Space-Efficient Data Structures for Polyominoes and Bar Graphs

We provide a compact data structure for representing polyominoes that supports neighborhood and visibility queries. Neighborhood queries concern reporting adjacent cells to a given cell, and visibility queries determine whether a straight line can be drawn within the polyomino that connects two specified cells. For an arbitrary small ϵ > 0, our data structure can encode a polyomino with n cells in (3 + ϵ)n + o(n) bits while supporting all queries in constant time. The space complexity can be improved to 3n + o(n), while supporting neighborhood queries in $\mathcal{O}(1)$ and visibility queries in $\mathcal{O}(t(n))$ for any arbitrary t(n) ∈ ω(1). Previous attempts at enumerating polyominoes have indicated that at least 2.00091n−o(n) bits are required to differentiate between distinct polyominoes, which shows our data structure is compact. In addition, we introduce a succinct data structure tailored for bar graphs, a specific subclass of polyominoes resembling histograms. We show that a bar graph comprising n cells can be encoded using n + o(n) bits, enabling constant-time query processing. Meanwhile, n − 1 bits are necessary to represent any bar graph, proving our data structure is succinct.