random sampling and greedy sparsification for matroid optimization problems random sampling is a powerful tool for gathering information about a group by considering only a small part of it we discuss some broadly applicable paradigms for using random sampling in combinatorial optimization and demonstrate the effectiveness of these paradigms for two optimization problems on matroids finding an optimum matroid basis and packing disjoint matroid bases application of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts an optimum matroid basis is typically found by agreedy algorithm that grows an independent set into an optimum basis one element at a time this continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm we simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a givenfixed basis is optimum showing that the two problems can be solved in roughly the same time another application of sampling is to packing matroid bases also known as matroid partitioning sampling reduces the number of bases that must be packed we combine sampling with a greedy packing strategy that reduces the size of the matroid together these techniques give accelerated packing algorithms we give particular attention to the problem of packing spanning trees in graphs which has applications in network reliability analysis our results can be seen as generalizing certain results from random graph theory the techniques have also been effective for other packing problems 1998 the mathematical programming society inc published by elsevier science b v

computational anisotropic hardening multiplicative elastoplasticity based on the corrector elastic logarithmic strain rate abstract in this paper we present a new computational framework for anisotropic elastoplasticity with mixed hardening which presents the following characteristics 1 it is motivated by a one dimensional rheological model where the main differences are due to geometric nonlinearities and three dimensional effects 2 it uses the lee multiplicative decomposition 3 it is valid for anisotropic yield functions 4 it is valid for any anisotropic stored energy either linear or nonlinear in logarithmic strains 5 it is valid for non moderate large elastic strains 6 it results in a six dimensional additive corrector update parallel to that of the infinitesimal theory 7 it does not explicitly employ plastic strain tensors or plastic metrics circumventing definitely the rate issue 8 the incremental plastic flow is isochoric using a simple backward euler scheme without explicitly using exponential mappings 9 no hypothesis is needed for the plastic spin in order to integrate the symmetric flow derived from the dissipation equation 10 the mandel stress tensor plays no role in the formulation 11 it yields a fully symmetric algorithmic linearization consistent with its associative nature and the principle of maximum dissipation and 12 it recovers the formulation of simo for isotropy as a particular case

cyclic plasticity using prager s translation rule and both nonlinear kinematic and isotropic hardening theory validation and algorithmic implementation abstract finite element analysis of structures under elasto plastic nonproportional cyclic loadings is useful in seismic engineering fatigue analysis and ductile fracture usual models with nonlinear stress strain curves in cyclic behavior are based on mroz multisurface plasticity bounding surface models or models derived from the armstrong frederick rule these models depart from the associative prager s rule with the main purpose of modeling aspects of cyclic nonlinear hardening in this paper we develop a model for cyclic plasticity within the framework of the associative classical plasticity theory using prager s rule accounting for anisotropic nonlinear kinematic hardening coupled with nonlinear isotropic hardening we include the validation of the theory against several uniaxial and multiaxial cyclic experiments and an efficient fully implicit radial return algorithm the parameters of the model are obtained directly by a discretization of the uniaxial stress strain behavior remarkably both the presented theory and the computational algorithm automatically recover classical bi linear plasticity and the krieg and key algorithm if the user prescribed stress strain curve is bilinear

sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs this paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network for both problems we determine the maximum and minimum weights that each edge can have so that a given path remains optimal for both problems we show how to determine these maximum and minimum values for all edges in o m k log k time where m is the number of edges in the network and k is the number of edges on the given optimal path

randomized approximation schemes for cuts and flows in capacitated graphs we describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs we give a near linear time randomized combinatorial construction that transforms any graph on n vertices into an o n log n edge graph on the same vertices whose cuts have approximately the same value as the original graph s in this new graph for example we can run the tilde o m 3 2 time maximum flow algorithm of goldberg and rao to find an s t minimum cut in tilde o n 3 2 time this corresponds to a 1 epsilon times minimum s t cut in the original graph a related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in tilde o m sqrt n time our algorithm can also be used to improve the running time of sparsest cut approximation algorithms from tilde o mn to tilde o n 2 and to accelerate several other recent cut and flow algorithms our algorithms are based on a general theorem analyzing the concent

minimum weight spanning tree construction in o log log n communication rounds we consider a simple model for overlay networks where all n processes are connected to all other processes and each message contains at most o log n bits for this model we present a distributed algorithm which constructs a minimum weight spanning tree in o log log n communication rounds where in each round any process can send a message to every other process if message size is theta n epsilon for some epsilon 0 then the number of communication rounds is o log 1 over epsilon

an optimal minimum spanning tree algorithm we establish that the algorithmic complexity of the minimumspanning tree problem is equal to its decision tree complexity specifically we present a deterministic algorithm to find aminimum spanning tree of a graph with n vertices andm edges that runs in timeo t m n wheret is the minimum number of edge weightcomparisons needed to determine the solution the algorithm isquite simple and can be implemented on a pointer machine althoughour time bound is optimal the exact function describing it is notknown at present the current best bounds known fort are t m n m and t m n o m m n where is acertain natural inverse of ackermann s function even under theassumption that t is superlinear we show thatif the input graph is selected from gn m our algorithm runs in linear time with high probability regardlessof n m or the permutation of edge weights theanalysis uses a new martingale for gn msimilar to the edge exposure martingale forgn p

verification and sensitivity analysis of minimum spanning trees in linear time komlos has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree the total computational work required by his method is much larger than linear however this paper describes a linear time algorithm for verifying a minimum spanning tree this algorithm combines the result of komlos with a preprocessing and table look up method for small subproblems and with a previously known almost linear time algorithm additionally an optimal deterministic algorithm and a linear time randomized algorithm for sensitivity analysis of minimum spanning trees are presented

a hyperelastic based large strain elasto plastic constitutive formulation with combined isotropic kinematic hardening using the logarithmic stress and strain measures summary this paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity the combined isotropic kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions i the formulation is hyperelastic based ii the stress strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the hencky strain tensor and its elastic work conjugate stress tensor and iii the multiplicative decomposition of the deformation gradient is adopted since no stress rates are present the formulation is of course numerically objective in the time integration it is shown that the model gives adequate physical behaviour and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor

implicit algorithms for multilayer j2 plasticity in this paper two implicit algorithms for multilayer deviatoric plasticity based on the closest point projection iteration are introduced the algorithmic elastoplastic tangents in order to preserve the second order convergence in newton algorithms based on converged steps are also developed an implicit mroz translation rule is formulated and serves as a basis for the construction of the traditional return mapping schemes the main differences between the proposed procedures lay on a forward update of the active surface and the construction of the return mapping only at the end of the step or each time there is a surface update the first option yields a scheme closer to that of a virtual bounding surface model and to that of explicit algorithms while the second one contributes with a mathematically more attractive and robust layout both of them allow a way to perform implicit unconditionally stable second order convergent multiaxial analysis using multilayer plasticity and they present an alternative to virtual bounding surface plasticity to which the examples are also compared

a minimum spanning tree algorithm with inverse ackermann type complexity a deterministic algorithm for computing a minimum spanning tree of a connected graph is presented its running time is 0 m m n where is the classical functional inverse of ackermann s function and n respectively m is the number of vertices respectively edges the algorithm is comparison based it uses pointers not arrays and it makes no numeric assumptions on the edge costs

approximating the minimum spanning tree weight in sublinear time we present a probabilistic algorithm that given a connected graph g represented by adjacency lists of average degree d with edge weights in the set 1 w and given a parameter 0 eps 1 2 estimates in time o dw varepsilon 2 log frac dw varepsilon the weight of the minimum spanning tree mst of g with a relative error of at most eps note that the running time does not depend on the number of vertices in g we also prove a nearly matching lower bound of omega dw varepsilon 2 on the probe and time complexity of any approximation algorithm for mst weight the essential component of our algorithm is a procedure for estimating in time o d eps 2 log frac d varepsilon the number of connected components of an unweighted graph to within an additive error of varepsilon n this becomes o eps 2 log frac 1 varepsilon for d o 1 the time bound is shown to be tight up to within the log frac d varepsilon factor our connected components algorithm picks o 1 varepsilon 2 vertices in the graph and then grows local spanning trees whose sizes are specified by a stochastic process from the local information collected in this way the algorithm is able to infer with high confidence an estimate of the number of connected components we then show how estimates on the number of components in various subgraphs of g can be used to estimate the weight of its mst

a shortest path algorithm for real weighted undirected graphs we present a new scheme for computing shortest paths on real weighted undirected graphs in the fundamental comparison addition model in an efficient preprocessing phase our algorithm creates a linear size structure that facilitates single source shortest path computations in o m log alpha time where alpha alpha m n is the very slowly growing inverse ackermann function m the number of edges and n the number of vertices as special cases our algorithm implies new bounds on both the all pairs and single source shortest paths problems we solve the all pairs problem in o mn log alpha m n time and if the ratio between the maximum and minimum edge lengths is bounded by n log n o 1 we can solve the single source problem in o m n log log n time both these results are theoretical improvements over dijkstra s algorithm which was the previous best for real weighted undirected graphs our algorithm takes the hierarchy based approach invented by thorup

on the consistency of nested surfaces models and their kinematic hardening rules abstract in this paper we briefly address the consistency of formulations for nested surfaces plasticity and their kinematic hardening translation rules some requirements for these formulations are then given it is shown and discussed that multilayer plasticity based on nested yield surfaces present some inconveniences when modelling multiaxial cyclic loading on the other hand the use of hardening surfaces instead of yield surfaces solves the problem partially it is also shown that multilayer plasticity based on the mroz kinematic rule yields clearly different multiaxial predictions when using the same uniaxial bilinear curve and different number of surfaces which is a priori inconsistent since the same monotonic stress strain curve should not result into a clearly different multiaxial behavior simply because of the discretization employed by the user it is demonstrated that in contrast with the mroz translation rule multilayer plasticity using the prager translation rule results in the same prediction regardless of the number of surfaces in which the hardening part of the bilinear curve is discretized this rule is also compliant with the principle of maximum dissipation it also allows for a vanishing elastic region without falling into theoretical or numerical difficulties hence it should not be discarded when developing advanced models

theory of minimum spanning trees ii exact graphical methods and perturbation expansion at the percolation threshold continuing the program begun by the authors in a previous paper we develop an exact low density expansion for the random minimum spanning tree mst on a finite graph and use it to develop a continuum perturbation expansion for the mst on critical percolation clusters in space dimension d the perturbation expansion is proved to be renormalizable in d 6 dimensions we consider the fractal dimension d sub p of paths on the latter mst our previous results lead us to predict that d sub p 2 for d d sub c 6 using a renormalization group approach we confirm the result for d 6 and calculate d sub p to first order in epsilon 6 d for d 6 using the connection with critical percolation with the result d sub p 2 epsilon 7 o epsilon sup 2

a randomized time work optimal parallel algorithm for finding a minimum spanning forest we present a randomized algorithm to find a minimum spanning forest msf in an undirected graph with high probability the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write erew pram this result is optimal w r t both work and parallel time and is the first provably optimal parallel algorithm for this problem under both measures we also give a simple general processor allocation scheme for tree like computations

random spanning trees and the prediction ofweighted graphs we investigate the problem of sequentially predicting the binary labels on the nodes of an arbitrary weighted graph we show that under a suitable parametrization of the problem the optimal number of prediction mistakes can be characterized up to logarithmic factors by the cutsize of a random spanning tree of the graph the cutsize is induced by the unknown adversarial labeling of the graph nodes in deriving our characterization we obtain a simple randomized algorithm achieving in expectation the optimal mistake bound on any polynomially connected weighted graph our algorithm draws a random spanning tree of the original graph and then predicts the nodes of this tree in constant expected amortized time and linear space experiments on real world data sets show that our method compares well to both global perceptron and local label propagation methods while being generally faster in practice

comments on the mroz multiple surface type plasticity models abstract several questions are addressed regarding the role of number of surfaces in the mroz type multiple surface models firstly increasing the number of surfaces improves the accuracy of the plastic modulus function but at the same time alters the translation behavior of surfaces two 90 out of phase axial torsion loading experiments are chosen to illustrate the number of surface influence on the model behavior the stresses exceed the experimental levels with increasing numbers of surfaces from 5 to 100 secondly for proportional loading the multiple surface models do not predict ratchetting progressive plastic strain accumulation in one direction however these models predict ratchetting for general nonproportional loading an ellipse shaped axial torsional loading path has been considered where predicted ratchetting rates far exceeded their experimental counter parts an explanation is forwarded to address these properties of the models it is further demonstrated that the multiple surface model of mroz and its modification by garud produce identical stress strain predictions when the number of surfaces exceeds a certain value for infinitesimal loading increment intersection of surfaces does not occur when using either the mroz or garud model however when finite loading increment is selected in numerical calculations the intersection problem arises in the mroz model

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

elliptic curves over a finite field and the trace formula we prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k points of the curves contain a chosen subgroup these formulas express the moments in terms of traces of hecke operators for certain congruence subgroups of sl 2 z as our main technical input we prove an eichler selberg trace formula for a family of congruence subgroups of sl 2 z which include as special cases the groups 1 n and n our formulas generalize results of birch and ihara the case of the trivial subgroup and the full modular group and previous work of the authors the subgroups z 2z and z 2z 2 and congruence subgroups 0 2 0 4 we use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field generalizing results of vladu gekeler howe and others

verification of minimum redundancy prefix codes we show that verifying a given prefix code for optimality requires spl omega nlogn time indicating that the verification problem is not asymptotically easier than the construction problem alternatively we give linear time verification algorithms for several special cases that are either typical in practice or theoretically interesting

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

deeper local search for parameterized and approximation algorithms for maximum internal spanning tree the maximum internal spanning tree problem asks for a spanning tree of a given graph that has the maximum number of internal vertices among all spanning trees of this graph in its parameterized version we are interested in whether the graph has a spanning tree with at least k internal vertices fomin et al 2013 4 crafted a very ingenious reduction rule and showed that a simple application of this rule is sufficient to yield a 3k vertex kernel implying an o 8 k time parameterized algorithm using depth 2 local search knauer and spoerhase 2015 9 developed a 5 3 approximation algorithm for the optimization version we try deeper local search we conduct a thorough combinatorial analysis on the obtained spanning trees and explore their algorithmic consequences we first observe that from the spanning tree obtained by depth 3 local search one can easily find a reducible structure and apply the reduction rule of fomin et al this gives an improved kernel of 2k vertices and as a by product a deterministic algorithm running in time o 4 k we then go even deeper by considering the spanning tree obtained by depth 5 local search it is shown that the number of internal vertices of this spanning tree is at least 2 3 of the maximum number a spanning tree can have thereby delivering an improved approximation algorithm with ratio 1 5 for the problem

soft heaps simplified in 1998 chazelle j acm 47 2000 pp 1012 1027 introduced a new kind of meldable heap priority queue called the soft heap soft heaps trade accuracy for speed the heap operations are allowed to increase the keys of certain items thereby making these items bad as long as the number of bad items in the data structure is at most varepsilon m where m is the total number of insertions performed so far and varepsilon is an error parameter the amortized time per heap operation is o lg frac 1 varepsilon reduced from o lg n where n is the number of items in the heap chazelle used soft heaps in several applications including a faster deterministic minimum spanning tree algorithm and a new deterministic linear time selection algorithm we give a simplified implementation of soft heaps that uses less space and avoids chazelle s dismantling operations we also give a simpler improved analysis that yields an amortized time bound of o lg frac 1 varepsilon for each deleti

random sampling in cut flow and network design problems we use random sampling as a tool for solving undirected graph problems we show that the sparse graph or skeleton that arises when we randomly sample a graph s edges will accurately approximate the value of all cuts in the original graph with high probability this makes sampling effective for problems involving cuts in graphs we present fast randomized monte carlo and las vegas algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted undirected graphs our cut approximation algorithms extend unchanged to weighted graphs while our weighted graph flow algorithms are somewhat slower our approach gives a general paradigm with potential applications to any packing problem it has since been used in a near linear time algorithm for finding minimum cuts as well as faster cut and flow algorithms our sampling theorems also yield faster algorithms for several other cut based problems including approximating the best balanced cut of a graph finding a k connected orientation of a 2k connected graph and finding integral multicommodity flows in graphs with a great deal of excess capacity our methods also improve the efficiency of some parallel cut and flow algorithms our methods also apply to the network design problem where we wish to build a network satisfying certain connectivity requirements between vertices we can purchase edges of various costs and wish to satisfy the requirements at minimum total cost since our sampling theorems apply even when the sampling probabilities are different for different edges we can apply randomized rounding to solve network design problems this gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large as a particular example we improve the best approximation bound for the minimum k connected subgraph problem from 1 85 to math not displayed

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

sparsification a technique for speeding up dynamic graph algorithms we provide data strutures that maintain a graph as edges are inserted and deleted and keep track of the following properties with the following times minimum spanning forests graph connectivity graph 2 edge connectivity and bipartiteness in time o n 1 2 per change 3 edge connectivity in time o n 2 3 per change 4 edge connectivity in time o n n per change k edge connectivity for constant k in time o n log n per change 2 vertex connectivity and 3 vertex connectivity in the o n per change and 4 vertex connectivity in time o n n per change further results speed up the insertion times to match the bounds of known partially dynamic algorithms all our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph which we call sparsification

linear time algorithms for parametric minimum spanning tree problems on planar graphs abstract a linear time algorithm for the minimum ratio spanning tree problem on planar graphs is presented the algorithm is based on a new planar minimum spanning tree algorithm the approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators

randomized obdd based graph algorithms abstract implicit graph algorithms deal with the characteristic function e of the edge set e of a graph g v e encoding the nodes by binary vectors e can be represented by an ordered binary decision diagram obdd which is a well known data structure for boolean functions obdd based graph algorithms solve graph optimization problems by mainly using functional operations and are a heuristic approach to cope with massive graphs these algorithms heavily rely on a compact representation of the underlying boolean functions which is why all previously known obdd based algorithms are deterministic since random functions are not compressible in general here the first randomized obdd based algorithms are presented where random functions with limited independence are used to overcome the large representation size on the theoretical part we investigate the size of obdds representing k wise independent random functions in addition we give a construction of almost k wise independent random functions by means of a random obdd generation and show an almost matching lower bound of the obdd size representing such random functions on the algorithmic part randomization often facilitates the design of simple algorithms which in the context of obdd based algorithms means a small number of functional operations and as few input variables of the used boolean functions as possible we present two randomized implicit algorithms a randomized minimum spanning tree algorithm and a maximal matching algorithm the matching algorithm needs o log 3 v functional operations in expectation and uses functions with at most 3 log v variables which is both better than the best known deterministic algorithms w r t functional operations and variables the algorithm may be of independent interest the experimental evaluation of the algorithms shows that they outperform known obdd based algorithms for the maximal matching problem and the minimum spanning tree problem

implicit j2 bounding surface plasticity using prager s translation rule a bounding surface j2 plasticity model that uses prager s translation rule is presented the model preserves masing s rules and is developed from the same ideas as classical infinitesimal j2 plasticity resulting in the same formulation with the exception of the algorithm for the computation of the hardening function instead of utilizing a loading surface as in a previous formulation hardening surfaces are introduced the formulation is similar to that of multilayer plasticity using prager s rule presented in previous work an implicit algorithm based on the radial return concept is used and the consistent elastoplastic tangent is also developed in closed form examples illustrating anisotropic behaviour are presented and compared to that predicted by a multilayer j2 plasticity model the model is also applied to a soil dynamics problem to show the robustness of the algorithm and its applicability to complex loading copyright 2002 john wiley sons ltd

the dfs fused lasso linear time denoising over general graphs the fused lasso also known as anisotropic total variation denoising is widely used for piecewise constant signal estimation with respect to a given undirected graph the fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure but for a special graph structure namely the chain graph the fused lasso or simply 1d fused lasso can be computed in linear time in this paper we revisit a result recently established in the online classification literature herbster et al 2009 cesa bianchi et al 2013 and show that it has important implications for signal denoising on graphs the result can be translated to our setting as follows given a general graph if we run the standard depth first search dfs traversal algorithm then the total variation of any signal over the chain graph induced by dfs is no more than twice its total variation over the original graph this result leads to several interesting theoretical and computational conclusions letting m and n denote the number of edges and nodes respectively of the graph in consideration it implies that for an underlying signal with total variation t over the graph the fused lasso properly tuned achieves a mean squared error rate of t2 3n 2 3 moreover precisely the same mean squared error rate is achieved by running the 1d fused lasso on the dfs induced chain graph importantly the latter estimator is simple and computationally cheap requiring o m operations to construct the dfs induced chain and o n operations to compute the 1d fused lasso solution over this chain further for trees that have bounded maximum degree the error rate of t2 3n 2 3 cannot be improved in the sense that it is the minimax rate for signals that have total variation t over the tree finally several related results also hold for example the analogous result holds for a roughness measure defined by the l0 norm of differences across edges in place of the total variation metric

on symbolic obdd based algorithms for the minimum spanning tree problem the minimum spanning tree problem is one of the most fundamental algorithmic graph problems and obdds are a very common dynamic data structure for boolean functions since in some applications graphs become larger and larger a research branch has emerged which is concerned with the design and analysis of so called symbolic algorithms for classical graph problems on obdd represented graph instances here a symbolic minimum spanning tree algorithm using o log 3 v functional operations is presented where v is the set of vertices of the input graph moreover the computation of the transitive closure is investigated and it is proved that there can be an exponential blow up from input to output size furthermore answering an open problem posed by sawitzki 37 it is shown that every symbolic obdd based algorithm for the minimum spanning tree problem needs exponential space with respect to the obdd size of the input graph this result even holds for planar input graphs

implicit plane stress algorithm for multilayer j2 plasticity using the prager ziegler translation rule a totally implicit algorithm for plane stress multilayer plasticity is presented the algorithm presents the plane stress version of the 3d plane strain model for multilayer plasticity presented in a previous work as in the 3d plane strain case the model is consistent with the principle of maximum dissipation and it may be considered as an extension of classical j2 plasticity for anisotropic non linear kinematic behaviour preserving masing s rules in order to obtain the asymptotic second order convergence of the newton algorithm the consistent tangent moduli are also given copyright 2003 john wiley sons ltd

pm pm patchmatch with potts model for object segmentation and stereo matching this paper presents a unified variational formulation for joint object segmentation and stereo matching which takes both accuracy and efficiency into account in our approach depth map consists of compact objects each object is represented through three different aspects 1 the perimeter in image space 2 the slanted object depth plane and 3 the planar bias which is to add an additional level of detail on top of each object plane in order to model depth variations within an object compared with traditional high quality solving methods in low level we use a convex formulation of the multilabel potts model with patchmatch stereo techniques to generate depth map at each image in object level and show that accurate multiple view reconstruction can be achieved with our formulation by means of induced homography without discretization or staircasing artifacts our model is formulated as an energy minimization that is optimized via a fast primal dual algorithm which can handle several hundred object depth segments efficiently performance evaluations in the middlebury benchmark data sets show that our method outperforms the traditional integer valued disparity strategy as well as the original patchmatch algorithm and its variants in subpixel accurate disparity estimation the proposed algorithm is also evaluated and shown to produce consistently good results for various real world data sets kitti benchmark data sets and multiview benchmark data sets

a randomized linear time algorithm to find minimum spanning trees we present a randomized linear time algorithm to find a minimum spanning tree in a connected graph with edge weights the algorithm uses random sampling in combination with a recently discovered linear time algorithm for verifying a minimum spanning tree our computational model is a unit cost random access machine with the restriction that the only operations allowed on edge weights are binary comparisons