An n-town, n@?N, is a group of n buildings, each occupying a distinct position on a 2-dimensional integer grid. If we measure the distance between two buildings along the axis-parallel street grid, then an n-town has optimal shape if the sum of all pairwise Manhattan distances is minimized. This problem has been studied for cities, i.e., the limiting case of very large n. For cities, it is known that the optimal shape can be described by a differential equation, for which no closed-form solution is known. We show that optimal n-towns can be computed in O(n^7^.^5) time. This is also practically useful, as it allows us to compute optimal solutions up to n=80.
Motivated by the problem of storing records in a 2-dimensional array, @cite_12 studied strategies that minimize average access time between successive queries; among other results, they described an optimal solution for the continuous version of our problem: What shape of area @math minimizes the average Manhattan distance between two interior points? Independently, @cite_2 solved this problem in the setting of a city, inspiring the subtitle of this paper. The optimal solution is described by a differential equation, and no closed-form solution is known.
@cite_12 @cite_2
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If one defines the distance between two points as the Manhattan distance (the sum of the horizontal distance along streets and the vertical distance along avenues) then one can define a city as being optimal if the average distance between pairs of points is a minimum. In this paper a nonlinear differential equation for the boundary curve of such a city is determined. The problem solved here is the continuous version of an optimization problem on how to design efficient allocation algorithms for massively parallel supercomputers. In the language of continuum mechanics, the shape of the optimal city is that taken by a blob of incompressible fluid composed of molecules whose pairwise interactions are described by an attractive potential proportional to the Manhattan distance between the particles. We consider the problem of placing records in a 2-dimensional storage array so that expected distance between consecutive references is minimized. A simple placement heuristic which uses only relative frequency of access for different records is shown to be within an additive constant of optimal when distance is measured by the Euclidean metric. For the rectilinear and maximum metrics, we show that there is no such heuristic. For the special case in which all access probabilities are equal, however, heuristics within an additive constant of optimal do exist, and their implementation requires solution of differential equations for which we give numerical solutions. Finding the maximals in a collection of vectors is relevant to many applications. The maximal set is related to the convex hull---and hence, linear optimization---and nearest neighbors. The maximal vector problem has resurfaced with the advent of skyline queries for relational databases and skyline algorithms that are external and relationally well behaved.The initial algorithms proposed for maximals are based on divide-and-conquer. These established good average and worst case asymptotic running times, showing it to be O(n) average-case. where n is the number of vectors. However, they are not amenable to externalizing. We prove, furthermore, that their performance is quite bad with respect to the dimensionality, k, of the problem. We demonstrate that the more recent external skyline algorithms are actually better behaved, although they do not have as good an apparent asymptotic complexity. We introduce a new external algorithm, LESS, that combines the best features of these. experimentally evaluate its effectiveness and improvement over the field, and prove its average-case running time is O(kn). Most methods for document image retrieval rely solely on text information to find similar documents. This paper describes a way to use layout information for document image retrieval instead. A new class of distance measures is introduced for documents with Manhattan layouts, based on a two-step procedure: First, the distances between the blocks of two layouts are calculated. Then, the blocks of one layout are assigned to the blocks of the other layout in a matching step. Different block distances and matching methods are compared and evaluated using the publicly available MARG database. On this dataset, the layout type can be determined successfully in 92.6 of the cases using the best distance measure in a nearest neighbor classifier. The experiments show that the best distance measure for this task is the overlapping area combined with the Manhattan distance of the corner points as block distance together with the minimum weight edge cover matching.