on james and jordan von neumann constants of lorentz sequence spaces the nonsquare or james constant j x and the jordan von neumann constant cnj x are computed for two dimensional lorentz sequence spaces d 2 w q in the case where 2 q the jordan von neumann constant is also calculated in the case where 1 q 2

non planar square orthogonal drawing with few bend edges we investigate square orthogonal drawings of non planar graphs with vertices represented as unit grid squares we present quadratic time algorithms to construct the square orthogonal drawings of 5 graphs 6 graphs and 8 graphs such that each edge in the drawing contains at most two two and three bends respectively in particular the novel analysis method we use to split a vertex so as to build some specific propagation channels in our algorithms is an interesting technique and may be of independent interest moreover we show that the decision problem of determining whether an 8 graph has a square orthogonal drawing without edge bends is np complete

on embedding between 2d meshes of the same size mesh is one of the most commonly used interconnection networks and therefore embedding between different meshes becomes a basic embedding problem not only does an efficient embedding between meshes allow one mesh connected computing system to efficiently simulate another but it also provides a useful tool for solving other embedding problems the authors study how to embed an s sub 1 spl times t sub 1 mesh into an s sub 2 spl times t sub 2 mesh where s sub i spl les t sub i i 1 2 s sub 1 spl les t sub 1 s sub 2 t sub 2 such that the minimum dilation and congestion can be achieved first they present a lower bound on the dilations and congestions of such embeddings for different cases then they propose an embedding with dilation s sub 1 s sub 2 2 and congestion s sub 1 s sub 2 4 for the case s sub 2 spl ges s sub 2 both of which almost match the lower bound s sub 1 s sub 2 finally for the case s sub 1 s sub 2 we present an embedding which has a dilation less than or equal to 2 spl radic s sub 1

cost trade offs in graph embeddings with applications an embedding of the graph g in the graph h is a one to one association of the vertices of g with the vertices of h there are two natural measures of the cost of a graph embedding namely the dilation cost of the embedding the maximum distance in h between the images of vertices that are adjacent in g and the expansion cost of the embedding the ratio of the size of h to the size of g the main results of this paper illustrate three situations wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost the first result establishes the following there is an embedding of n node complete ternary trees in complete binary trees with dilation cost 2 and expansion cost n where log3 4 3 but any embedding of these ternary trees in binary trees that has expansion cost 2 must infinitely often have dilation cost const log log log n the second result provides a stronger but less easily stated example of the same type of tradeoff the third result concerns generic binary trees that is complete binary trees into which all n node binary trees are efficiently embeddable there is a generic binary tree into which all n node binary trees are embeddable with dilation cost o 1 and expansion cost o nc for some fixed constant c if one insists on embeddings whose dilation cost is exactly 1 then these embeddings must have expansion cost n log n 2 if one insists on embeddings whose expansion cost is 2 then these embeddings must infinitely often have dilation cost const log log log n an interesting application of the polynomial size generic binary tree in the first part of this three part result is to yield simplified proofs of several results concerning computational systems with an intrinsic notion of computation tree such as alternating and nondeterministic turing machines and context free grammars

recursive circulants and their embeddings among hypercubes we propose an interconnection structure for multicomputer networks called recursive circulant recursive circulant g n d is defined to be a circulant graph with n nodes and jumps of powers of d g n d is node symmetric and has some strong hamiltonian properties g n d has a recursive structure when n cdm 1 c d we develop a shortest path routing algorithm in g cdm d and analyze various network metrics of g cdm d such as connectivity diameter mean internode distance and visit ratio g 2m 4 whose degree is m compares favorably to the hypercube qm g 2m 4 has the maximum possible connectivity and its diameter is 3m 1 4 recursive circulants have interesting relationship with hypercubes in terms of embedding we present expansion one embeddings among recursive circulants and hypercubes and analyze the costs associated with each embedding the earlier version of this paper appeared in park and chwa proc internat symp parallel architectures algorithms and networks ispan 94 kanazawa japan december 1994 pp 73 80

a note on jordan von neumann constant and james constant abstract let x be a non trivial banach space l maligranda conjectured c nj x 1 j x 2 4 for james constant j x and von neumann jordan constant c nj x of x satit saejung gave a proof of it in 2006 in this note we show that the last step in satit saejung s proof is not valid using his proof the result should be c nj x 1 j x 2 4 j x 4 2 j x 2 2 4 on the other hand we give a new proof of c nj x 1 j x 2 4 as an application we give a relation between j x and j l p x

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

on sum coloring and sum multi coloring for restricted families of graphs we consider the sum coloring chromatic sum problem and the sum multi coloring problem for restricted families of graphs in particular we consider the graph classes of proper intersection graphs of axis parallel rectangles proper interval graphs and unit disk graphs all the above mentioned graph classes belong to a more general graph class of k 1 clawfree graphs respectively for k 4 2 5 we prove that sum coloring is np hard for penny graphs and unit square graphs which implies np hardness for unit disk graphs and proper intersection graphs of axis parallel rectangles we show a 2 approximation algorithm for unit square graphs with the assumption that the geometric representation of the graph is given for sum multi coloring we confirm that the greedy first fit coloring after ordering vertices by their demands achieves a k approximation for the preemptive version of sum multi coloring on k 1 clawfree graphs finally we study priority algorithms as a model for greedy algorithms for the sum coloring problem and the sum multi coloring problem we show various inapproximation results under several natural input representations

a model of computation for vlsi with related complexity results a new model of computation for vlsi based on the assumption that time for propagating information is at least linear in the distance is proposed while accommodating for basic laws of physics the model is designed to be general and technology independent thus from a complexity viewpoint it is especially suited for deriving lower bounds and trade offs new results for a number of problems including fan in transitive functions matrix multiplication and sorting are presented as regards upper bounds it must be noted that because of communication costs the model clearly favors regular and pipelined architectures e g systolic arrays

placement of the processors of a hypercube the authors formalize the problem of minimizing the length of the longest interprocessor wire as the problem of embedding the processors of a hypercube onto a rectangular mesh so as to minimize the length of longest wire where neighboring nodes of the mesh are taken as being at unit distance from one another and where wires are constrained to be laid out as horizontal and vertical wires the length of the wire joining nodes u and v of the mesh equals the graph theoretic distance between u and v the problem of minimizing delays due to interprocessor communication is then modeled as the problem of embedding the vertices of a hypercube onto the nodes of a mesh so as to minimize dilation two embeddings which achieve dilations that for large n are within 26 of the lower bound for square meshes and within 12 for meshes with aspect ratio 2 are presented

resource bounded immunity and simplicity revisiting the 30 years old notions of resource bounded immunity and simplicity we investigate the structural characteristics of various immunity notions strong immunity almost immunity and hyperimmunity as well as their corresponding simplicity notions we also study limited immunity and simplicity called k immunity and feasible k immunity and their simplicity notions finally we propose the k immune hypothesis as a working hypothesis that guarantees the existence of simple sets in np

the planar k means problem is np hard in the k means problem we are given a finite set s of points in m and integer k 1 and we want to find k points centers so as to minimize the sum of the square of the euclidean distance of each point in s to its nearest center we show that this well known problem is np hard even for instances in the plane answering an open question posed by dasgupta 2007 7

adherences of d0l languages abstract procedures for constructing representations of each of the finite number of sequences i e words that belong to the adherence of an arbitrary d0l language are given a procedure is then given for deciding whether or not the languages generated by an arbitrary pair of d0l systems have the same adherence from an arbitrary d0l system simpler systems are constructed which have the same adherence as the original system representations of the sequences in the adherence of these simpler systems are then constructed such sequences either have the form uv for finite strings u and v or they have a form widely discussed by salomaa wsh s h 2 s h n s where h w ws the problem of deciding equality of two sequences of the latter type was recently solved by culik and harju 1981 and their algorithm is a major tool used here for deciding the adherence equivalence of d0l languages

on infinite words obtained by iterating morphisms abstract the paper investigates infinite words and sets of them associated with dol and dtol systems main emphasis is on characterization and decidability results

the size of higman haines sets we show that for the family of church rosser languages the higman haines sets which are the sets of all scattered subwords of a given language and the sets of all words that contain some word of a given language as a scattered subword cannot be effectively constructed although both these sets are regular for any language this nicely contrasts the result on the effectiveness of the higman haines sets for the family of context free languages the non effectiveness is based on a non recursive trade off result between the language description mechanism of church rosser languages and the corresponding higman haines sets which in turn is also valid for all supersets of the language family under consideration and in particular for the family of recursively enumerable languages finally for the family of regular languages we prove an upper and a matching lower bound on the size of the higman haines sets in terms of nondeterministic finite automata

power consumption in packet radio networks in this paper we study the problem of assigning transmission ranges to the nodes of a multi hop packet radio network so as to minimize the total power consumed under the constraint that adequate power is provided to the nodes to ensure that the network is strongly connected i e each node can communicate along some path in the network to every other node such assignment of transmission ranges is called complete we also consider the problem of achieving strongly connected bounded diameter networks

separators for sphere packings and nearest neighbor graphs a collection of n balls in d dimensions forms a k ply system if no point in the space is covered by more than k balls we show that for every k ply system there is a sphere s that intersects at most o k 1 d n 1 1 d balls of and divides the remainder of into two parts those in the interior and those in the exterior of the sphere s respectively so that the larger part contains at most 1 1 d 2 n balls this bound of o k 1 d n 1 1 d is the best possible in both n and k we also present a simple randomized algorithm to find such a sphere in o n time our result implies that every k nearest neighbor graphs of n points in d dimensions has a separator of size o k 1 d n 1 1 d in conjunction with a result of koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk packing our result not only gives a new geometric proof of the planar separator theorem of lipton and tarjan but also generalizes it to higher dimensions the separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space

advances in the theory and practice of graph drawing the visualization of conceptual structures is a key component of support tools for complex applications in science and engineering foremost among the visual representations used are drawings of graphs and ordered sets in this talk we survey recent advances in the theory and practice of graph drawing specific topics include bounds and tradeoffs for drawing properties three dimensional representations methods for constraint satisfaction and experimental studies

improved approximations of crossings in graph drawings and vlsi layout areas we give improved approximations for two classical embedding problems i minimizing the number of crossings in a drawing on the plane of a bounded degree graph and ii minimizing the vlsi layout area of a graph of maximum degree four these improved algorithms can be applied to improve a variety of vlsi layout problems our results are as follows i we compute a drawing on the plane of a bounded degree graph in which the sum of the numbers of vertices and crossings is o log3 n times the optimal minimum sum this is a logarithmic factor improvement relative to the best known result ii we compute a vlsi layout of a graph of maximum degree four in a square grid whose area is o log4 n times the minimum layout area this is an o log2 n improvement over the best known long standing result

graphs that are almost binary trees this paper studies embeddings of graphs in binary trees the cost of such an embedding is the maximum distance in the binary tree between images of adjacent graph vertices several techniques for bounding the costs of such embeddings from above are derived notable among these is an algorithm for embedding any outerplanar graph in a binary tree with a cost that is within a factor of 3 of optimal a number of techniques for bounding the costs of such embeddings from below are developed notable here are two techniques for inferring the presence of large separators in graphs finally a number of characterizations are established of those families of graphs that are almost binary trees in the sense that every graph in the family is embeddable in a binary tree within bounded cost

some properties concerning milman s moduli abstract in this paper some properties concerning milman s moduli j t x and s t x including equivalent representations upper and lower bounds as well as relationship between some geometric properties and these two moduli are studied in addition some sufficient conditions for the fixed point property are presented

liar s domination in unit disk graphs abstract in this article we study a variant of the minimum dominating set problem known as the minimum liar s dominating set mlds problem we prove that the mlds problem is np hard in unit disk graphs we point out that the approximation guarantee of 11 2 for the approximation algorithm for unit disk graphs proposed by banerjee and bhore 2019 11 is not correct and we propose a simple o n m time 7 31 factor approximation algorithm where n and m are the number of vertices and edges respectively in the given unit disk graph finally we prove that the mlds problem admits a polynomial time approximation scheme

mapping cycles and trees on wrap around butterfly graphs we give a new algebraic representation for the wrap around butterfly interconnection network this new representation is based on the direct product of groups and finite fields and allows an algebraic expression of the network connectivity the abstract algebraic tools may then be employed to explore the structural properties of the butterfly in this paper we exploit this model to map guest graphs on the butterfly in particular we provide designs of unit dilation mappings of all possible length cycles on butterflies we also map the largest possible binary trees on butterfly networks with a dilation 2 if the network degree is less than 16 3 if it is less than 32 and 4 if it is less than 64 this is a great improvement over previous results

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

on a new geometric constant related to the von neumann jordan constant abstract the von neumann jordan constant c nj x is computed for x being l 2 l 1 and l l 1 space by introducing a new geometric constant x t these partly give an answer to an open question posed by kato et al some basic properties of this new coefficient are investigated moreover we obtain a new class of banach spaces with uniform normal structure

finding the minimum bandwidth of an interval graph abstract an assignment of unique integers to the vertices of a graph is called a linear layout the bandwidth minimization problem bandwidth is the following given a graph g v e and an integer k determine whether there exists a linear layout of g such that the maximum difference between adjacent vertices is bounded by k interval graphs are the intersection graphs of a family of intervals of the real line bandwidth remains np complete even when restricted to special subclasses of trees we show that bandwidth can be solved in time o n2 for interval graphs moreover for a given interval graph a linear layout with minimum bandwidth can be constructed in time o n2logn as a by product we get that this construction can be done for proper interval graphs in time o nlog n m

many to one embeddings from grids into cylinders tori and hypercubes we describe novel methods for embedding 2 dimensional grid graphs into cylinders one way wrap around grids tori and hypercubes where the guest grid g is larger than the host graph h implying a many to one embedding we call lceil mid g mid mid h mid rceil the optimal load denoted l we consider optimal embeddings with respect to dilation the stretching of guest edges and load i e edges are mapped to edges or onto one node and the number of grid nodes mapped onto any hypercube node is not greater than l we show by construction that for loads of at least 4 optimal embeddings into the hypercube always exist subject only to modest restrictions on the relative dimensions of guest and host if the problem instances are grouped by grid height the restrictions imply that only some finite number of instances in each group may not be solvable by the given methods the essence of the method is a mapping from grids into cylinders of height at least one half of but not greater than the grid height and so it can also be used to construct embeddings into cylinders and tori previous work has gone so far as to show that if the optimal load l is a power of 2 then dilation 1 load l 1 embeddings into the hypercube can be constructed optimal results for loads 2 and 3 are not known

extremal structure of absolute normalized norms on r2 and the james constant abstract we denote by an2 the set of all absolute normalized norms on r 2 the set an2 has a convex structure with respect to the usual operation in this paper we calculate the james constant of r 2 when is an extreme point of an2

on embedding rectangular grids in square grids the main results in this paper demonstrate that there exist pairs of integers e d for area expansion and edge dilation respectively such that any n vertex rectangular grid can be embedded into a square grid having at most en vertices in such a way that images in the square grid of vertices that are adjacent in the rectangular grid are at most distance d apart several techniques for squaring up rectangular grids are presented sample values for the parameter pair e d are e 1 2 d 15 e 1 45 d 9 e 1 8 d 3 note that these values of e and d hold for all rectangular grids independent of number of vertices the quest for these results was motivated by the question of whether or not one could automatically square up circuit layouts having aspect ratios very far from unity without compromising the efficiency of the layout in terms of area and length of the longest run of wire the results reported here yield an affirmative answer to this question at least in an idealized setting one corollary of the embeddings presented here is that the side 2n square king s move grid contains as a subgraph every n vertex rectangular grid another way to think of this result is that this embellished grid can be programmed or personalized by setting switches to represent any n vertex rectangular grid

on james and von neumann jordan constants and sufficient conditions for the fixed point property abstract in this paper we prove that a banach space x and its dual space x have uniform normal structure if c nj x 1 3 2 the garcia falset coefficient r x is estimated by the c nj x constant and the weak orthogonality coefficient introduced by b sims finally we present an affirmative answer to a conjecture by l maligranda concerning the relation between the james and c nj x constants for a banach space

the ptolemy and zbăganu constants of normed spaces abstract in every inner product space h the ptolemy inequality holds the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides in other words x y z w x z y w z y x w for any points w x y z in h it is known that for each normed space x there exists a constant c such that for any w x y z x we have x y z w c x z y w z y x w the smallest such c is called the ptolemy constant of x and is denoted by c p x we study the relationships between this constant and the geometry of the space x and hence with metric fixed point theory in particular we relate the ptolemy constant c p to the zb ganu constant c z and prove that if x is a banach space with c z x 1 3 2 then x has uniform normal structure and therefore the fixed point property for nonexpansive mappings we derive general lower and upper bounds for both c p and c z and calculate the precise values of these two constants for several normed spaces we also present a number of conjectures and open problems

on infinite words determined by l systems abstract a deterministic l system generates an infinite word if each word in its derivation sequence is a prefix of the next yielding as a limit we generalize this notion to arbitrary l systems via the concept of prefix languages a prefix language is a language l such that for all x y l x is a prefix of y or y is a prefix of x every infinite prefix language determines a single infinite word where c is a class of l systems e g 0l dt0l we denote by c the class of infinite words determined by the prefix languages in c this allows us to speak of infinite 0l words infinite dt0l words etc we categorize the infinite words determined by a variety of l systems showing that the whole hierarchy collapses to just three distinct classes of infinite words pd0l d0l and cd0l our results are obtained with the help of a pumping lemma which we prove for t0l systems

the fixed point property in banach spaces with the nus property abstract in this paper we show that the weak nearly uniform smooth banach spaces have the fixed point property for nonexpansive mappings

universality considerations in vlsi circuits the problem of embedding the interconnection pattern of a circuit into a two dimensional surface of minimal area is discussed since even for some natural patterns graphs containing m connections may require m2 area in order to achieve compact embeddings restricted classes of graphs have to be considered for example arbitrary trees of bounded degree can be embedded in linear area without edges crossing over planar graphs can be embedded efficiently only if crossovers are allowed in the embedding

a fast distributed shortest path algorithm for a class of hierarchically clustered data networks a distributed algorithm is presented that can be used to solve the single destination shortest path sdsp problem or the all pairs shortest path apsp problem for a class of clustered data networks the network graph is assumed to be characterized with a balanced hierarchically clustered bhc topology the bhc topology is introduced in this paper and is shown to be a realistic characterization for a large class of interconnected data networks for certain types of bhc topologies the sdsp problem can be solved with computation and communication time complexities of o log n assuming one processor is available at each of the n number of nodes assuming p processors are available at each node computation and communication time complexities of o n p log n and o n log n are achievable respectively for solving the apsp problem it is also shown that the algorithm converges in an asynchronous environment

efficient embeddings of trees in hypercubes the boolean hypercube is a particularly versatile network for parallel computing it is well known that multi dimensional grid machines can be simulated on a hypercube with no communications overhead in this paper we show that every bounded degree tree can be simulated on the hypercube with constant communications overhead our proof in fact shows that every bounded degree graph with an 0 1 separator can be embedded in a hypercube of the same size with dilation and congestion both 0 1 we prove also that not all bounded degree graphs can be efficiently embedded within the hypercube

a ptas for the minimum weight connected vertex cover p 3 problem on unit disk graphs let g v e be a weighted graph i e with a vertex weight function w v r we study the problem of determining a minimum weight connected subgraph of g that has at least one vertex in common with all paths of length two in g it is known that this problem is np hard for general graphs we first show that it remains np hard when restricted to unit disk graphs our main contribution is a polynomial time approximation scheme for this problem if we assume that the problem is c local and the unit disk graphs have minimum degree of at least two

embedding complete binary trees into butterfly networks the authors present embeddings of complete binary trees into butterfly networks with or without wrap around connections let m be an even integer and q m log m 1 the authors show how to embed a 2 sup q 1 1 node complete binary tree t q into a m 1 2 sup m 1 node wrap around butterfly b sub w m 1 with a dilation of 4 and how to embed t q into a m 2 2 sup m 2 node wrap around butterfly b sub w m 2 with an optimal dilation of 2 they also present an embedding of a wrap around butterfly b sub w m into a m 1 2 sup m node no wrap around butterfly b m with a dilation of 3 using this embedding it is shown that t q can be embedded into a no wrap butterfly b m 1 resp b m 2 with a dilation of 8 resp 5

optimal dynamic embedding of x trees into arrays abstract we present a dynamic embedding of the contents of a storage medium organized as an x tree into a storage medium organized as a d dimensional array let x be a multihead bounded activity machine with a single storage tape organized as an x tree a multihead bounded activity machine d with a single tape organized as a d dimensional array is used to simulate x on line if x runs in time t n on inputs of length n then d simulates x in time o t n 1 1 d log t n furthermore this simulation is optimal

a polynomial algorithm for the min cut linear arrangement of trees an algorithm is presented that finds a min cut linear arrangement of a tree in o n log n time an extension of the algorithm determines the number of pebbles needed to play the black and white pebble game on a tree

embedding meshes and tori on double loop networks of the same size double loop networks are extensions of ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures however embedding of other types of networks on double loop networks has not been well studied due to the topological complexity of double loop networks the traditional l shape designed to compute the diameter of double loop networks is not effective to solve the embedding problem we propose a novel tessellation approach to partition the geometric plane of double loop networks into a set of parallelogram tiles called p shape based on the characteristics of p shape we design a simple embedding scheme namely p shape embedding that embeds meshes and tori on double loop networks in a systematic way under p shape embedding we evaluate the embedding metrics of dilation average dilation and congestion which depend heavily on the parameters of p shape a main merit of p shape embedding is that a large fraction of embedded mesh torus edges have edge dilation 1 resulting in a low average dilation these are the first results to our knowledge for embedding meshes and tori on double loop networks which is of great significance due to the popularity of these architectures our p shape construction bridges between regular graphs and double loop networks and provides a powerful tool for studying double loop networks

a linear time algorithm for embedding hypercube into cylinder and torus in this paper we solve two conjectures proposed by manuel et al 2011 7 to obtain exact wirelength of embedding an r dimensional hypercube into cylinder c 2 r 1xp 2 r 2 and torus c 2 r 1xc 2 r 2 where r 1 r 2 r and r email protected r 2 we provide a linear time algorithm to compute the exact wirelength of embedding hypercube into cylinder and torus further we extend the result for higher dimensional cylinder and torus

spirality and optimal orthogonal drawings we deal with the problem of constructing the orthogonal drawing of a graph with the minimum number of bends along the edges the problem has been recently shown to be np complete in the general case in this paper we introduce and study the new concept of spirality which is a measure of how an orthogonal drawing is rolled up and develop a theory on the interplay between spirality and number of bends of orthogonal drawings we exploit this theory to present polynomial time algorithms for two significant classes of graphs series parallel graphs and 3 planar graphs series parallel graphs arise in a variety of problems such as scheduling electrical networks data flow analysis database logic programs and circuit layout also they play a central role in planarity problems furthermore drawings of 3 planar graphs are a classical field of investigation

a new pumping lemma for indexed languages with an application to infinite words we prove a pumping lemma in order to characterize the infinite words determined by indexed languages an infinite language l determines an infinite word if every string in l is a prefix of if l is regular or context free it is known that must be ultimately periodic we show that if l is an indexed language then is a morphic word i e can be generated by iterating a morphism under a coding since the other direction that every morphic word is determined by some indexed language also holds this implies that the infinite words determined by indexed languages are exactly the morphic words the pumping lemma which we use to obtain this result generalizes one recently proved for the class et0l

the computational complexity of the self avoiding walk on random lattices it is proved that finding a minimum energy n step self avoiding walk on a 2d square lattice with assigned site energies is an np complete problem however if the fraction of lowest energy sites exceeds the percolation threshold then it is argued the problem can almost always be solved in polynomial time

on the computational complexity of upward and rectilinear planarity testing a directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction and no two edges cross an undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment and no two edges cross testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures for example upward planarity is useful for the display of order diagrams and subroutine call graphs while rectilinear planarity is useful for the display of circuit schematics and entity relationship diagrams we show that upward planarity testing and rectilinear planarity testing are np complete problems we also show that it is np hard to approximate the minimum number of bends in a planar orthogonal drawing of an n vertex graph with an o n 1 epsilon error for any epsilon 0

embedding of grids into optimal hypercubes this paper addresses the following graph embedding question given a grid and the smallest hypercube with at least as many nodes as the grid has grid points how can grid points be assigned to hypercube nodes with at most one grid point per node so as to keep grid neighbors as near each other as possible in the hypercube for two dimensional grids a simple embedding strategy is introduced which ensures that grid neighbors are always mapped to hypercube nodes that are within a distance of two edges of each other moreover this embedding algorithm takes time linear to the number of grid points extending and adding further concepts an embedding strategy is formulated for d dimensional grids which ensures that grid neighbors are distanced by o d edges in the hypercube this algorithm takes o d times text the number of grid points time

embedding rectangular grids into square grids it is shown that two dimensional rectangular grids of large aspect ratio can be embedded into rectangular grids of smaller aspect ratios with small expansion and dilation in particular width can be reduced by a factor of up to two with optimal expansion i e when the host rectangle is the smallest sufficient to contain the guest and optimal dilation i e two a width reduction factor of three can be obtained with optimal expansion and dilation three in general any rectangular grid can be embedded into a square grid that is no more than unity larger on the side than the minimum possible with dilation no more than three these results improve on those previously obtained in which dilation of better than 18 could not be guaranteed they might be applicable to more complex grid embedding problems such as embedding multidimensional grids into hypercubes

banach space properties sufficient for normal structure in this paper we establish lower bounds for the weakly convergent sequence coefficient wcs x of a banach space x in terms of some well known moduli and coefficients by mean of these bounds we identify several properties of geometrical nature which imply normal structure we show that these properties are strictly more general than other previously known sufficient conditions for normal structure