Related papers: Counting LEGO configurations
We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. While previous work within this model of assembly has focused on…
The wavelet tree has become a very useful data structure to efficiently represent and query large volumes of data in many different domains, from bioinformatics to geographic information systems. One problem with wavelet trees is their…
This paper studies automatic prototyping using LEGO. To satisfy individual needs and self-sustainability, this paper presents a framework that learns the assembly and disassembly sequences from human demonstrations. In addition, a digital…
We show how to answer spatial multiple-set intersection queries in O(n(log w)/w + kt) expected time, where n is the total size of the t sets involved in the query, w is the number of bits in a memory word, k is the output size, and c is any…
We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et…
We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional…
In this paper we develop the theory of how to count, in thin concurrent games, the configurations of a strategy witnessing that it reaches a certain configuration of the game. This plays a central role in many recent developments in…
We consider a class of cut-and-project sets $\Lambda = \Lambda_F \times \zahl$ in the plane. Let $L=\Lambda+w\real$, $w\in\real^2$, be a countable union of parallel lines. Then either (1) $L$ is a discrete family of lines, (2) $L$ is a…
Define a building blocks set to be a collection of n cubes (each with six sides) where each side is assigned one letter and one color from a palette of m colors. We propose a novel problem of assigning letters and colors to each face so as…
We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a…
We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…
We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent…
We prove new upper and lower bounds on the number of iterations the $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) requires until stabilization. For $k \geq 3$, we show that $k$-WL stabilizes after at most $O(kn^{k-1}\log n)$…
The KLEE'S MESURE of $n$ axis-parallel boxes in $\mathbb{R}^d$ is the volume of their union. It can be computed in time within $O(n^{d/2})$ in the worst case. We describe three techniques to boost its computation: one based on some type of…
With the growing demand for safeguarding sensitive user information in recommender systems, recommendation attribute unlearning is receiving increasing attention. Existing studies predominantly focus on single-attribute unlearning. However,…
How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$,…
How hard is it to program $n$ robots to move about a long narrow aisle while making a series of $r-2$ intermediate stops, provided only $w$ of the robots can fit across the width of the aisle? In this paper, we answer this question by…
What is the maximum number of holes enclosed by a $d$-dimensional polyomino built of $n$ tiles? Represent this number by $f_d(n)$. Recent results show that $f_2(n)/n$ converges to $1/2$. We prove that for all $d \geq 2$ we have $f_d(n)/n…
Polymetric walls are walls built from bricks in more than one size. Architects and builders want to built polymetric walls that satisfy certain structural and aesthetical constraints. In a recent paper by de Jong, Vinduska, Hans and Post…
In self-assembly, a $k$-counter is a tile set that grows a horizontal ruler from left to right, containing $k$ columns each of which encodes a distinct binary string. Counters have been fundamental objects of study in a wide range of…