The National Airspace System (NAS) is designed to accommodate a large number of flights over North America. For purposes of workload limitations for air traffic controllers, the airspace is partitioned into approximately 600 sectors; each sector is observed by one or more controllers. In order to satisfy workload limitations for controllers, it is important that sectors be designed carefully according to the traffic patterns of flights, so that no sector becomes overloaded. We formulate and study the airspace sectorization problem from an algorithmic point of view, modeling the problem of optimal sectorization as a geometric partition problem with constraints. The novelty of the problem is that it partitions data consisting of trajectories of moving points, rather than static point set partitioning that is commonly studied. First, we formulate and solve the 1d version of the problem, showing how to partition a line into “sectors ” (intervals) according to historical trajectory data. Then, we apply the 1D solution framework to design a 2D sectorization heuristic based on binary space partitions. We also devise partitions based on balanced “pie partitions ” of a convex polygon. We evaluate our 2D algorithms experimentally. We conduct experiments using actual historical flight track data for the NAS as the basis of our partitioning. We compare the workload balance of our methods to that of the existing set of sectors for the NAS and find that our resectorization yields competitive and improved workload balancing. In particular, our methods yield an improvement by a factor between 2 and 3 over the current sectorization in terms of the time-average and the worst-case workloads of the maximum workload sector. An even better improvement is seen in the standard deviations (over all sectors) of both timeaverage and worst-case workloads. 75 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited 1

In this paper we consider several variations of the following basic tiling problem: given a sequence of real numbers with two size bound parameters, we want to find a set of tiles of maximum total weight such that each tiles satisfies the size bounds. A solution to this problem is important to a number of computational biology applications such as selecting genomic DNA fragments for PCR-based amplicon microarrays and performing homology searches with long sequence queries. Our goal is to design efficient algorithms with linear or near-linear time and space in the normal range of parameter values for these problems. For this purpose, we first discuss the solution to a basic online interval maximum problem via a sliding window approach and show how to use this solution in a non-trivial manner for many of the tiling problems introduced. We also discuss NP-hardness results and approximation algorithms for generalizing our basic tiling problem to higher dimensions. Finally, computational results from applying our tiling algorithms to genomic sequences of five model eukaryotes are reported.

We consider the following question: Given a subdivision of space into n convex polyhedral cells, what is the worst-case complexity of a binary space partition (BSP) for the subdivision? We show that if the subdivision is rectangular and axis-aligned, then the worstcase complexity of an axis-aligned BSP is #(n ) and O(n n), where # = 1+log 2 (4/3) = 1.4150375 . . . . By contrast, it is known that the BSP of a collection of n rectangular cells not forming a subdivision has worstcase complexity #(n ). We also show that the worstcase complexity of a BSP for a general convex polyhedral subdivision of total complexity O(n) is #(n ).

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We provide improved approximation algorithms for the minmax generalization problems considered by Du, Eppstein, Goodrich, and Lueker [1]. In min-max generalization problems, the input consists of data items with weights and a lower bound wlb, and the goal is to partition individual items into groups of weight at least wlb, while minimizing the maximum weight of a group. The rules of legal partitioning are specific to a problem. Du et al. consider several problems in this vein: (1) partitioning a graph into connected subgraphs, (2) partitioning unstructured data into arbitrary classes and (3) partitioning a 2-dimensional array into non-overlapping contiguous rectangles (subarrays) that satisfy the above size requirements. We significantly improve approximation ratios for all the problems considered by Du et al., and provide additional motivation for these problems. Moreover, for the first problem, while Du et al. give approximation algorithms for specific graph families, namely, 3-connected and 4-connected planar graphs, no approximation algorithm that works for all graphs was known prior to this work.

Abstract—We address the problem of load balancing for parallel joins. We show that the distribution of input data received and the output data produced by worker machines are both important for performance. As a result, previous work, which optimizes either for input or output, stands ineffective for load balancing. To that end, we propose a multi-stage load-balancing algorithm which considers the properties of both input and output data through sampling of the original join matrix. To do this efficiently, we propose a novel category of equi-weight histograms. To build them, we exploit state-of-the-art computational geometry algorithms for rectangle tiling. To our knowledge, we are the first to employ tiling algorithms for join load-balancing. In addition, we propose a novel, join-specialized tiling algorithm that has drastically lower time and space complexity than existing algorithms. Experiments show that our scheme outperforms state-of-the-art techniques by up to a factor of 15. I.