Settling the bound on the rectilinear link radius of a simple rectilinear polygon

We improve the best known bound on the rectilinear link radius of a simple rectilinear polygon with respect to its rectilinear link diameter. The new bound is tight and is compatible with the known bound on the (regular) link radius of a (regular) simple polygon with respect to its (regular) link diameter. The previous bound on the rectilinear link radius of a simple rectilinear polygon was proven by Nilsson and Schuierer in 1991.     

The rectilinear steiner arborescence problem

The Rectilinear Steiner Arborescence (RSA) problem is Given a setN ofn nodes lying in the first quadrant of E², find the shortest directed tree rooted at the origin, containing all nodes inN, and composed solely of horizontal and vertical arcs oriented only from left to right or from bottom to top. In this paper we investigate many fundamental properties of the RSA problem, propose anO(n logn)-time heuristic algorithm giving an RSA whose length has an upper bound of twice that of the minimum length RSA, and show that a polynomial-time algorithm that was earlier reported in the literature for this problem is incorrect.     

An efficient algorithm for line and polygon clipping

We present an algorithm for clipping a polygon or a line against a convex polygonal window. The algorithm demonstrates the practicality of various ideas from computational geometry. It spendsO(logp) time on each edge of the clipped polygon, wherep is the number of window edges, while the Sutherland-Hodgman algorithm spendsO(p) time per edge. Theoretical and experimental analyses show that the constants involved are small enough to make the algorithm competitive even for windows with four edges. The algorithm enables image-space clipping against windows whose boundaries are convex spline curves. The paper contains detailed pseudo-code implementation of the algorithm and an adaptation of the simulation of simplicity method for handling degenerate cases.     

Discussion on some convergence problems in buckling optimisation

This note reports and briefly discusses some of the numerous reasons for bad convergence in linear buckling optimisation. Above all, it highlights that erratic convergence history can be avoided when the design optimisation problem includes enough buckling modes (and not only the first ones as it is the usual case), which keep the whole structure sensitive to the design restriction. This strategy is illustrated with an example and shows that possible significant improvement in the convergence speed can also be achieved by simply considering a large number of buckling modes in the optimisation problem. The selection of a suitable approximation scheme is also discussed.     

Optimal parallel algorithms for point-set and polygon problems

In this paper we give parallel algorithms for a number of problems defined on point sets and polygons. All our algorithms have optimalT(n) * P(n) products, whereT(n) is the time complexity andP(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. Our algorithms provide parallel analogues to well-known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently than point-set problems, and that nearest-neighbor problems can be solved without explicitly constructing a Voronoi diagram.The research of R. Cole was supported in part by NSF Grants CCR-8702271, CCR-8902221, and CCR-8906949, by ONR Grant N00014-85-K-0046, and by a John Simon Guggenheim Memorial Foundation fellowship. M. T. Goodrich's research was supported by the National Science Foundation under Grant CCR-8810568 and by the National Science Foundation and DARPA under Grant CCR-8908092.     

Proximity problems for points on a rectilinear plane with rectangular obstacles

We consider the following four problems for a setS ofk points on a plane, equipped with the rectilinear metric and containing a setR ofn disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles inR): (a) find aclosest pair of points inS, (b) find anearest neighbor for each point inS, (c) compute the rectilinearVoronoi diagram ofS, and (d) compute a rectilinearminimal spanning tree ofS. We describeO ((n+k) log(n+k))-time sequential algorithms for (a) and (b) based onplane-sweep, and the consideration of geometrically special types of shortest paths, so-calledz-first paths. For (c) we present anO ((n+k) log(n+k) logn)-time sequential algorithm that implements a sophisticateddivide-and-conquer scheme with an addedextension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-calledz-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved inO ((n+k) log(n+k) logn) time as well. All our algorithms arenear-optimal, as well as easy to implement. An extended abstract appeared inProc. 13th Conf. on the Foundations of Software Technology and Theoretical Computer Science, Bombay, 1993, Springer-Verlag, pp. 218–227. Sumanta Guha was supported in part by a UW-Milwaukee Graduate School Research Committee Award. Ichiro Suzuki was supported in part by the National Science Foundation under Grants CCR-9004346 and IRI-9307506, the Office of Naval Research under Grant N00014-94-1-0284, and an endowed chair supported by Hitachi Ltd. at the Faculty of Engineering Science, Osaka University.     

Efficient implementations of construction heuristics for the rectilinear block packing problem

The rectilinear block packing problem is a problem of packing a set of rectilinear blocks into a larger rectangular container, where a rectilinear block is a polygonal block whose interior angle is either 90° or 270°. There exist many applications of this problem, such as VLSI design, timber/glass cutting, and newspaper layout. In this paper, we design efficient implementations of two construction heuristics for rectilinear block packing. The proposed algorithms are tested on a series of instances, which are generated from nine benchmark instances. The computational results show that the proposed algorithms are especially efficient for large instances with repeated shapes.     

Parallel triangulation of a polygon in two calls to the trapezoidal map

We give a parallel method for triangulating a simple polygon by two (parallel) calls to the trapezoidal map computation. The method is simpler and more elegant than previous methods. Along the way we obtain an interesting partition of one-sided monotone polygons. Using the best-known trapezoidal map algorithm, ours run in timeO(logn) usingO(n) CREW PRAM processors.This research was supported by NSF Grants No. DCR-84-01898 and No. DCR-84-01633, and ONR Contract N00014-85-K-0046.     

Implicit branching and parameterized partial cover problems

Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems has been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems.     

Finding rectilinear least cost paths in the presence of convex polygonal congested regions

This paper considers the problem of finding the least cost rectilinear distance path in the presence of convex polygonal congested regions. We demonstrate that there are a finite, though exponential number of potential staircase least cost paths between a specified pair of origin–destination points. An upper bound for the number of entry/exit points of a rectilinear path between two points specified a priori in the presence of a congested region is obtained. Based on this key finding, a “memory-based probing algorithm” is proposed for the problem and computational experience for various problem instances is reported. A special case where polynomial time solutions can be obtained has also been outlined.     

Convergence of semidiscrete approximations to optimal control problems in hilbert spaces: A counterexample

In this paper we give and discuss a counterexample related to the approximation of non-quadratic optimal control problems for linear distributed parameter systems. More precisely, we show that the assumption (1.7), which has been proved in a previous paper to be sufficient, is also necessary for the convergence (at least in the most general setting of the problem).     

最佳逼近圆周的正多边形算法

本文定义了圆周的最佳逼近正n边形，在上述意义下，解决了圆周的最佳逼近正n边形半径的计算问题。     

Heuristic solutions for transshipment problems in a multiple door cross docking warehouse

Cross docking is a practice in logistics with the main operations of unloading products from an incoming truck, regrouping them with respect to their destinations and loading them directly into an outbound truck with minimum storage in between these operations. In this article, we study the transshipment scheduling problem in a multiple inbound and outbound dock configuration. The operations manager has several decisions to make: he can decide to transship products directly from inbound to an outbound truck, if an outbound truck is available; he can temporarily store certain products and have them loaded later on; or he can replace an outbound truck to facilitate direct loading. The objective is to find the best schedule of transshipment operations to minimize the sum of inventory holding and truck replacement costs. In this article, we present several heuristics to attain this objective. Numerical experiments are presented and the results are compared with the optimal solution to evaluate the performance of the heuristics.     

On approximation behavior of the greedy triangulation for convex polygons

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's (n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n ²) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).     

An efficient variable neighborhood search heuristic for very large scale vehicle routing problems

In this paper, we present an efficient variable neighborhood search heuristic for the capacitated vehicle routing problem. The objective is to design least cost routes for a fleet of identically capacitated vehicles to service geographically scattered customers with known demands. The variable neighborhood search procedure is used to guide a set of standard improvement heuristics. In addition, a strategy reminiscent of the guided local search metaheuristic is used to help escape local minima. The developed solution method is specifically aimed at solving very large scale real-life vehicle routing problems. To speed up the method and cut down memory usage, new implementation concepts are used. Computational experiments on 32 existing large scale benchmarks, as well as on 20 new very large scale problem instances, demonstrate that the proposed method is fast, competitive and able to find high-quality solutions for problem instances with up to 20,000 customers within reasonable CPU times.     

An efficient approach to approximate solutions of eighth-order boundary-value problems

In this paper, the approximate solutions to the eighth-order boundary-value differential equations are solved by using the Adomian decomposition method (ADM). The numerical solutions of the problem are calculated in the form of a series with easily computable components. The numerical illustrations show that this technique is more reliable, efficient and accurate than the traditional schemes.     

On asymptotically efficient solutions for a class of supervisorycontrol problems

We consider restricted sequences of forbidden-string and forbidden-state problems and show the limiting problem instance defined by a sequence has a solution if and only if each problem instance in the sequence has a solution. Furthermore, the supervisor that enforces the desired behavior in the limiting problem instance also enforces the desired behavior in each problem instance in the sequence. In essence, the storage requirement for supervisory control of each instance is bounded above by that of the limiting problem instance. In addition, if the limiting problem instance is available, a similar observation can be made regarding the complexity of supervisory control. This observation coupled with recent results on the solvability of the supervisory control problem involving certain classes of nonregular behaviors and finite-representations of infinite-state systems is suggested as an effective procedure to combat the computational and storage issues in supervisory control. We conclude with an incomplete list of future research directions     

A unifying approach for a class of problems in the computational geometry of polygons

A generalized problem is defined in terms of functions on sets and illustrated in terms of the computational geometry of simple planar polygons. Although its apparent time complexity is O(n ²), the problem is shown to be solvable for several cases of interest (maximum and minimum distance, intersection detection and rerporting) in O(n logn), O(n) or O(logn) time, depending on the nature of a specialized selection function. (Some of the cases can also be solved by the Voronoi diagram method; but time complexity increases with that approach.) A new use of monotonicity and a new concept of locality in set mappings contribute significantly to the derivation of the results.     

Visibility between two edges of a simple polygon

One of the most recurring themes in many computer applications such as graphics automated cartography, image processing and robotics is the notion of visibility. We are concerned with the visibility between two edges of a simplen-vertex polygon. Four natural definitions of edge-to-edge visibility are proposed. There existO(nlogn) algorithms and complicatedO(nlog logn) algorithms to solve this problem partially and indirectly. A linear running time, and thus optimal algorithm is presented to determine edge-to-edge visibility under any of the four definitions. This simple, efficient, and direct algorithm without computing the triangulation of the simple polygon also identifies the visibility region if it exists.     

Complexities of some interesting problems on spanning trees

This paper deals with the complexity issues of some new interesting spanning tree problems. Here we define some new spanning tree problems by imposing various constraints and restrictions on graph parameters and present relevant results. Also we introduce a new notion of “set version” of some decision problems having integer K<|V| as a parameter in the input instance, where we replace K by a set X⊆|V|. For example, the set version of Maximum Leaf Spanning Tree problem asks whether there exists a spanning tree in G that contains X as a subset of the leaf set. We raise the issue of whether the set versions of NP-complete problems are as hard as the original problems and prove that although in some cases the set versions are easier to solve, this is not necessarily true in general.