This paper addresses the vertical partitioning of a set of logical records or a relation into fragments. The rationale behind vertical partitioning is to produce fragments, groups of attribute columns, that “closely match ” the requirements of transactions. Vertical partitioning is applied in three contexts: a database stored on devices of a single type, a database stored in different memory levels, and a distributed database. In a two-level memory hierarchy, most transactions should be processed using the fragments in primary memory. In distributed databases, fragment allocation should maximize the amount of local transaction process-ing. Fragments may be nonoverlapping or overlapping. A two-phase approach for the determination of fragments is proposed; in the first phase, the design is driven by empirical objective functions which do not require specific cost information. The second phase performs cost optimization by incorporating the knowledge of a specific application environment. The algorithms presented in this paper have been implemented, and examples of their actual use are shown. 1.

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jEj and assume without loss of generality that n 1 n 2 . We call a bipartite network unbalanced if n 1 ø n 2 and balanced otherwise. (This notion is necessarily imprecise.) We show that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a two-edge push rule that allows us to "charge" most computation to vertices in V 1 , and hence develop algorithms whose running times depend on n 1 rather than n. For example, we show that the two-edge push version of Goldberg and Tarjan's FIFO preflow push algorithm runs in O(n 1 m + n 3 1 ) time and that the analogous version of Ahuja and Orlin's excess scaling algori...

The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for different λ’s. An example is a weighting between data and regularization terms in image segmentation or stereo: it is desirable to vary it both during training (to learn λ from ground truth data) and testing (to select best λ using high-knowledge constraints, e.g. user input). We review algorithmic aspects of this parametric maximum flow problem previously unknown in vision, such as the ability to compute all breakpoints of λ and corresponding optimal configurations in finite time. These results allow, in particular, to minimize the ratio of some geometric functionals, such as flux of a vector field over length (or area). Previously, such functionals were tackled with shortest path techniques applicable only in 2D. We give theoretical improvements for “PDE cuts ” [5]. We present experimental results for image segmentation, 3D reconstruction, and the cosegmentation problem. 1.

We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...

We study a practical implementation of the parametric flow algorithm of Gallo, Grigoriadis, and Tarjan. We describe an efficient implementation of the algorithm and compare it with a simpler algorithm.

Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an n-node, m-edge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...

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Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [Multiple-source shortest paths in planar graphs. In Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest non-contractible or non-separating cycle in embedded, undirected graphs in O(g² n log n) time.