Elena Barcucci

TL;DR: The ECO method as discussed by the authors enumerates some classes of combinatorial objects by means of an operator that performs a "local expansion" on the objects, and uses these constructions to deduce some new funtional equations verified by classes' generating functions.

TL;DR: Some operations for recontructing convex polyominoes by means of vectors H's and V's partial sums allows a new algorithm to be defined whose complexity is less than O(n2m2).

TL;DR: The basic idea is to translate ECO method from a method for the enumeration of combinatorial objects into a random generation method based on the concepts of succession rule and generating tree.

Abstract: In a previous report, we studied the problem of reconstructing a discrete set from its horizontal and vertical projections. We defined an algorithm that decides whether there is a convex polyomino whose horizontal and vertical projections are given by (H, V), with H ∈ ℕm and V ∈ ℕn. If there is at least one convex polyomino with these projections, the algorithm reconstructs one of them in O(n4m4) time. In this article, we introduce the geometrical concept of a discrete set's medians. Starting out from this geometric property, we define some operations for reconstructing convex polyominoes from their projections (H, V). We are therefore able to define a new algorithm whose complexity is less than O(n2m2). Hence, this algorithm is much faster than the previous one. At the moment, however, we only have experimental evidence that this algorithm decides if there is a convex polyomino whose projections are equal to (H, V), for all (H, V) instances. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 69–77, 1998

TL;DR: This work introduces an exhaustive generation algorithm, which is general for the classes of succession rules considered in [1], and shows that the algorithm is efficient in an amortized sense; it actually uses only a constant amount of computation per object.