Roberto Giaccio

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.

On-line algorithms for satisfiability problems with uncertainty

In this paper the problem of the on-line satisfiability of a Horn formula with uncertainty is addressed; we show how to represent a significant class of formulae by weighted directed hypergraphs and we present two algorithms that solve the on-line SAT problem and find a minimal interpretation for the formula working on the dynamic hypergraph representation. These algorithms make increasing assumptions on the formula and we will find that the second one solves the on-line SAT problem with a total time linear in the size of the formula, matching the optimal result for boolean Horn formulae.

Decremental Maintenance of Reachability in Hypergraphs and Minimum Models of Horn Formulae

In this paper we present a decremental algorithm for maintaining minimum rank hyperpaths in a directed hypergraph from a source vertex s to all other vertices, under the assumption of unit hyperedge weights. Given a hypergraph H with n vertices and m hyperedges, the total time needed to perform a sequence of m hyperedge deletions is O(n · Size(H)), where Size(H) is the sum of the sizes of the hyperedges of H; the total space needed is O(n + Size(H)). In the case of integer hyperedge weights in [1, C] our solution requires O(C · n · Size(H)) total time and O(C + n + Size(H)) space.

Maintaining Maxima under Boundary Updates

Given a set of point P ∈ ℝ2, we consider the well-known maxima problem, consisting of reporting the maxima (not dominated) points of P, in the dynamic setting of boundary updates. In this setting we allow insertions and deletions of points at the extremities of P: this permits to move a resizable vertical window on the point set.

Incremental Hive Graph

The hive graph is a rectangular graph satisfying some additional condition widely used in computational geometry for solving several kinds of fundamental queries. It has been introduced by Chazelle

INTERSECTION PROBLEMS ON SEGMENTS UNDER BOUNDARY UPDATES WITH APPLICATION TO PERSISTENT LISTS

We consider some intersection problems on segments of in the partially dynamic setting called boundary update, where updates occur at the boundary of a given region. In particular, we maintain a set S of line segments under (boundary) insertions and deletions, such that, given a line segment l either of fixed slope or originating from a fixed point and given a point p∈S∩l, we can efficiently and orderly report all segments intersecting l; insertions/deletions of segments occur at the boundaries of a vertical infinite slab. We provide practical algorithms requiring space, time per update and time per query, where k is the number of reported segments. Our results allow both modeling a moving window over a larger data set and answering segment intersection queries at an extra query cost of ; also, they provide a methodology for designing access methods to temporal databases as well as a new kind of partially persistent lists.

On-line Algorithms for Satisfiability Problems with Uncertainty

In this paper the problem of the on-line satisfiability of a Horn formula with uncertainty is addressed; we show how to represent a significant class of formulae by weighted directed hypergraphs and we present two algorithms that solve the on-line SAT problem and find a minimal interpretation for the formula working on the dynamic hypergraph representation. These algorithms make increasing assumptions on the formula and we will find that the second one solves the on-line SAT problem with a total time linear in the size of the formula, matching the optimal result for boolean Horn formulae.