No abstract

A pattern database (PDB) is a heuristic function stored as a lookup table. This paper considers how best to use a fixed amount (m units) of memory for storing pattern databases. In particular, we examine whether using n pattern databases of size m/n instead of one pattern database of size m improves search performance. In all the state spaces considered, the use of multiple smaller pattern databases reduces the number of nodes generated by IDA*. The paper provides an explanation for this phenomenon based on the distribution of heuristic values that occur during search.

In this paper, we show that any scaled-up version of any discrete self-similar tree fractal does not strictly self-assemble, at any temperature, in Winfree's abstract Tile Assembly Model.

Let b (n) denote the number of -regular partitions of n. Recently Andrews, Hirschhorn, and Sellers proved that b 4 (n) satisfies two infinite families of congruences modulo 3, and Webb established an analogous result for b 13 (n). In this paper we prove similar families of congruences for b (n) for other values of .

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for all N ∈ N, there is a tile set that uniquely self-assembles into an N × N square shape at temperature 1 with optimal program-size complexity of O(log N/ log log N ) (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it works even when the placement of tiles is restricted to the z = 0 and z = 1 planes. This result affirmatively answers an open question from Cook, Fu, Schweller (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SICOMP 2007), and perhaps of independent interest.