Efficient algorithms for three-dimensional axial and planar random assignment problems

Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural "Axial" and "Planar" versions, both of which are NP-hard. For 3-dimensional Axial random assignment instances of size $n$, the cost scales as $\Omega(1/n)$, and a main result of the present paper is a linear-time algorithm that, with high probability, finds a solution of cost $O(n^{-1+o(1)})$. For 3-dimensional Planar assignment, the lower bound is $\Omega(n)$, and we give a new efficient matching-based algorithm that with high probability returns a solution with cost $O(n \log n)$

Community involvement in urban design: help or hindrance?

This paper examines the realities of community involvement in urban design in the context of a study of the West Itchen Neighbourhood of Southampton, a diverse inner city area accommodating some 7,000 households and 18,000 people. The findings are based on a literature review of community involvement in urban design and case study research into three government supported regeneration projects all located within the study area: a Neighbourhood Renewal Area - designated in 1994; an Estate Action Scheme - implemented between 1993 and 1996; and a Single Regeneration Budget programme - following a successful bid in 1995. The research was undertaken by Helen Gregory in 1997/8 as the basis of a dissertation, supervised by Alan Rowley, submitted for the award of an MPhil in Environmental Planning and Development from The University of Reading

Approximating Subadditive Hadamard Functions on Implicit Matrices

An important challenge in the streaming model is to maintain small-space approximations of entrywise functions performed on a matrix that is generated by the outer product of two vectors given as a stream. In other works, streams typically define matrices in a standard way via a sequence of updates, as in the work of Woodruff (2014) and others. We describe the matrix formed by the outer product, and other matrices that do not fall into this category, as implicit matrices. As such, we consider the general problem of computing over such implicit matrices with Hadamard functions, which are functions applied entrywise on a matrix. In this paper, we apply this generalization to provide new techniques for identifying independence between two vectors in the streaming model. The previous state of the art algorithm of Braverman and Ostrovsky (2010) gave a $(1 \pm \epsilon)$-approximation for the $L_1$ distance between the product and joint distributions, using space $O(\log^{1024}(nm) \epsilon^{-1024})$, where $m$ is the length of the stream and $n$ denotes the size of the universe from which stream elements are drawn. Our general techniques include the $L_1$ distance as a special case, and we give an improved space bound of $O(\log^{12}(n) \log^{2}({nm \over \epsilon})\epsilon^{-7})$

Memory and Pluralism on a Property Law Frontier

This article explores the limits of legal victory and the problem of legitimacy of legal outcomes. It chronicles the decades-long dispute between Hispano settlers on northern New Mexico’s Sangre de Cristo land grant and the succession of entrepreneur owners of the grant in the last decades of the 19th century and the first decades of the 20th century. The dispute occurred on a legal and cultural frontier defined by the transition from Mexican to U.S. dominion in the years following the end of the Mexican War and by the opening of the region to larger scale economic development at the end of the 19th century. The new Dutch and American owners of the grant sought to displace patterns of land and resource use developed during the Mexican colonial period, and suited to the frontier circumstances and subsistence local economy of that earlier period, with patterns of use intended to encourage new colonial settlement and the intensive development of the region’s natural resources. In spite of winning every legal challenge to their new ownership, the new entrepreneurs struggled to establish effective control over land and resources to which they held formal title. The resistance of the Hispano settlers produced continual litigation and exacted efforts from the entrepreneurs to appease the sense of right of the Hispano settlers faced with extinction of their customary rights of access to the land and resources. The settlers’ resistance, grounded in a sense of right based on the circumstances of their settlement and in an unwillingness to be displaced from their homes, proved enduring, and stymied the efforts of the entrepreneurs to develop the land as they would have preferred. Using correspondence and litigation records, this article reconstructs the dispute between the U.S. Freehold Land & Emigration Co. and its successors and the Hispano settlers, who organized themselves as the Defensive Association of the Land Settlers of the Rio de Costilla. The article situates the Costilla episode as one of several in American legal history where popular property norms diverge from the property rules offered by the legal system. The article is strongly grounded in the specifics of the Costilla dispute, but it uses its sense of locality to explore larger questions about law, dissent, and the challenges that are posed when a legal system is asked to absorb and reflect pluralist values. The author may be contacted at the following e-mail address: [email protected]