Paolo Giulio Franciosa

Efficient searching with linear constraints

We show how to preprocess a set S of points in Rd into an external memory data structure that efficiently supports linear-constraint queries. Each query is in the form of a linear constraint xd?a0+?d?1i=1aixi; the data structure must report all the points of S that satisfy the constraint. This problem is called halfspace range searching in the computational geometry literature. Our goal is to minimize the number of disk blocks required to store the data structure and the number of disk accesses (I/Os) required to answer a query. For d=2, we present the first data structure that uses linear space and answers linear-constraint queries using an optimal number of I/Os in the worst case. For d=3, we present a near-linear-size data structure that answers queries using an optimal number of I/Os on the average. We present linear-size data structures that can answer d-dimensional linear-constraint queries (and even more general d-dimensional simplex queries) efficiently in the worst case. For the d=3 case, we also show how to obtain trade-offs between space and query time.

Directed Hypergraphs: Problems, Algorithmic Results, and a Novel Decremental Approach

The purpose of this paper is twofold. First, we review several basic combinatorial problems that have been stated in terms of directed hypergraphs and have been studied in the literature in the framework of different application domains. Among them, transitive closure, transitive reduction, flow and cut problems, and minimum weight traversal problems. For such problems we illustrate some of the most important algorithmic results in the context of both static and dynamic applications. Second, we address a specific dynamic problem which finds several interesting applications, especially in the framework of knowledge representation: the maintenance of minimum weight hyperpaths under hyperarc weight increases and hyperarc deletions. For such problem we provide a new efficient algorithm applicable for a wide class of hyperpath weight measures.

On the optimal binary plane partition for sets of isothetic rectangles

Abstract We present a binary space partition algorithm for a set of disjoint isothetic rectangles. It recursively splits the set by means of isothetic cutting lines, until no two rectangles belong to the same portion of the plane. Rectangles intersected by a cutting line are split. We provide an algorithm that given n rectangles builds a linear sized Binary Space Partition with no empty regions by means of at most 4 n -1 cuts. The algorithm runs in O( n log n ) worst-case space and time. We generalize and improve a previous result achieved on isothetic line segments by Paterson and Yao.

Semi-Dynamic Shortest Paths and Breadth-First Search in Digraphs

We show how to maintain a shortest path tree of a general directed graph G with unit edge weights and n vertices, during a sequence of edge deletions or a sequence of edge insertions, in O(n) amortized time per operation using linear space. Distance queries can be answered in constant time, while shortest path queries can be answered in time linear in the length of the retrieved path. These results are extended to the case of integer edge weights in [1,C], with a bound of O(Cn) amortized time per operation.

Enclosing Many Boxes by an Optimal Pair of Boxes

We look at the problem: Given a set

The incremental maintenance of a depth-first-search tree in directed acyclic graphs

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A CONVEX HULL ALGORITHM FOR POINTS WITH APPROXIMATELY KNOWN POSITIONS

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An Optimal Algorithm for Approximating a Set of Rectangles by Two Minimum Area Rectangles

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Semi-dynamic breadth-first search in digraphs

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