Rectilinear Norm Research Articles

Probabilistic analysis of an enhanced partitioning algorithm for the steiner tree problem in Rd

AbstractThe Geometric Steiner Minimum Tree problem (GSMT) is to connect at minimum cost n given points (called terminals) in d‐dimensional Euclidean space. We generalize a GSMT approximation partitioning algorithm by Komlós and Shing (KS) and analyze its performance under more relaxed conditions. These generalizations have practical applications for multilayer VLSI routing. Whereas KS assumed d = 2, our algorithm works for any dimension d ⩾ 2. Moreover, whereas their analysis assumed the rectilinear norm and a uniform distribution of input points, our analysis holds for any norm on Rd and whenever the terminals are any independent identically distributed random variables taking on values in any bounded subset of Rd. Both algorithms depend on a parameter t through which the user can trade off time for solution quality. We evaluate our algorithm in terms of its performance ratio–the ratio of the cost of the Steiner tree computed by the algorithm divided by the cost of a Steiner minimum tree. Applying a probability theorem on subadditive Euclidean functionals by Steele, we prove the following: Under the aforementioned distribution of inputs, the limit as n → ∞ of the supremum of the performance ratio of our algorithm is 1 + Ot−1/d(d‐1), almost surely. This result generalizes the corresponding 1 + O(t−1/2) bound proven by KS. Along the way, we prove a useful combinatorial lemma about d‐dimensional rectangle slicings. We prove that the worst‐case time and space complexity of our algorithm is θ(n lg (n/t) + TMST(v, v) + nTSMT(t)/t) and θ(SMST(v, v) + SSMT(t)), respectively, where v ⩽ n + 2d+1 (n/t) σ(t) is the number of vertices in the resulting Steiner tree. Here, TSMT(t) and SSMT(t) are the time and space required to solve exactly any GSMT problem of size less than t; TMST(n, m) and SMST(n, m) are the time and space required to find a minimum spanning tree of a graph with n nodes and m edges; and σ(t) is the maximum number of Steiner points for any Steiner minimum tree with t terminals. For example, for R2 and the rectilinear norm, the time is O(n lg(n/t) + n lg*n + nTSMT(t)/t) and the space is O(n lg *n + SSMT(t)). © 1994 by John Wiley & Sons, Inc.

An Algorithm for the p-Center Problem in the Plane

The problem of locating p facilities to service m demand points in a plane, such that each demand point is as close as possible to some facility, is considered. An algorithm based on geometry is given, and some computational results presented. Extensions to the weighted case, the surface of a sphere, and the rectilinear norm are also provided.

Geometrical properties of the Fermat-Weber problem

This paper is devoted to the Fermat-Weber problem with mixed gauges in order to take into account nonsymmetric distances. A geometrical description of the set of solutions is obtained. Then the concept of metric hull is defined; general sufficient conditions are given such that the metric hull of existing facilities intersects the set of optimal locations. Particularly, it is shown that an optimal location can be found in the metric hull, in dimension two, whatever the gauges are and in dimension n with the rectilinear norm. The connection between metric hull, convex hull and octagonal hull is studied.

An O(n) algorithm for the linear multiple choice knapsack problem and related problems

We present an O(n) algorithm for the Linear Multiple Choice Knapsack Problem and its d-dimensional generalization which is based on Megiddo's (1982) algorithm for linear programming. We also consider a certain type of convex programming problems which are common in geometric location models. An application of the linear case is an O(n) algorithm for finding a least distance hyperplane in R d according to the rectilinear norm. The best previously available algorithm for this problem was an O(n log 2n) algorithm for the two-dimensional case. A simple application of the nonlinear case is an O(n) algorithm for finding the point at which a ‘pursuer’ minimizes its distance from the furthest among n ‘targets’, when the trajectories involved are straight lines in R d.

Generalized Weber Problem with Rectangular Regions

The problem of locating one or more new facilities relative to a number of existing rectangular regions is treated for the case where the rectilinear norm is used. The new facilities are to be located such that the total weighted distance is minimized. If there is interfacility interaction among the new facilities, a gradient-free nonlinear search algorithm is utilized. Computational experience suggests that this algorithm is expedient even in the solution of large problems.

An efficient algorithm for the location‐allocation problem with rectangular regions

AbstractThe location‐allocation problem for existing facilities uniformly distributed over rectangular regions is treated for the case where the rectilinear norm is used. The new facilities are to be located such that the expected total weighted distance is minimized. Properties of the problem are discussed. A branch and bound algorithm is developed for the exact solution of the problem. Computational results are given for different sized problems.

Technical Note—Location Theory, Dominance, and Convexity: Some Further Results

This note presents some generalizations of results on the single norm-one facility location problem due to Wendell and Hurter. It is first shown that a solution to the multifacility location problem with a single norm can be found in the convex hull of sources and destinations. In the particular case of the rectilinear norm, only intersection points belonging to this convex hull need be considered. When different 1p-norms are envisaged, the octagonal hull contains one solution to the single facility location problem. This result is also extended to the multifacility case.

An Efficient Algorithm for the Solution of the Weber Problem with Mixed Norms

An algorithm is developed for the solution of the Weber problem when the distances are measured as arbitrary linear combinations of $l_p $-norms. Euclidean and rectilinear norms are used to illustrate the development. The algorithm applies the Dantzig–Wolfe decomposition method to the geometric programming dual of the Weber problem. The algorithm can be extended to the solution of the multifacilities location problem with constraints.