Gadi Aleksandrowicz

Counting d -dimensional polycubes and nonrectangular planar polyominoes

A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. Similarly, a d-dimensional polycube (for d ≥2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d–1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeier’s algorithm for counting two-dimensional rectangular polyominoes [Re81], which counts all the above types of polyominoes. For example, our program computed the number of distinct 3-D polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.

COUNTING d-DIMENSIONAL POLYCUBES AND NONRECTANGULAR PLANAR POLYOMINOES

A planar polyomino of size n is an edge-connected set of n squares on a rectangular two-dimensional lattice. Similarly, a d-dimensional polycube (for d ≥ 2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d - 1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeiers algorithm for counting two-dimensional rectangular polyominoes, which counts all the above types of polyominoes. For example, our program computed the number of distinct three-dimensional polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.

The growth rate of high-dimensional tree polycubes

A d-dimensional polycube is a face-connected set of cubes in ddimensions, where the faces are (d-1) dimensional. Fixed polycubes are distinct if they differ in shape or orientation. The cube-adjacency graph of a tree polycube does not have cycles. In this paper we estimate the asymptotic growth rate of d-dimensional tree polycubes at (2d-3.5)e+O(1/d).

The Growth Rate of High-Dimensional Tree Polycubes

Abstract A d-dimensional polycube is a ( d − 1 ) -face-connected set of cubes in d dimensions. Fixed polycubes are considered distinct if they differ in shape or orientation. The cube-adjacency graph of a tree polycube is a tree. In this paper we investigate the asymptotic growth rate of d-dimensional tree polycubes, and estimate it at ( 2 d − 3.5 ) e + O ( 1 / d ) .

Counting Polycubes without the Dimensionality Curse

A d-D polycube of size nis a connected set of ncells (hypercubes) of an orthogonal d-dimensional lattice, where connectivity is through (di¾? 1)-dimensional faces of the cells. Computing A d (n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present a method whose order of memory consumption is a (very low) polynomialin both nand d. Furthermore, we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously. This enables us to find A d (n) for values of dand nfar beyond any previous attempt.

Permutations with forbidden patterns and polyominoes on a twisted cylinder of width 3

Abstract In a cubical lattice on a “twisted cylinder,” connectivity of cells form a spiral rather than a cylindrical shape. It was recently proven that for any fixed w , the enumerating sequence for the number of polyominoes on the twisted cylinder of perimeter w satisfies a linear recursion. This recursion is determined by the minimal polynomial of the transfer matrix that models the growth of polyominoes on the cylinder. In this paper we present a direct construction of the recursion for the case w = 3 . In addition, we make a connection between enumeration of polyominoes and of permutations: We find bijections between three classes of permutations, each defined in terms of eight forbidden patterns, and the set of polyominoes on a twisted cylinder of width 3. In particular, it follows that all three classes of permutations belong to the same Wilf class, obeying the same recurrence formula (with respect to size) as that of the considered polyominoes, that is, a n = 2 a n − 1 + a n − 2 + 2 a n − 3 , with a 1 = 1 , a 2 = 2 , and a 3 = 6 .

Parallel enumeration of lattice animals

Lattice animals are connected sets of lattice cells. When the lattice is in d dimensions, connectedness is through (d - 1)-dimensional features of the lattice. For example, connectedness of two-dimensional animals (e.g., on the rectangular, triangular, and hexagonal lattices) are through edges, connectedness of 3-dimensional polycubes is through faces, etc. Much attention has been given in the literature to algorithms for counting animals of a given size (number of cells) on different lattices. One such algorithm was suggested in 1981 by Redelmeier for counting polyominoes (animals on the 2D orthogonal lattice). This was the first algorithm that generated polyominoes without repetitions. In previous works we extended this algorithm to other lattices and showed how to avoid its (originally) huge memory consumption. In the current paper we describe how to parallelize the extended algorithm. Our implementation runs on the Internet, effectively using an unlimited number of computers running portions of the computation. Thus, we were able to extend the known counts of animals on many types of lattices with values which were previously out of reach.

Recovering highly-complex linear recurrences of integer sequences

Abstract We suggest a variant of the Berlekamp–Massey algorithm, originally used for recovering a linear shift register from known output bits, for recovering a linear recurrence satisfied by a sequence of natural numbers from known values of the sequence. We present an application of the algorithm to recovering extremely complex recurrences satisfied by the sequences enumerating polyominoes on twisted cylinders.

Redelmeier's algorithm for counting lattice animals

In this video we present Redelmeiers algorithm for counting polyominoes, its generalization for counting animals on any lattice, and our implementation of a parallel version of it.