015 2 studied a type of sequences known as k decomposable sequences in this context where k parametrizes easiness of the sequence using tools from forbidden submatrix theory they showed that takes n 2 o k 2 time on this sequence and explicitly raised the question of improving this bound in this paper we show that takes o n log k time on k decomposable sequences in contrast to the previous approach ours is based on first principles one of the main ingredients of our result is a new construction of a lower bound certificate on the performance of any algorithm this certificate is constructed using the execution of and is more nuanced and possibly more flexible than the previous independent set certificate of demaine et al this result which is applicable to all sequences may be of independent interest and may lead to further progress in analyzing on k decomposable as well as general sequences

the pseudoflow algorithm a new algorithm for the maximum flow problem we introduce the pseudoflow algorithm for the maximum flow problem that employs only pseudoflows and does not generate flows explicitly the algorithm solves directly a problem equivalent to the minimum cut problem the maximum blocking cut problem once the maximum blocking cut solution is available the additional complexity required to find the respective maximum flow is o m log n a variant of the algorithm is a new parametric maximum flow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximum flow problem the pseudoflow algorithm has also a simplex variant pseudoflow simplex that can be implemented to solve the maximum flow problem one feature of the pseudoflow algorithm is that it can initialize with any pseudoflow this feature allows it to reach an optimal solution quickly when the initial pseudoflow is close to an optimal solution the complexities of the pseudoflow algorithm the pseudoflow simplex and the parametric variants of pseudoflow and pseudoflow simplex algorithms are all o mn log n on a graph with n nodes and m arcs therefore the pseudoflow simplex algorithm is the fastest simplex algorithm known for the parametric maximum flow problem the pseudoflow algorithm is also shown to solve the maximum flow problem on s t tree networks in linear time where s t tree networks are formed by joining a forest of capacitated arcs with nodes s and t adjacent to any subset of the nodes

optimal biweighted binary trees and the complexity of maintaining partial sums let a be an array the partial sum problem concerns the design of a data structure for implementing the following operations the operation update j x has the effect a j leftarrow a j x and the query operation ssum j returns the partial sum sum i 1 j a i our interest centers upon the optimal efficiency with which sequences of such operations can be performed and we derive new upper and lower bounds in the semigroup model of computation our analysis relates the optimal complexity of the partial sum problem to optimal binary trees relative to a type of weighting scheme that defines the notion of biweighted binary tree

incremental convex planarity testing an important class of planar straight line drawings of graphs are convex drawings in which all the faces are drawn as convex polygons a planar graph is said to be convex planar if it admits a convex drawing we give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components we then consider the problem of testing convex planarity in an incremental environment where a biconnected planar graph is subject to on line insertions of vertices and edges we present a data structure for the on line incremental convex planarity testing problem with the following performance where n denotes the current number of vertices of the graph strictly convex planarity testing takes o 1 worst case time insertion of vertices takes o log n worst case time insertion of edges takes o log n amortized time and the space requirement of the data structure is o n

dynamic optimality almost we present an o lg lg n competitive online binary search tree improving upon the best previous trivial competitive ratio of o lg n this is the first major progress on sleator and tarjan s dynamic optimality conjecture of 1985 that o 1 competitive binary search trees exist

optimal binary search trees abstract we consider the problem of building optimal binary search trees the binary search tree is a widely used data structure for information storage and retrieval a binary search tree t for a set of keys from a total order is a binary tree in which each node has a key value and all the keys of the left subtree are less than the key at the root and all the keys of the right subtree are greater than the key at the root this property holding recursively for the left and right subtrees of the tree t suppose we are given n keys and the probabilities of accessing each key and those occurring in the gap between two successive keys the optimal binary search tree problem is to construct a binary search tree on these n keys that minimizes the expected access time one variant of this problem is when only the gaps have nonzero access probabilities and is called the optimal alphabetic tree problem another related problem is when there is no order between the keys and there are probabilities associated only with the gaps and the objective is to build a binary tree with minimum expected weighted path length from the root this is called the huffman tree problem in this survey we assess known results on the structural properties of the optimal trees algorithms and lower bounds to construct and to verify optimal trees and heuristics to construct nearly optimal trees and other related results

lower bounds for accessing binary search trees with rotations two methods are given for obtaining lower bounds on the cost of accessing a sequence of nodes in a symmetrically ordered binary search tree in a model where rotations can be done on the tree and the entire sequence is known before accessing begins but the accesses must be done in the order given for example it can be proven that the bit reversal permutation requires theta n log n time to access in this model it is also shown that the expected cost of accessing random sequences in this model is the same as it is for the case where the tree is static

a unified access bound on comparison based dynamic dictionaries we present a dynamic comparison based search structure that supports insertions deletions and searches within the unified bound the unified bound specifies that it is quick to access an element that is near a recently accessed element more precisely if w y distinct elements have been accessed since the last access to element y and d x y denotes the rank distance between x and y among the current set of elements then the amortized cost to access element x is o minylog w y d x y 2 this property generalizes the working set and dynamic finger properties of splay trees

resolvability and fault tolerant resolvability structures of convex polytopes abstract in this paper we study resolvability and fault tolerant resolvability of convex polytopes and related geometric graphs imran et al 2010 18 raised an open problem asserting that whether or not every family of convex polytope has a constant metric dimension raza et al 2018 28 studied the fault tolerant metric dimension of certain families of convex polytopes they also concluded their study with an open problem asking whether or not every family of convex polytope has a constant fault tolerant metric dimension in this paper we provide negative answers to both of the aforementioned open problems we answer the first question by constructing a family of convex polytopes with an unbounded metric dimension by proving a result between resolvability and fault tolerant resolvability structures of a graph we show that the family of convex polytope with an unbounded metric dimension also possesses an unbounded fault tolerant resolvability structure moreover we construct three more infinite families of graphs which are closely related to convex polytopes having an unbounded metric dimension

the list update problem and the retrieval of sets we consider the list update problem under a sequence of requests of sets of items and for this problem we investigate the competitiveness features of two algorithms we prove that algorithm move set to front msf is 1 s competitive where is the size of the largest requested set and that a lower bound is roughly 2 we provide an upper bound to the msf competitive ratio by relating it to the size n of the list obtaining that the algorithm is 1 n 4 competitive in general and o n competitive if we add a not too restrictive constraint to the sizes of the requested sets

two hard spheres in a spherical pore exact analytic results in two and three dimensions the partition function and the one and two body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore the equivalent problem for hard disks is addressed too we establish a relation valid for any dimension between these partition functions second virial coefficient for inhomogeneous systems in a spherical pore and third virial coefficients for polydisperse hard spheres mixtures using the established relation we were able to evaluate the cluster integral b2 v related with the second virial coefficient for the hard disc system into a circular pore finally we analyse the behaviour of the obtained expressions near the maximum density

a study on splay trees abstract we study the dynamic optimality conjecture which predicts that splay trees are a form of universally efficient binary search tree for any access sequence we reduce this claim to a regular access bound which seems plausible and might be easier to prove this approach may be useful to establish dynamic optimality

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

biased search trees we consider the problem of storing items from a totally ordered set in a search tree so that the access time for a given item depends on a known estimate of the access frequency of the item we describe two related classes of biased search trees whose average access time is within a constant factor of the minimum and that are easy to update under insertions deletions and more radical update operations we present and analyze efficient update algorithms for biased search trees we list several applications of such trees

on the fault tolerant metric dimension of convex polytopes abstract a convex polytopes is a polytope that is also a convex set of points in the n dimensional euclidean space r n by preserving the same adjacency relation between vertices of a convex polytope its graph is constructed the metric dimension problem has been extensively studied for convex polytopes and other families of graphs in this paper we study the fault tolerant metric dimension problem for convex polytopes by using a relation between resolving sets and fault tolerant resolving sets of graphs we prove that certain infinite families of convex polytopes are the families of graphs with constant fault tolerant metric dimension we conclude the paper with some open problems

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 cost of offline binary search tree algorithms and the complexity of the request sequence in evaluating the performance of online algorithms for search trees one wants to compare them to the best offline algorithm available in this paper we lower bound the cost of an optimal offline binary search tree using the kolmogorov complexity of the request sequence we obtain several applications for this result first any offline binary search tree algorithm can be at most a constant factor away from the entropy of the process producing the request sequence second for a fraction 1 1 2 m of request sequences of length m on n items the cost of any offline algorithm is w m logn 1 third the expected cost of splay trees is within a constant factor of the expected cost of an optimal offline binary search tree algorithm in a subset of markov chains

on the edge version atom bond connectivity and geometric arithmetic indices of certain graph operations let dl g e be the degree of an edge e in line graph l g of a graph g the edge versions of atom bond connectivity abce and geometric arithmetic gae indices of g are defined as efe l g dl g e dl g f 2dl g e dl g f and gae g efe l g 2dl g e dl g f dl g e dl g f in this paper we study the abce and gae indices for joint graphs and certain graph operations

robust balancing in b trees robust balancing is a technique that substantially decreases the worst case cost of rebalancing b trees when intermixed insertions and deletions occur from theta k log k to 0 k for k insertions and deletions it requires a slight extension to the standard data structure robust b trees but it uses the same simple operations as the standard algorithm to rebalance robust and standard b tree algorithms have identical behavior and 0 k cost when only insertions occur on intermixed insertions and deletions a robust algorithm substantially decreases rebalancing cost and increases space utilization and search cost slightly this is most significant when the cost of rebalancing dominates the total cost as in applications with localized accesses and finger access points in the active localities in applications that use top down searching search cost and total cost are 0 k log k and a robust algorithm improves total cost by a large constant factor because of the larger coefficient for rebalancing than for searching this dissertation makes a thorough study of robust balancing and its impact on maintenance cost we examine the choices for rebalancing that robust b trees introduce and define new algorithms to exploit these choices we develop powerful and uniform techniques for deriving upper bounds on rebalancing cost using the bank account paradigm a method for analyzing the cost of a sequence of operations the power of these techniques is demonstrated by strong lower bounds including several tight bounds and by simple extensions which show the dependence of rebalancing cost on different types of operations the mix of insertions and deletions and the height of rebalancing further refinements show strong bounds on subasymptotic rebalancing cost in a nonempty initial tree and on rebalancing cost in b trees a b tree generalization with improved space utilization robust algorithms for b trees can significantly improve rebalancing cost and total cost even when only insertions occur we use this analysis to find the best algorithms for several applications these include some algorithms whose rebalancing cost is sublinear in the sense of having maximum cost k c for some c 1 previous work done independently by this author mehlhorn and maier and salveter considered only one robust algorithm with a linear rebalancing cost our original contributions include new algorithms for robust b trees and b trees development of the accounting techniques and their use in identifying the best algorithms

dynamic trees as search trees via euler tours applied to the network simplex algorithm thedynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges linking and splitting by deleting edges cutting while at the same time allowing reporting of certain combinations of vertex or edge values for many applications of dynamic trees values must be combined along paths for other applications values must be combined over entire trees for the latter situation an idea used originally in parallel graph algorithms to represent trees by euler tours leads to a simple implementation with a time of o logn per tree operation wheren is the number of tree vertices we apply this representation to the implementation of two versions of the network simplex algorithm resulting in a time of o logn per pivot wheren is the number of vertices in the problem network

efficient simulation of the random cluster model the simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down a number of cluster algorithms usually based on the fortuin kasteleyn representation of the potts model and suitable generalizations for continuous spin models have been used to increase simulation efficiency the first algorithm making use of this representation suggested by sweeny in 1983 has not found widespread adoption due to problems in its implementation however it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well known algorithm due to swendsen and wang here we present an efficient implementation of sweeny s approach for the random cluster model using recent algorithmic advances in dynamic connectivity algorithms

convergence of cluster and virial expansions for repulsive classical gases we study the convergence of cluster and virial expansions for systems of particles subject to positive two body interactions our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure our treatment of the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes and generalize to soft repulsions previous approaches for models with hard exclusions our main theorem holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces our virial results stated only for homogeneous single space systems rely on an approach due to ramawadh and tate the virial coefficients are computed using lagrange inversion techniques but only at the level of formal power series thereby yielding diagrammatic expressions in terms of trees rather than the doubly connected diagrams traditionally used we obtain a new criterion that strengthens for repulsive interactions the best criterion previously available proposed by groeneveld and proven by ramawadh and tate we illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii

optimal expected case planar point location point location is the problem of preprocessing a planar polygonal subdivision s of size n into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point we consider this problem from the perspective of expected query time we are given the probabilities p z that the query point lies within each cell z in s the entropy h of the resulting discrete probability distribution is the dominant term in the lower bound on the expected case query time we show that it is possible to achieve query time h o sqrt h 1 with space o n which is optimal up to lower order terms in the query time we extend this result to subdivisions with convex cells assuming a uniform query distribution within each cell in order to achieve space efficiency we introduce the concept of entropy preserving cuttings

on the sharp lower bounds of zagreb indices of graphs with given number of cut vertices abstract the first zagreb index of a graph g is the sum of the square of every vertex degree while the second zagreb index is the sum of the product of vertex degrees of each edge over all edges in our work we solve an open question about zagreb indices of graphs with given number of cut vertices the sharp lower bounds are obtained for these indices of graphs in v n k where v n k denotes the set of all n vertex graphs with k cut vertices and at least one cycle as consequences those graphs with the smallest zagreb indices are characterized

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

on the sequential access theorem and deque conjecture for splay trees we give a new simple proof for the sequential access theorem for splay trees for an n node splay tree our bound on the number of rotations is 4 5n with a smaller constant than the bound of 10 8n concluded by tarjan we extend our proof to prove the deque conjecture for output restricted deques our proofs provide additional insights into the workings of splay trees

computing the metric dimension of wheel related graphs abstract an ordered set w w 1 w k v g of vertices of g is called a resolving set or locating set for g if every vertex is uniquely determined by its vector of distances to the vertices in w a resolving set of minimum cardinality is called a basis for g and this cardinality is the metric dimension or location number of g denoted by g in this paper we study the metric dimension of certain wheel related graphs namely m level wheels an infinite class of convex polytopes and antiweb gear graphs denoted by w n m q n and awj 2 n respectively we prove that these infinite classes of wheel related graphs have unbounded metric dimension the study of an infinite class of convex polytopes generated by wheel denoted by q n also gives a negative answer to an open problem proposed by imran et al 2012 in 8 open problem is it the case that the graph of every convex polytope has constant metric dimension it is natural to ask for characterization of graphs with unbounded metric dimension

on families of convex polytopes with constant metric dimension let g be a connected graph and d x y be the distance between the vertices x and y a subset of vertices w w 1 w 2 w k is called a resolving set for g if for every two distinct vertices x y v g there is a vertex w i w such that d x w i d y w i a resolving set containing a minimum number of vertices is called a metric basis for g and the number of vertices in a metric basis is its metric dimension dim g a family g of connected graphs is a family with constant metric dimension if dim g is finite and does not depend upon the choice of g in g in this paper we study the metric dimension of some classes of convex polytopes which are obtained by the combinations of two different graph of convex polytopes it is shown that these classes of convex polytopes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes

minimal doubly resolving sets and the strong metric dimension of some convex polytopes abstract in this paper we consider two similar optimization problems on graphs the strong metric dimension problem and the problem of determining minimal doubly resolving sets we prove some properties of strong resolving sets and give an integer linear programming formulation of the strong metric dimension problem these results are used to derive explicit expressions in terms of the dimension n for the strong metric dimension of two classes of convex polytopes d n and t n on the other hand we prove that minimal doubly resolving sets of d n and t n have constant cardinality for n 7

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

computation of topological indices of certain networks abstract there are certain types of topological indices such as degree based topological indices distance based topological indices and counting related topological indices etc among degree based topological indices the so called atom bond connectivity abc geometric arithmetic ga are of vital importance these topological indices correlate certain physico chemical properties such as boiling point stability and strain energy etc of chemical compounds in this paper we compute abc 4 and ga 5 indices for certain networks like silicate chain silicate hexagonal oxide and honeycomb networks the atom bond connectivity index abc index and geometric arithmetic index ga index for oxide and chain silicate networks are also computed in this paper

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

virial coefficients of hard spheres and hard disks up to the ninth a technique for topological analysis of the ree hoover diagrams is developed with the aim to calculate the ree hoover weights up to the ninth order with moderate demands on computer storage and cpu time the ninth virial coefficients of hard spheres and disks are calculated and the lower virial coefficients are accurately recalculated the calculations require several spanning diagrams the most important spanning chains are generated by reptation other spanning diagrams by the standard metropolis monte carlo algorithm the tenth and eleventh virial coefficients for hard spheres are estimated