Reconstructing hv-Convex Polyominoes from Orthogonal Projections

TL;DR

This paper presents a fast polynomial-time algorithm for reconstructing hv-convex polyominoes from row and column projections, significantly improving previous methods, and also addresses a linear-time solution for centered hv-convex polyominoes.

Contribution

It introduces a more efficient polynomial algorithm for reconstructing hv-convex polyominoes and provides a linear-time solution for centered hv-convex polyominoes.

Findings

Polynomial algorithm is several orders faster than previous methods.

Reconstruction of centered hv-convex polyominoes can be done in linear time.

The approach significantly improves efficiency in tomography-based reconstruction tasks.

Abstract

Tomography is the area of reconstructing objects from projections. Here we wish to reconstruct a set of cells in a two dimensional grid, given the number of cells in every row and column. The set is required to be an hv-convex polyomino, that is all its cells must be connected and the cells in every row and column must be consecutive. A simple, polynomial algorithm for reconstructing hv-convex polyominoes is provided, which is several orders of magnitudes faster than the best previously known algorithm from Barcucci et al. In addition, the problem of reconstructing a special class of centered hv-convex polyominoes is addressed. (An object is centered if it contains a row whose length equals the total width of the object). It is shown that in this case the reconstruction problem can be solved in linear time.