We describe a new approach for cluster-based drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) time, where n is the number of points. We show that the BAR tree can be used to recursively divide a graph embedded in the plane into subgraphs of roughly equal size, such that the drawing of each subgraph has a balanced aspect ratio. As a result, we obtain a representation of a graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. The overall running time of the algorithm is O(n log n+m+D0(G)), where n and m are the number of vertices and edges of the graph G, andD0(G) is the time it takes to obtain an initial embedding of G in the plane. In particular, if the graph is planar each layer is a graph drawn with straight lines and without crossings on the n×n grid and the running time reduces to O(n log n).

The last twenty years have seen enormous progress in the design of algorithms, but little of it has been put into practice. Because many recently developed algorithms are hard to characterize theoretically and have large running-time coefficients, the gap between theory and practice has widened over these years. Experimentation is indispensable in the assessment of heuristics for hard problems, in the characterization of asymptotic behavior of complex algorithms, and in the comparison of competing designs for tractable problems. Implementation, although perhaps not rigorous experimentation, was characteristic of early work in algorithms and data structures. Donald Knuth has throughout insisted on testing every algorithm and conducting analyses that can predict behavior on actual data; more recently, Jon Bentley has vividly illustrated the difficulty of implementation and the value of testing. Numerical analysts have long understood the need for standardized test suites to ensure robustness, precision and efficiency of numerical libraries. It is only recently, however, that the algorithms community has shown signs of returning to implementation and testing as an integral part of algorithm development. The emerging disciplines of experimental algorithmics and algorithm engineering have revived and are extending many of the approaches used by computing pioneers such as Floyd and Knuth and are placing on a formal basis many of Bentley's observations. We reflect on these issues, looking back at the last thirty years of algorithm development and forward to new challenges: designing cache-aware algorithms, algorithms for mixed models of computation, algorithms for external memory, and algorithms for scientific research.

Mean-variance analysis of the performance of spatial ordering

This paper develops analytical expressions and algorithms for deriving the number of clusters that an average query of a given size and orientation retrieves using a given spatial access method. These algorithms run very fast and can be used by query schedulers and optimizers on the fly to determine best processing plans. Current techniques for determining the number of clusters are based on simulation, which is slow. Our algorithms are discussed in the context of the Hilbert method for ease of exposition, but they do generalize in most cases to other space filling methods. Moreover, they apply for any given blocking factor. We prove that the proposed approach provides exact results for the case of blocking factor of 1, and empirically demonstrate that it yields near exact results for other blocking factors. Keywords: Spatial clustering, spatial query processing, multi-dimensional indexing, performance evaluation, Hilbert ordering. 1 Introduction Spatial query processing involves proc...