Drawings of graphs on surfaces with few crossings

We give drawings of a complete graphK _n withO(n ⁴ log² g/g) many crossings on an orientable or nonorientable surface of genusg 2. We use these drawings ofK _n and give a polynomial-time algorithm for drawing any graph withn vertices andm edges withO(m ² log² g/g) many crossings on an orientable or nonorientable surface of genusg 2. Moreover, we derive lower bounds on the crossing number of any graph on a surface of genusg 0. The number of crossings in the drawings produced by our algorithm are within a multiplicative factor ofO(log² g) from the lower bound (and hence from the optimal) for any graph withm 8n andn ²/m g m/64.The research of the third and the fourth authors was partially supported by Grant No. 2/1138/94 of the Slovak Academy of Sciences and by EC Cooperative action IC1000 Algorithms for Future Technologies (Project ALTEC). A preliminary version of this paper was presented at WG93 and published in Lecture Notes in Computer Science, Vol. 790, 1993, pp. 388–396.     

Communication—Processor Tradeoffs in a Limited Resources PRAM

   Abstract. We consider a simple restriction of the PRAM model (called PPRAM), where the input is arbitrarily partitioned between a fixed set of p processors and the shared memory is restricted to m cells. This model allows for investigation of the tradeoffs/ bottlenecks with respect to the communication bandwidth (modeled by the shared memory size m ) and the number of processors p . The model is quite simple and allows the design of optimal algorithms without losing the effect of communication bottlenecks. We have focused on the PPRAM complexity of problems that have

Large deviations for distributions of sums of random variables: Markov chain method

Let {ξ _k } _k=0^∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} , n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| ^p , p > 0, and exponential random variables with g(x) = x for x ≥ 0.     

Efficient implementation of a generalized polak-ribière algorithm for nonlinear optimization

We have recently developed a new conjugate gradient type method, the Generalized Polak-Ribière (GPR) method, for unconstrained minimization. It is based on search directions that are parallel to the Newton direction of the restriction of the objective function f on the two dimensional subspace span{?g p}, with p a suitable direction in span{? g,s^?}, where g and s ^? are the current gradient and previous search direction respectively. The new approach proved to be considerably more efficient than the original Polak-Ribière method.

In this paper, various implementations of the GPR method are compared with a currently available standard NAG software routine and also with the Nocedal, Buckley-LeNir and Shanno's limited memory algorithms. The results demonstrate the general effectiveness of the new algorithm. We also give a very brief discussion of extensions of the GPR method that generate search directions parallel to Newton directions in subspaces of dimension greater than two.     

Global controllability of nonlinear systems in two dimensions

LetM be a connected real-analytic 2-dimensional manifold. Consider the system (t) = f(x(t)) + u(t)g(x(t)),x(0) =x ₀ M, wheref andg are real-analytic vector fields onM which are linearly independent at some point ofM, andu is a real-valued control. Sufficient conditions on the vector fieldsf andg are given so that the system is controllable fromx ₀. Suppose that every nontrivial integral curve ofg has a pointp wheref andg are linearly dependent,g(p) is nonzero, and that the Lie bracket [f,g] andg are linearly independent atp. Then the system is controllable (with the possible exception of a closed, nowhere dense set which is not reachable) from any pointx ₀ such that the vector space dimension of the Lie algebraL _A generated byf,g and successive Lie brackets is 2 atx ₀.Research supported in part by the National Science Foundation under NSF Grant MCS76-05267-A01 and by the Joint Services Electronics Program under ONR Contract 76-C-1136.     

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p ₁ , p ₂ , \ldots, p _n } be a set of n {\lilsf terminal points} in the Euclidean plane, where point p _i has a {\lilsf service request of grade} g(p _i ) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ⋅s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points p _i and p _j there is a path whose minimum grade of service is at least as large as \min(g(p _i ), g(p _j )) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+ε) -approximation to the minimum cost network under a given topology or its degeneracies in O(n ^1.5 (log n + log (1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.     

Testing Periodicity

We study the string-property of being periodic and having periodicity smaller than a given bound. Let Σ be a fixed alphabet and let p,n be integers such that p ￡ \fracn2p\leq \frac{n}{2} . A length-n string over Σ, α=(α ₁,…,α _n ), has the property Period(p) if for every i,j∈{1,…,n}, α _i =α _j whenever i≡j (mod p). For an integer parameter g ￡ \fracn2,g\leq \frac{n}{2}, the property Period(≤g) is the property of all strings that are in Period(p) for some p≤g. The property Period( ￡ \fracn2)\mathit{Period}(\leq \frac{n}{2}) is also called Periodicity.     

Entropy-Inspired Competitive Clustering Algorithms

In this paper, the well-known competitive clustering algorithm (CA) is revisited and reformulated from a point of view of entropy minimization. That is, the second term of the objective function in CA can be seen as quadratic or second-order entropy. Along this novel explanation, two generalized competitive clustering algorithms inspired by Renyi entropy and Shannon entropy, i.e. RECA and SECA, are respectively proposed in this paper.Simulation results show that CA requires a large number of initial clusters to obtain the right number of clusters, while RECA and SECA require small and moderate number of initial clusters respectively. Also the iteration steps in RECA and SECA are less than that of CA.Further CA and RECA are generalized to CA-p and RECA-p by using the p-order entropy and Renyi's p-order entropy in CA and RECA respectively. Simulation results show that the value of phas a great impact on the performance of CA-p, whereas it has little in uence on that of RECA-p.     

On the optimal number of features in the classification of multivariate Gaussian data

We consider the classification problem in the context of two equiprobable multivariate Gaussian densities with a common covariance matrix. The use of Anderson's W statistic for classification results in the existence of an optimum number of features, p_opt such that, for a given sample size, the average probability of misclassification decreases at first as the number of features are increased, attains a minimum at p_opt and then starts increasing. We have examined this peaking phenomenon for several cases and provide expressions which relate p_opt to the number of available training samples and the Mahalanobis distance between the two populations. We also show that to prevent peaking, each additional feature's contribution to the Mahalanobis distance must be a certain proportion of the accumulated Mahalanobis distance.     

Optimal hypercube simulation on the partitioned optical passive stars network

Partitioned Optimal Passive Stars network, POPS(d,g), is an optical interconnection network of N processors (N=dg) with g ² optical passive star couplers. In this network, there are g groups of d processors each and the g ² couplers are used for connecting each group with each of the groups, including itself. In this paper, we present a technique for optimally simulating a frequently arising hypercube communication pattern on this network for all combinations of values of d and g. Specifically, we show that one-hop movements on the hypercube along the same dimension can be simulated on the POPS(d,g) network in slots for d≠g and in 2 slots for d=g.     

Searching a polygonal region by a group of stationary k-searchers

We study the problem of searching for mobile intruders in a polygonal region by stationary searchers having various levels of vision given by the number of flashlights that a searcher carries. We show that (2g−1) 1-searchers (i.e., 2g−1 searchers with one flashlight each) are always sufficient, and sometimes necessary, to search a simple polygonal region having a guard number g, which is the size of a minimum guard set. We also show that g(h+1)-searchers (i.e., g searchers with h+1 flashlights each), and consequently g(h+1) 1-searchers as well, can always search a polygonal region with h?1 holes having a guard number g.     

A meta-analysis of research on digital game-based science learning

This meta-analysis investigates the relative effectiveness of game-based science learning against other instructional methods (Gameplay design) as well as against science game variants enriched with mechanisms (Game-mechanism design). An overall medium effect size for Gameplay design (k = 14, N_es = 14, g_RE = 0.646, p = .000), and an overall small-to-medium effect size for Game-mechanism design (k = 12, N_es = 13, adjusted g_RE = 0.270, p = .001) are reported. Further, the results of subgroup analyses suggest that students across educational levels all significantly benefit from game-based science learning although there is no significant difference between the subgroup mean effects. Further, learning and gaming mechanisms play equal roles significantly increasing students' scientific knowledge gains. With these promising results, however, high variance within the subgroups of educational levels and those of gaming mechanisms indicate that gaming mechanisms should be developed with care to meet students' different needs in different educational levels.     

Sufficient conditions for the existence and asymptotic behaviour of solution to a quasi-linear elliptic problem

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form?Δ _p u=λ a(x)g(u) in D′(Ω), u>0 in Ω, lim _{x→? Ω} u(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ₀ such that for all λ∈(0, Λ₀], the problem has a solution u _λ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution.     

On the characterization of linear and linearizable automata by a superposition principle

This paper deals with the superponability of linear and linearizable automata. Automata linear in GF(p),p2, with input and output sets {0, 1,,p-1} and with arbitrary initial state are characterized as the class of automata superponable, i.e. permutable, with respect to a functionax+(1-a)x. The casep=2 requires special considerations. The inputoutput behaviour of this automata can be described by generalized impulse responses. Using instead of GF(p) an arbitrary commutative field on the same basic set {0,1,,p-1} some of the functions nonlinear in GF(p) are linear in this field. These functions are called linearizable. Linearizable functions are characterized by a functional equation and solvability conditions and as polynomials in GF(p). Thus the well known theory of linear automata can be extended to linearizable automata.     

Sylow's theorem in polynomial time

Given a set Γ of permutations of an n-set, let G be the group of permutations generated by Γ. If p is a prime, a Sylow p-subgroup of G is a subgroup whose order is the largest power of p dividing |G|. For more than 100 years it has been known that a Sylow p-subgroup exists, and that for any two Sylow p-subgroups P₁, P₂ of G there is an element g?G such that P₂ = g^?1P₁g. We present polynomial-time algorithms that find (generators for) a Sylow p-subgroup of G, and that find g?G such that P₂ = g^?1P₁g whenever (generators for) two Sylow p-subgroups P₁, P₂ are given. These algorithms involve the classification of all finite simple groups.     

On efficient implementation of an approximation algorithm for the Steiner tree problem

Summary This paper studies the design and implementation of an approximation algorithm for the Steiner tree problem. Given any undirected distance graph G and a set of Steiner points S, the algorithm produces a Steiner tree with total weight on its edges no more than 2(1–1/L) times the total weight on the optimal Steiner tree, where L is the number of leaves in the optimal Steiner tree. Our implementation of the algorithm, in the worst case, makes it run in 0(¦E _g¦+¦V _g–S¦log¦V _g–S¦+¦S¦log ¦S¦) time for general graph G and in 0(¦S¦ log¦S¦+M log (M,¦V _g–S¦)) time for sparse graph G, where E _g is the set of edges in G, V_g is the set of vertices in G, M = min {¦E _g, (¦V _g–S¦–1)²/2} and (x,y) = min {i¦log⁽ⁱ⁾ y x/y}.The implementation is not likely to be improved significantly without the improvement of the shortest paths algorithm and the minimum spanning tree algorithm as the algorithm essentially composes of the computation of the multiple sources shortest paths of a graph with ¦V _g¦ vertices and ¦E _g¦ edges and the minimum spanning tree of a graph with ¦V _g–S¦ vertices and M edges.     

On the extension of solutions of linear constant-coefficient PDE

We solve the following over-determined boundary value problem (the “extension problem”): Let R(∂) be a matrix whose entries are linear partial differential operators, with constant coefficients. Let Ω be a non-empty, open, bounded, convex set. We consider the homogeneous system R(∂)f=0 in a neighborhood of , subject to the boundary condition f=g in a neighborhood of ∂Ω. For a given g, we give a criterion for the (unique) existence of a smooth solution f to this problem. There are two obvious necessary conditions: g is smooth and R(∂)g=0 in a neighborhood of ∂Ω. We characterize the class of differential operators R(∂) for which the problem is solvable for any g satisfying the necessary conditions. Finally, in the case where the solution is non-unique, we consider the possibility of obtaining uniqueness by fixing several components of the desired solution.     

Searching for Mobile Intruders in a Polygonal Region by a Group of Mobile Searchers

The problem of searching for mobile intruders in a polygonal region by mobile searchers is considered. A searcher can move continuously inside a polygon holding a flashlight that emits a single ray of light whose direction can be changed continuously. The vision of a searcher at any time instant is limited to the points on the ray. The intruders can move continuously with unbounded speed. We denote by ps(P) the polygon search number of a simple polygon P , which is the number of searchers necessary and sufficient to search P . Let n , r , b , and g be the number of edges, the number of reflex vertices, the bushiness, and the size of a minimum guard set of P , respectively. In this paper we present matching upper and (worst case) lower bounds of 1 + \lfloor log ₃ (2b+1) \rfloor on ps(P) . Also upper bounds on ps(P) in terms of n,r , and g are presented;ps(P) ≤ 1 + \lfloor log ₃ (n-3) \rfloor, ps(P) ≤ 1 + \lfloor log ₃ r \rfloor , and ps(P) ≤ 2 + \lceil log ₂ g \rceil . These upper bounds are tight or almost tight in the worst case, since we show that for any natural number s \geq 2 , there is a polygon P such that ps(P) = log ₃ (n+1) = log ₃ (2r+3) = 1 + log ₃ (2g-1) = s . Received September 4, 1998; revised May 25, 1999.     

RECONSTRUCTABILITY ANALYSIS FOR GENERAL FUNCTIONS

The new discipline of reconstructability analysis has provided a powerful framework for the study of the relationships between parts and wholes. The concentration of effort has been on systems with probabilistic or possibilistic behavior functions. This paper extends aspects of reconstructability analysis to general functions which need not be behavior functions. We refer to a system with such a function as a g-system.

First, a g-system is transformed to a dimensionless form (borrowing a term from partial differential equations). A mathematical structure is then induced via a type of isomorphism onto this system which renders it amenable to analysis by established techniques in reconstructability analysis. Absolutely no restrictions are placed on the units or mathematical structure of the g-system. We refer to the system induced from the g-system as a Klir system or k-system. These systems are named in honor of the reconstructability analysis founder.

Finally, we explicate some uses of k-systems induced from given g-systems. k-systems have easy immediate application (e.g. minimal storage of system information), but, more importantly, they render accessible new tracts of system dynamics studies.