We introduce the problem of shape replication in the Wang tile self-assembly model. Given an input shape, we consider the problem of designing a self-assembly system which will replicate that shape into either a specific number of copies, or an unbounded number of copies. Motivated by practical DNA implementations of Wang tiles, we consider a model in which tiles consisting of DNA or RNA can be dynamically added in a sequence of stages. We further permit the addition of RNase enzymes capable of disintegrating RNA tiles. Under this model, we show that arbitrary genus-0 shapes can be replicated infinitely many times using only O(1) distinct tile types and O(1) stages. Further, we show how to replicate precisely n copies of a shape using O(log n) stages and O(1) tile types.
For a set of points in the plane and a fixed integer k > 0, the Yao graph Y k partitions the space around each point into k equiangular cones of angle θ = 2π/k, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of Y5, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd k ≥ 5, the spanning ratio of Y k is at most 1/(1 − 2 sin(3θ/8)), which gives the first constant upper bound for Y5, and is an improvement over the previous bound of 1/(1 − 2 sin(θ/2)) for odd k ≥ 7. We further reduce the upper bound on the spanning ratio for Y5 from 10.9 to 2 + √ 3 ≈ 3.74, which falls slightly below the lower bound of 3.79 established for the spanning ratio of Θ5 (Θ-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). number of cones. We also give a lower bound of 2.87 on the spanning ratio of Y5. Finally, we revisit the Y6 graph, which plays a particularly important role as the transition between the graphs (k > 6) for which simple inductive proofs are known, and the graphs (k ≤ 6) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of Y6 from 17.6 to 5.8, getting closer to the spanning ratio of 2 established for Θ6.
We show that the Yao graph Y 4 in the L 2 metric is a spanner with stretch factor 8(29+23 √ 2). Enroute to this, we also show that the Yao graph Y ∞ 4 in the L ∞ metric is a planar spanner with stretch factor 8.In the appendix, we improve the stretch factor and show that, in fact, Y k is a spanner for any k ≥ 7. Recently, Molla [4] showed that Y 2 and Y 3 are not
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