Incremental Algorithms Based on Discrete Green Theorem

Abstract

By using the discrete version of Green's theorem and bivariate difference calculus we provide incremental algorithms to compute various statistics about polyominoes given, as input, by 4-letter words describing their contour. These statistics include area, coordinates of the center of gravity, moment of inertia, higher order moments, size of projections, hook lengths, number of pixels in common with a given set of pixels and also q-statistics.

Discrete Versions of Stokes’ Theorem Based on Families of Weights on Hypercubes

“…For the special case k = n = 2, taking Φ(P) = P f (x, y) dx dy and using the graphical convention of Figure 1, Proposition 2 generalizes the vertical algorithm (V -algo) given in [1,2,3]: for a lattice polyomino P ⊆ R 2 given by its contour…”

“…, q n ] where K is a field or A = Z/(2). As in [1,2,3], it is more useful to write (Δw)(P) in the form W (P), where W is a given weight function and P is an arbitrary hypercubic configuration. Then, the right member w(∂P) can be described as an algorithm to compute W (P) using another weight function w, applied to the boundary ∂P.…”

“…Finally, weight functions can take values in an arbitrary commutative ring. In particular, taking values in the boolean ring B = {0, 1, and, xor}, our results provide algorithms to decide, for example, whether a given hypercube (or pixel, in the bi-dimensional case) belongs to an hypercubic configuration (or polyomino) using a suitable weight on its boundary (see [1,2,3], for similar examples in the case of polyominoes).…”

Discrete sets with minimal moment of inertia

“…The goal of this paper is to study the roundest discrete sets S of N pixels (or N points), over the class of all discrete sets with the same number of pixels (or points), in the sense of having minimal moment of inertia I(S), relative to the center of gravity. This problem was raised in previous papers [4,5] in the context of the study of incremental $ With the support of NSERC (Canada).…”

Combinatorial View of Digital Convexity

“…Debled-Rennesson et al [8] already provided a linear time algorithm deciding if a given polyomino is convex. Their method uses arithmetical tools to compute series of digital line segments of decreasing slope: optimal time is achieved with a moving digital straight line recognition algorithm [9,10] Recently, Brlek et al looked at discrete geometry from the combinatorics of words point of view, showing for instance how the discrete Green theorem provides a series of optimal algorithms for diverse statistics on polyominoes [11,12]. This method is extended to study minimal moment of inertia polyominoes in [13].…”